Stphane Goutte
email: goutte@math.univ-paris13.fr

Nadia Oudjane
email: nadia.oudjane@edf.fr

Francesco Russo
email: russo@math.univ-paris13.fr

Variance Optimal Hedging for continuous time processes with independent increments and applications

Keywords: Variance-optimal hedging, Fllmer-Schweizer decomposition, Lévy process, Cumulative generating function, Characteristic function, Normal Inverse Gaussian process, Electricity markets, Process with independent increments. 2000 AMS-classification: 60G51, 60H05, 60J25, 60J75 JEL-classification: C02, G11, G12, G13

For a large class of vanilla contingent claims, we establish an explicit Föllmer-Schweizer decomposition when the underlying is a process with independent increments (PII) and an exponential of a PII process. This allows to provide an efficient algorithm for solving the mean variance hedging problem. Applications to models derived from the electricity market are performed.

Introduction

There are basically two main approaches to define the mark to market of a contingent claim: one relying on the no-arbitrage assumption and the other related to a hedging portfolio, those two approaches converging in the specific case of complete markets. A simple introduction to the different hedging and pricing models in incomplete markets can be found in chapter 10 of [START_REF] Cont | Financial modelling with jump processes[END_REF].

The fundamental theorem of Asset Pricing [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF] implies that a pricing rule without arbitrage that moreover satisfies some usual conditions (linearity non anticipativity . . . ) can always be written as an expectation under a martingale measure. In general, the resulting price is not linked with a hedging strategy except in some specific cases such as complete markets. More precisely, it is proved [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF] that the market completeness is equivalent to uniqueness of the equivalent martingale measure. Hence, when the market is not complete, there exist several equivalent martingale measures (possibly an infinity) and one has to specify a criterion to select one specific pricing measure: to recover some given option prices (by calibration) [START_REF] Goll | Minimal distance martingale measures and optimal portfolios consistent with observed market proces[END_REF]; to simplify calculus and obtain a simple process under the pricing measure; to maintain the structure of the real world dynamics; to minimize a distance to the objective probability (entropy [START_REF] Frittelli | The minimal entropy martingale measure and the valuation problem in incomplete markets[END_REF]) . . . In this framework, the difficulty is to understand in a practical way the impact of the choice of the martingale measure on the resulting prices.

If the resulting price is in general not connected to a hedging strategy, yet it is possible to consider the hedging question in a second step, optimizing the hedging strategy for the given price. In this framework, one approach consists in deriving the hedging strategy minimizing the global quadratic hedging error under the pricing measure where the martingale property of the underlying highly simplifies calculations. This approach, is developed in [START_REF] Cont | Financial modelling with jump processes[END_REF], in the case of exponential-Lvy models: the optimal quadratic hedge is then expressed as a solution of an integro-differential equation involving the Lvy measure. Unfortunately, minimizing the quadratic hedging error under the pricing measure has no clear interpretation since the resulting hedging strategy can lead to huge quadratic error under the objective measure.

Alternatively, one can define option prices as a by product of the hedging strategy. In the case of complete markets, any option can be replicated perfectly by a self-financed hedging portfolio continuously rebalanced, then the option hedging value can be defined as the cost of the hedging strategy. When the market is not complete, it is not possible, in general, to hedge perfectly an option. One has to specify a risk criteria, and consider the hedging strategy that minimizes the distance (in terms of the given criteria) between the pay-off of the option and the terminal value of the hedging portfolio. Then, the price of the option is related to the cost of this imperfect hedging strategy to which is added in practice another prime related to the residual risk induced by incompleteness. Several criteria can be adopted. The aim of super-hedging is to hedge all cases. This approach yields in general prices that are too expensive to be realistic [START_REF] El Karoui | Dynamic programming and pricing of contingent claims in an incomplete market[END_REF]. Quantile hedging modifies this approach allowing for a limited probability of loss [START_REF] Fllmer | Quantile hedging[END_REF]. Indifference Utility pricing introduced by [START_REF] Hodges | Optimal replication of contingent claims under transactions costs[END_REF] defines the price of an option to sell (resp. to buy) as the minimum initial value s.t. the hedging portfolio with the option sold (resp. bought) is equivalent (in term of utility) to the initial portfolio. Quadratic hedging is developed in [START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF], [START_REF] Schweizer | Variance-optimal hedging in discrete time[END_REF].

The quadratic distance between the hedging portfolio and the pay-off is minimized. Then, contrarily to the case of utility maximization, losses and gains are treated in a symmetric manner, which yields a fair price for both the buyer and the seller of the option. In this paper, we follow this last approach and our developments can be used in both the no-arbitrage value and the hedging value framework: either to derive the hedging strategy minimizing the global quadratic hedging error under the objective measure, for a given pricing rule; or to derive both the price and the hedging strategy minimizing the global quadratic hedging error under the objective measure. We spend now some words related to the global quadratic hedging approach which is also called meanvariance hedging or global risk minimization. Given a square integrable r.v. H, we say that the pair (V 0 , ϕ) is optimal if (c, v) = (V 0 , ϕ) minimizes the functional E Hc -T 0 vdS 2 . The price V 0 represents the price of the contingent claim H and ϕ is the optimal strategy. Technically speaking, the global risk minimization problem, is based on the so-called Föllmer-Schweizer decomposition (or FS decomposition) of a square integrable random variable (representing the contingent claim) with respect to an (F t )-semimartingale S = M + A modeling the asset price. M is an (F t )-local martingale and A is a bounded variation process with A 0 = 0. Mathematically, the FS decomposition, constitutes the generalisation of the martingale representation theorem (Kunita-Watanabe representation) when S is a Brownian motion or a martingale. Given square integrable random variable H, the problem consists in expressing H as H 0 + T 0 ξdS + L T where ξ is predictable and L T is the terminal value of an orthogonal martingale L to M , i.e. the martingale part of S. The seminal paper is [START_REF] Fllmer | Hedging of contingent claims under incomplete information[END_REF] where the problem is treated in the case that S is continuous. In the general case S is said to have the structure condition (SC) condition if there is a predictable process α such that A t = t 0 α s d M s and T 0 α 2 s d M s < ∞ a.s. In the sequel most of contributions were produced in the multidimensional case. Here for simplicity we will formulate all the results in the one-dimensional case.

An interesting connection with the theory of backward stochastic differential equations (BSDEs) in the sense of [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF], was proposed in [START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF]. [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF] considered BSDEs driven by Brownian motion; in [START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF] the Brownian motion is in fact replaced by M . The first author who considered a BSDE driven by a martingale was [START_REF] Buckdahn | Backward stochastic differential equations driven by a martingale[END_REF]. Suppose that V t = t 0 α s d M s . The BSDE problem consists in finding a triple (V, ξ, L) where

V t = H - T t ξ s dM s - T t ξ s α s d M s -(L T -L t ),
and L is an (F t )-local martingale orthogonal to M . In fact, this decomposition provides the solution to the so called local risk minimization problem, see [START_REF] Fllmer | Hedging of contingent claims under incomplete information[END_REF].

In this case, V t represents the price of the contingent claim at time t and the price V 0 constitutes in fact the expectation under the so called variance optimal measure (VOM), as it will be explained at Section 2.5, with references to [START_REF] Schweizer | A guided tour through quadratic hedging approaches[END_REF], [START_REF] Arai | An extension of mean-variance hedging to the discontinuous case[END_REF] and [START_REF] Arai | Some properties of the variance-optimal martingale measure for discontinuous semimartingales[END_REF]. In the framework of FS decomposition, a process which plays a significant role is the so-called mean variance tradeoff (MVT) process K. This notion is inspired by the theory in discrete time started by [START_REF] Schl | On quadratic cost criteria for options hedging[END_REF]; in the continuous time case K is defined as K t = t 0 α 2 s d M s , t ∈ [0, T ]. [START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF] shows the existence of the meanvariance hedging problem if the MVT process is deterministic. In fact, a slight more general condition was the (ESC) condition and the EMVT process but we will not discuss here further details. We remark that in the continuous case, treated by [START_REF] Fllmer | Hedging of contingent claims under incomplete information[END_REF], no need of any condition on K is required. When the MVT process is deterministic, [START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF] is able to solve the global quadratic variation problem and provides an efficient relation, see Theorem 5.2 with the FS decomposition. He also shows that, for the obtention of the mentioned relation, previous condition is not far from being optimal. The next important step was done in [START_REF] Monat | Fllmer-Schweizer decomposition and mean-variance hedging for general claims[END_REF] where under the only condition that K is uniformly bounded, the FS decomposition of any square integrable random variable admits existence and uniqueness and the global minimization problem admits a solution. More recently has appeared an incredible amount of papers in the framework of global (resp. local) risk minimization, so that it is impossible to list all of them and it is beyond our scope. Two significant papers containing a good list of references are [START_REF] Schweizer | A guided tour through quadratic hedging approaches[END_REF], [START_REF] Benth | Explicit representation of the minimal variance portfolio in markets driven by Lvy processes[END_REF] and [START_REF] Černỳ | On the structure of general man-variance hedging strategies[END_REF]. The present paper puts emphasis on processes with independent increments (PII) and exponential of those processes. It provides explicit FS decompositions when the process S is of that type when the contingent claims are provided by some Fourier transform (resp. Laplace-Fourier transform) of a finite measure. Some results of [START_REF] Hubalek | Variance-optimal hedging for processes with stationary independent increments[END_REF] concerning exponential of Lévy processes are generalized trying to investigate some peculiar properties behind and to consider the case of PII with possibly non stationary increments. The motivation came from hedging problems in the electricity market.

Because of non-storability of electricity, the hedging instrument is in that case, a forward contract with value S 0 t = e -r(T d -t) (F T d t -F T d 0 ) where F T d t is the forward price given at time t ≤ T d for delivery of 1MWh at time T d . Hence, the dynamic of the underlying S 0 is directly related to the dynamic of forward prices. Now, forward prices show a volatility term structure that requires the use of models with non stationary increments and motivates the generalization of the pricing and hedging approach developed in [START_REF] Hubalek | Variance-optimal hedging for processes with stationary independent increments[END_REF] for Lvy processes to the case of PII with possibly non stationary increments. The paper is organized as follows. After this introduction and some generalities about semimartingales, we introduce the notion of FS decomposition and describe local and global risk minimization. Then, we examine at Chapter 3 (resp. 4) the explicit FS decomposition for PII processes (resp. exponential of PII processes). Chapter 5 is devoted to the solution to the global minimization problem and Chapter 6 to the case of a model intervening in the electricity market. Chapter 7 is devoted to simulations. This paper will be followed by a companion paper, i. e. [START_REF] Goutte | Variance Optimal Hedging for discrete time processes with independent increments[END_REF] which concentrates on the discrete time case leaving more space to numerical implementations.

2 Generalities on semimartingales and Fllmer-Schweizer decomposition

In the whole paper, T > 0, will be a fixed terminal time and we will denote by (Ω, F , (F t ) t∈[0,T ] , P ) a filtered probability space, fulfilling the usual conditions.

Generating functions

Let X = (X t ) t∈[0,T ] be a real valued stochastic process.

Definition 2.1. The characteristic function of (the law of ) X t is the continuous mapping

ϕ Xt : R → C with ϕ Xt (u) = E[e iuXt ] .
In the sequel, when there will be no ambiguity on the underlying process X, we will use the shortened notation ϕ t for ϕ Xt .

Definition 2.2. The cumulant generating function of (the law of ) X t is the mapping z

→ Log(E[e zXt ])
where Log(w) = log(|w|) + iArg(w) where Arg(w) is the Argument of w, chosen in ]π, π]; Log is the principal value logarithm. In particular we have

κ Xt : D → C with e κX t (z) = E[e zXt ] ,
where

D := {z ∈ C | E[e Re(z)Xt ] < ∞, ∀t ∈ [0, T ]}.
In the sequel, when there will be no ambiguity on the underlying process X, we will use the shortened notation κ t for κ Xt .

We observe that D includes the imaginary axis.

Remark 2.3.

1. For all z ∈ D, κ t (z) = κ t (z) , where z denotes the conjugate complex of z ∈ C. Indeed, for any z ∈ D,

exp(κ t (z)) = E[exp(zX t )] = E[exp(zX t )] = E[exp(zX t )] = exp(κ t (z)) = exp(κ t (z)) .

For all

z ∈ D ∩ R , κ t (z) ∈ R .

Semimartingales

An (F t )-semimartingale X = (X t ) t∈[0,T ] is a process of the form X = M + A, where M is an (F t )-local martingale and A is a bounded variation adapted process vanishing at zero. ||A|| T will denote the total variation of A on [0, T ]. Given two (F t )-local martingales M and N , M, N will denote the angle bracket of M and N , i.e. the unique bounded variation predictable process vanishing at zero such that M N -M, N is an (F t )-local martingale. If X and Y are (F t )-semimartingales, [X, Y ] denotes the square bracket of X and Y , i.e. the quadratic covariation of X and Y . In the sequel, if there is no confusion about the underlying filtration (F t ), we will simply speak about semimartingales, local martingales, martingales. All the local martingales admit a cdlg version. By default, when we speak about local martingales we always refer to their cdlg version.

More details about previous notions are given in chapter I.1. of [START_REF] Jacod | Limit theorems for stochastic processes[END_REF].

Remark 2.4.

1. All along this paper we will consider C-valued martingales (resp. local martingales,

semimartingales). Given two C-valued local martingales M 1 , M 2 then M 1 , M 2 are still local martin- gales. Moreover M 1 , M 2 = M 1 , M 2 .
2. If M is a C-valued martingale then M, M is a real valued increasing process.

Theorem 2.5. (X t ) t∈[0,T ] is a real semimartingale iff the characteristic function, t → ϕ t (u), has bounded variation over all finite intervals, for all u ∈ R.

Remark 2.6. According to Theorem I.4.18 of [START_REF] Jacod | Limit theorems for stochastic processes[END_REF], any local martingale M admits a unique (up to indistinguishability) decomposition,

M = M 0 + M c + M d ,
where M c 0 = M d 0 = 0, M c is a continuous local martingale and M d is a purely discontinuous local martingale in the sense that N, M d = 0 for all continuous local martingales N . M c is called the continuous part of M and M d the purely discontinuous part. Definition 2.7. An (F t )-special semimartingale is an (F t )-semimartingale X with the decomposition X = M + A, where M is a local martingale and A is a bounded variation predictable process starting at zero. Remark 2.8. The decomposition of a special semimartingale of the form X = M + A is unique, see [START_REF] Jacod | Limit theorems for stochastic processes[END_REF] definition 4.22.

For any special semimartingale X we define

||X|| 2 δ 2 = E [[M, M ] T ] + E ||A|| 2 T .
The set δ 2 is the set of (F t )-special semimartingale X for which ||X|| 2 δ 2 is finite. A truncation function defined on R is a bounded function h : R → R with compact support such that h(x) = x in a neighbourhood of 0.

An important notion, in the theory of semimartingales, is the notion of characteristics, defined in definition II.2.6 of [START_REF] Jacod | Limit theorems for stochastic processes[END_REF]. Let X = M + A be a real-valued semimartingale. A characteristic is a triplet, (b, c, ν), depending on a fixed truncation function, where 1. b is a predictable process with bounded variation; 2. c = M c , M c , M c being the continuous part of M according to Remark 2.6; 3. ν is a predictable random measure on R + × R, namely the compensator of the random measure µ X associated to the jumps of X.

Fllmer-Schweizer Structure Condition

Let X = (X t ) t∈[0,T ] be a real-valued special semimartingale with canonical decomposition,

X = M + A .
For the clarity of the reader, we formulate in dimension one, the concepts appearing in the literature, see e.g. [START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF] in the multidimensional case. Definition 2.9. For a given local martingale M , the space

L 2 (M ) consists of all predictable R-valued pro- cesses v = (v t ) t∈[0,T ] such that E T 0 |v s | 2 d M s < ∞ .
For a given predictable bounded variation process A, the space L 2 (A) consists of all predictable R-valued

processes v = (v t ) t∈[0,T ] such that E ( T 0 |v s |d||A|| s ) 2 < ∞ .

Finally, we set

Θ := L 2 (M ) ∩ L 2 (A) .
For any v ∈ Θ, the stochastic integral process

G t (v) := t 0 v s dX s , for all t ∈ [0, T ] ,
is therefore well-defined and is a semimartingale in δ 2 with canonical decomposition

G t (v) = t 0 v s dM s + t 0 v s dA s , for all t ∈ [0, T ] .
We can view this stochastic integral process as the gain process associated with strategy v on the underlying process X.

Definition 2.10. The minimization problem we aim to study is the following. Given H ∈ L 2 , an admissible strategy pair (V 0 , ϕ) will be called optimal if (c, v) = (V 0 , ϕ) minimizes the expected squared hedging error

E[(H -c -G T (v)) 2 ] , (2.1)
over all admisible strategy pairs (c, v) ∈ R × Θ. V 0 will represent the price of the contingent claim H at time zero.

Definition 2.11. Let X = (X t ) t∈[0,T ] be a real-valued special semimartingale. X is said to satisfy the structure condition (SC) if there is a predictable R-valued process α = (α t ) t∈[0,T ] such that the following properties are verified.

1. A t = t 0 α s d M s , for all t ∈ [0, T ], so that dA ≪ d M . 2. T 0 α 2 s d M s < ∞ , P -a.s.
Definition 2.12. From now on, we will denote by K = (K t ) t∈[0,T ] the cdlg process

K t = t 0 α 2 s d M s , for all t ∈ [0, T ] .
This process will be called the mean-variance tradeoff (MVT) process.

Remark 2.13. In [START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF], the process

(K t ) t∈[0,T ] is denoted by ( K t ) t∈[0,T ] .
We provide here a technical proposition which allows to make the class Θ of integration of X explicit.

Proposition 2.14.

If X satisfies (SC) such that E[K T ] < ∞, then Θ = L 2 (M ). Proof. Assume that E[K T ] < ∞, we will prove that L 2 (M ) ⊆ L 2 (A). Let us consider a process v ∈ L 2 (M ), then E T 0 |v s |d||A|| s 2 = E T 0 |v s ||α s |d M s 2 ≤ E T 0 |v s | 2 d M s E T 0 |α| 2 d M s 1 2 , = E T 0 |v s | 2 d M s E(K T ) 1 2 < ∞ .
Schweizer in [START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF] also introduced the extended structure condition (ESC) on X and he provided the Fllmer-Schweizer decomposition in this more extended framework. We recall that notion (in dimension 1).

Given a real cdlg stochastic process X, the quantity ∆X t will represent the jump X t -X t-. Definition 2.15. Let X = (X t ) t∈[0,T ] be a real-valued special semimartingale. X is said to satisfy the extended structure condition (ESC) if there is a predictable R-valued process α = (α t ) t∈[0,T ] with the following properties.

1. A t = t 0 α s d M s , for all t ∈ [0, T ] , so that dA ≪ dM .

The quantity

T 0 α 2 s 1 + α 2 s ∆ M s d M s is finite.
If condition (ESC) is fulfilled, then the process

K t := t 0 α 2 s 1 + α 2 s ∆ M s , for all t ∈ [0, T ] ,
is well-defined. It is called extended mean-variance tradeoff (EMVT) process.

Remark 2.16.

(SC) implies (ESC).

2. If M is continuous then (ESC) and (SC) are equivalent and K = K.

Kt

= t 0 |α s | 2 1 + ∆K s d M s = t 0 1 1 + ∆K s dK s , for all t ∈ [0, T ] . 4. K t = t 0 1 1 -∆ Ks d Ks , for all t ∈ [0, T ] . 5. If K is deterministic then K is deterministic.
The structure condition (SC) appears quite naturally in applications to financial mathematics. In fact, it is mildly related to the no arbitrage condition. In fact (SC) is a natural extension of the existence of an equivalent martingale measure from the case where X is continuous. Next proposition will show that every adapted continuous process X admitting an equivalent martingale measure satisfies (SC). In our applications (ESC) will be equivalent to (SC) since in Section 3.2 and Section 4.2, M will always be continuous.

Proposition 2.17. Let X be a (P, F t ) continuous semimartingale. Suppose the existence of a locally equivalent probability Q ∼ P under which X is an (Q, F t )-local martingale, then (SC) is verified.

Proof. Let (D t ) t∈[0,T ] be the strictly positive continuous Q-local martingale such that dP = D T dQ. By Theorem VIII.1.7 of [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF], M = X -X, L is a continuous P -local martingale, where L is the continuous Q-local martingale associated to the density process i.e.

D t = exp{L t - 1 2 L t } , for all t ∈ [0, T ] .
According to Lemma IV.4.2 in [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF], there is a progressively measurable process R such that for all t ∈ [0, T ],

L t = t 0 R s dX s + O t and T 0 R 2 s d X s < ∞ , Q -a.s. ,
where O is a Q-local martingale such that X, O = 0. Hence,

X, L t = t 0 R s d X s and X t = M t + t 0 R s d[X] s , for all t ∈ [0, T ].
We end the proof by setting

α t = d X, L t d X t = R t .

Fllmer-Schweizer Decomposition and variance optimal hedging

Throughout this section, as in Section 2.3, X is supposed to be an (F t )-special semimartingale fulfilling the (SC) condition.

We recall here the definition stated in Chapter IV. Proof. Let S(M ) be the stable subspace generated by M. S(M ) includes the space of martingales of the form

M f t := t 0 f (s)dM s , for all t ∈ [0, T ] ,
where f ∈ L 2 (dM ) is deterministic. According to Lemma IV.3.2 of [START_REF] Protter | Stochastic integration and differential equations[END_REF], it is enough to show that, for any f ∈ L 2 (dM ), g ∈ L 2 (dN ), M f and N g are weakly orthogonal in the sense that E[M f T N g T ] = 0. This is clear since previous expectation equals

E[ M f , N g T ] = E T 0 f gd M, N = 0
if M, N = 0. This shows the converse implication.

The direct implication follows from the fact that M N is a martingale, the definition of the angle bracket and uniqueness of special semimartingale decomposition.

Definition 2.21. We say that a random variable H ∈ L 2 (Ω, F , P ) admits a Fllmer-Schweizer (FS) decomposition, if it can be written as

H = H 0 + T 0 ξ H s dX s + L H T , P -a.s. , (2.2
)

where H 0 ∈ R is a constant, ξ H ∈ Θ and L H = (L H t ) t∈[0,T ] is a square integrable martingale, with E[L H 0 ] = 0 and strongly orthogonal to M .
We formulate for this section one basic assumption. Assumption 1. We assume that X satisfies (SC) and that the MVT process K is uniformly bounded in t and ω.

The first result below gives the existence and the uniqueness of the Fllmer-Schweizer decomposition for a random variable H ∈ L 2 (Ω, F , P ). The second affirms that subspaces G T (Θ) and {L 2 (F 0 ) + G T (Θ)} are closed subspaces of L 2 . The last one provides existence and uniqueness of the solution of the minimization problem (2.1). We recall Theorem 3.4 of [START_REF] Monat | Fllmer-Schweizer decomposition and mean-variance hedging for general claims[END_REF].

Theorem 2.22. Under Assumption 1, every random variable H ∈ L 2 (Ω, F , P) admits a FS decomposition. Moreover, this decomposition is unique in the following sense: If

H = H 0 + T 0 ξ H s dX s + L H T = H ′ 0 + T 0 ξ ′ H s dX s + L ′ H T ,
where

(H 0 , ξ H , L H ) and (H ′ 0 , ξ ′ H , L ′ H
) satisfy the conditions of the FS decomposition, then

     H 0 = H ′ 0 , P -a.s. , ξ H = ξ ′ H in L 2 (M ) , L H T = L ′ H
T , Pa.s. . We recall Theorem 4.1 of [START_REF] Monat | Fllmer-Schweizer decomposition and mean-variance hedging for general claims[END_REF]. So we can project any random variable H ∈ L 2 on G T (Θ). By Theorem 2.22, we have the uniqueness of the solution of the minimization problem (2.1). This is given by Theorem 4.6 of [START_REF] Monat | Fllmer-Schweizer decomposition and mean-variance hedging for general claims[END_REF], which is stated below. Theorem 2.24. We suppose Assumption 1.

1. For every H ∈ L 2 (Ω, F , P ) and every c ∈ L 2 (F 0 ), there exists a unique strategy ϕ (c) ∈ Θ such that

E[(H -c -G T (ϕ (c) )) 2 ] = min v∈Θ E[(H -c -G T (v)) 2 ] .
(2.3)

2.
For every H ∈ L 2 (Ω, F , P) there exists a unique (c

(H) , ϕ (H) ) ∈ L 2 (F 0 ) × Θ such that E[(H -c (H) -G T (ϕ (H) )) 2 ] = min (c,v)∈L 2 (F0)×Θ E[(H -c -G T (v)) 2 ] .
Next theorem gives the explicit form of the optimal strategy ϕ (c) , which is valid even in the case where X satisfies the extended structure condition (ESC). For the purpose of the present work, this will not be useful, see considerations following Remark 2. [START_REF] Cont | Financial modelling with jump processes[END_REF]

t = ξ H t + α t 1 + ∆K t (H t--c -G t-(ϕ (c) )) , for all t ∈ [0, T ] (2.4)
where the process (H t ) t∈[0,T ] is defined by

H t := H 0 + t 0 ξ H s dX s + L H t , (2.5)
and the process α is the process appearing in Definition 2.11 of (SC).

Proof. Theorem 3 of [START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF] states the result under the (ESC) condition. We recall that (SC) implies (ESC), see Remark 2.16 and the result follows.

To obtain the solution to the minimization problem (2.1), we use Corollary 10 of [START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF] that we recall.

Corollary 2.26. Under the assumption of Theorem 2.25, the solution of the minimization problem (2.1) is given by the pair (H 0 , ϕ (H0) ) .

In the sequel of this paper we will only refer to the structure condition (SC) and to the MVT process K. The definition below can be found in section II.8 p. 85 of [START_REF] Protter | Stochastic integration and differential equations[END_REF].

Definition 2.27. The Dolans-Dade exponential of a semimartingale X is defined to be the unique cdlg adapted solution Y to the stochastic differential equation,

dY t = Y t-dX t , for all t ∈ [0, T ] with Y 0 = 1 .

This process is denoted by E(X).

This solution is a semimartingale given by

E(X) t = exp (X t -X 0 -[X] t /2) s≤t (1 + ∆X s ) exp(-∆X s ) .
Theorem below is stated in [START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF].

Theorem 2.28. Under the assumptions of Theorem 2.25, for any c ∈ R, we have

min v∈Θ E[(H -c -G T (v)) 2 ] = E(-KT ) (H 0 -c) 2 + E[(L H 0 ) 2 ] + T 0 1 E(-Ks ) d E[ L H s ] . ( 2
min v∈Θ E[(H -c -G T (v)) 2 ] = exp(-K T ) (H 0 -c) 2 + E[(L H 0 ) 2 ] +E T 0 exp{-(K T -K s )}d L H s .
(2.7)

Proof. Remark 2.16 implies that K = K. Since K is continuous and with bounded variation, its Dolans-Dade exponential coincides with the classical exponential. The result follows from Theorem 2.28.

In the sequel, we will find an explicit expression of the FS decomposition for a large class of square integrable random variables, when the underlying process is a process with independent increments, or is an exponential of process with independent increments. For this, the first step will consist in verifying (SC) and the boundedness condition on the MVT process, see Assumption 1.

2.5 Link with the equivalent signed martingale measure 2.5.1 The Variance optimal martingale (VOM) measure Definition 2.30.

1. A signed measure, Q, on (Ω, F T ), is called a signed Θ-martingale measure, if (a) Q(Ω) = 1 ; (b) Q ≪ P with dQ dP ∈ L 2 (P ) ; (c) E[ dQ dP G T (v)] = 0 for all v ∈ Θ.
We denote by P s (Θ), the set of all such signed Θ-martingale measures. Moreover, we define P e (Θ) := {Q ∈ P s (Θ) | Q ∼ P and Q is a probability measure} , and introduce the closed convex set,

D d := {D ∈ L 2 (P ) | D = dQ dP f or some Q ∈ P s (Θ)} .
2. A signed martingale measure P ∈ P s (Θ) is called variance-optimal martingale (VOM) measure if

D = argmin D∈D d V ar[D 2 ] = argmin D∈D d E[D 2 ] -1 , where D = d P dP .
The space G T (Θ) := {G T (v) | v ∈ Θ} is a linear subspace of L 2 (P ). Then, we denote by G T (Θ) ⊥ its orthogonal complement, that is,

G T (Θ) ⊥ := {D ∈ L 2 (P ) | E[DG T (v)] = 0 for any v ∈ Θ} .
Furthermore, G T (Θ) ⊥⊥ denotes the orthogonal complement of G T (Θ) ⊥ , which is the L 2 (P )-closure of G T (Θ). A simple example when P e (Θ) is non empty is given by the following proposition, that anticipates some material treated in the next section.

Proposition 2.31. Let X be a process with independent increments such that • X t has the same law as -X t , for any t ∈ [0, T ];

• 1 2 belongs to the domain D of the cumulative generating function (t, z) → κ t (z).

Then, there is a probability Q ∼ P such that S t = exp(X t ) is a martingale.

Proof. For all t ∈ [0, T ], we set

D t = exp{-Xt 2 -κ t (- 1
2 )}. Notice that D is a martingale so that the measure Q on (Ω, F T ) defined by dQ = D T dP is an (equivalent) probability to P . On the other hand, the symmetry of the law of X t implies for all t ∈ [0, T ],

S t D t = exp{ X t 2 -κ t (- 1 2 )} = exp{ X t 2 -κ t ( 1 2 )} .
So SD is also a martingale. According to [START_REF] Jacod | Limit theorems for stochastic processes[END_REF], chapter III, Proposition 3.8 a), S is a Q-martingale and so S is a Q-martingale.

Example 2.32. Let Y be a process with independent increments. We consider two copies

Y 1 of Y and Y 2 of -Y . We set X = Y 1 + Y 2 .
Then X has the same law of -X.

For simplicity, we suppose from now that Assumption 1 is verified, even if one could consider a more general framework, see [START_REF] Arai | An extension of mean-variance hedging to the discontinuous case[END_REF] Therorem 1.28. This ensures that the linear space

G T (Θ) is closed in L 2 (Ω), therefore G T (Θ) = G T (Θ) = G T (Θ) ⊥⊥ .
Moreover, Proposition 2.14 ensures that Θ = L 2 (M ). We recall an almost known fact cited in [START_REF] Arai | An extension of mean-variance hedging to the discontinuous case[END_REF]. For completeness, we give a proof.

Proposition 2.33.

P s (Θ) = ∅ is equivalent to 1 / ∈ G T (Θ) .
Proof. Let us prove the two implications.

• Let Q ∈ P s (Θ). If 1 ∈ G T (Θ), then Q(Ω) = E Q (1) = 0 which leads to a contradiction since Q is a probability. Hence 1 / ∈ G T (Θ).
• Suppose that 1 / ∈ G T (Θ). We denote by f the orthogonal projection of 1 on

G T (Θ). Since E[f (1-f )] = 0, then E[1 -f ] = E[(1 -f ) 2 ]. Recall that 1 = f ∈ G T (Θ), hence we have E[f ] = 1.
Therefore, we can define the signed measure P by setting

P (A) = A DdP , with D = 1 -f 1 -E[f ] . (2.8)
We check now that P ∈ P s (Θ).

-Trivially P (Ω) = E( D) = 1 ;

-P ≪ P , by construction.

-Let v ∈ Θ, E[ DG T (v)] = 1 1 -E[f ] (E[(1 -f )G T (v)]) = 0 , since 1 -f ∈ G T (Θ) ⊥ .
Hence, P ∈ P s (Θ) which concludes the proof of the Proposition.

Remark 2.34. If 1 is orthogonal to G T (Θ), then f = 0 and P ∈ P s (Θ) so P s (Θ) = ∅.

In fact, P constructed in the proof of Proposition 2.33 coincides with the VOM measure.

Proposition 2.35. Let P be the signed measure defined in (2.8). Then,

D = arg min D∈D d E[D 2 ] = arg min D∈D d V ar[D] .
Proof. Let D ∈ D d and Q such that dQ = DdP . We have to show that

E[D 2 ] ≥ E[ D 2 ]. We write E[D 2 ] = E[(D -D) 2 ] + E[ D 2 ] + 2 1 -E[f ] E[(D -D)(1 -f )] . Moreover, since f ∈ G T (Θ) yields E[(D -D)(1 -f )] = E[D] -E[ D] -E[Df ] + E[ Df ] , = Q(Ω) -Q(Ω) . = 0 .
Remark 2.36. 1. Arai [START_REF] Arai | Some properties of the variance-optimal martingale measure for discontinuous semimartingales[END_REF] gives sufficient conditions under which the VOM measure is a probability, see Theorem 3.4 in [START_REF] Arai | Some properties of the variance-optimal martingale measure for discontinuous semimartingales[END_REF].

2. Taking in account Proposition 2.33, the property 1 / ∈ G T (Θ) may be viewed as non-arbitrage condition.

In fact, in [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF], the existence of a martingale measure which is a probability is equivalent to a no free lunch condition.

Next proposition can be easily deduced for a more general formulation, see [START_REF] Schweizer | A guided tour through quadratic hedging approaches[END_REF].

Proposition 2.37. We assume Assumption 1. Let H ∈ L 2 (Ω) and consider the solution (c H , ϕ H ) of the minimization problem (2.1). Then, the price c H equals the expectation under the VOM measure P of H.

Proof. We have

H = c H + G T (ϕ H ) + R ,
where R is orthogonal to G T (Θ) and E[R] = 0. Since P ∈ P s (Θ), taking the expectation with respect to P ,

denoted by E we obtain

E[H] = c H + E[R] .
From the proof of Proposition 2.33, we have

E[R] = E[(1 -f )R] 1 -E[f ] = 1 1 -E[f ] (E[R] -E[f R]) . Since f ∈ G T (Θ) and R is orthogonal to G T (Θ), we get E[R] = 0 .
3 Processes with independent increments (PII)

This section deals with the case of Processes with Independent Increments. The preliminary part recalls some useful properties of such processes. Then, we obtain a sufficient condition on the characteristic function for the existence of the FS decomposition. Moreover, an explicit FS decomposition is derived.

Beyond its own theoretical interest, this work is motivated by its possible application to hedging and pricing energy derivatives and specifically electricity derivatives. Indeed, one way of modeling electricity forward prices is to use arithmetic models such as the Bachelier model which was developed for standard financial assets. The reason for using arithmetic models, is that the usual hedging intrument available on electricity markets are swap contracts which give a fixed price for the delivery of electricity over a contracted time period. Hence, electricty swaps can be viewed as a strip of forwards for each hour of the delivery period.

In this framework, arithmetic models have the significant advantage to yield closed pricing formula for swaps which is not the case of geometric models.

However, in whole generality, an arithmetic model allows negative prices which could be underisable.

Nevertheless, in the electricity market, negative prices may occur because it can be more expensive for a producer to switch off some generators than to pay someone to consume the resulting excess of production.

Still, in [START_REF] Benth | A non-Gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing[END_REF], is introduced a class of arithmetic models where the positivity of spot prices is ensured, using a specific choice of increasing Lvy process. The parameters estimation of this kind of model is studied in [START_REF] Meyer-Brandis | Multi-factor jump-diffusion models of electricity prices[END_REF].

Preliminaries

Definition 3.1. X = (X t ) t∈[0,T ] is a (real) process with independent increments (PII) iff 1. X is adapted to the filtration (F t ) t∈[0,T ] and has cdlg paths.

X

0 = 0. 3. X t -X s is independent of F s for 0 ≤ s < t ≤ T .
Moreover we will also suppose 4. X is continuous in probability, i.e. X has no fixed time of discontinuties.

We recall Theorem II.4.15 of [START_REF] Jacod | Limit theorems for stochastic processes[END_REF].

Theorem 3.2. Let (X t ) t∈[0,T ] be a real-valued special semimartingale, with X 0 = 0. Then, X is a process with independent increments, iff there is a version (b, c, ν) of its characteristics that is deterministic. Remark 3.3. In particular, ν is a (deterministic non-negative) measure on the Borel σ-field of [0, T ] × R.

From now on, given two reals a, b, we denote by a ∨ b (resp. a ∧ b) the maximum (resp. minimum) between a and b. Proposition 3.4. Suppose X is a semimartingale with independent increments with characteristics (b, c, ν), then there exists an increasing function t → a t such that

db t ≪ da t , dc t ≪ da t and ν(dt, dx) = F t (dx)da t , (3.1
)

where F t (dx) is a non-negative kernel from [0, T ], B([0, T ]) into (R, B) verifying R (|x| 2 ∧ 1) Ft (dx) ≤ 1 , ∀t ∈ [0, T ]. (3.2)
and

a t = ||b|| t + c t + R (|x| 2 ∧ 1)ν([0, t], dx) , (3.3)
Proof. The existence of (a t ) as a process fulfilling (3.3) and F fulfilling (3.2) is provided by the statement and the proof of Proposition II. 2.9 of [START_REF] Jacod | Limit theorems for stochastic processes[END_REF]. (3.3) and Theorem 3.2 guarantee that (a t ) is deterministic.

Remark 3.5. In particular, (b t ), (c t ) and t → [0,t]×B (|x| 2 ∧ 1)ν(ds, dx) has bounded variation for any B ∈ B.

The proposition below provides the so called Lvy-Khinchine Decomposition.

Proposition 3.6. Assume that (X t ) t∈[0,T ] is a process with independent increments. Then

ϕ t (u) = e Ψt(u) , for all u ∈ R , (3.4)
Ψ t , is given by the Lvy-Khinchine decomposition of the process X,

Ψ t (u) = iub t - u 2 2 c t + R (e iux -1 -iuh(x))F t (dx) , for all u ∈ R , (3.5
)

where B → F t (B) is the positive measure ν([0, t] × B) which integrates 1 ∧ |x| 2 for any t ∈ [0, T ].
We introduce here a simplifying hypothesis for this section.

Assumption 2. For any t > 0, X t is never deterministic.

Remark 3.7. We suppose Assumption 2.

1. Up to a 2πi addition of κ t (e), we can write Ψ t (u) = κ t (iu), ∀u ∈ R. From now on we will always make use of this modification.

2. ϕ t (u) is never a negative number. Otherwise, there would be u ∈ R * , t > 0 such that E(cos(uX t )) = -1.

Since cos(uX t ) + 1 ≥ 0 a.s. then cos(uX t ) = -1 a.s. and this is not possible since X t is nondeterministic.

Previous point implies that all the differentiability properties of

u → ϕ t (u) are equivalent to those of u → Ψ t (u). 4. If E[|X t | 2 ] < ∞, then for all u ∈ R, Ψ ′ t (u) and Ψ ′′ t (u) exist.
We come back to the cumulant generating function κ and its domain D.

Remark 3.8. In the case where the underlying process is a PII, then

D := {z ∈ C | E[e Re(z)Xt ] < ∞, ∀t ∈ [0, T ]} = {z ∈ C | E[e Re(z)XT ] < ∞} . In fact, for given t ∈ [0, T ], γ ∈ R we have E(e γXT ) = E(e γXt )E(e γ(XT -Xt) ) < ∞.
Since each factor is positive, and if the left-hand side is finite, then E(e γXt ) is also finite.

We need now a result which extends the Lévy-Khinchine decomposition to the cumulant generating function. Similarly to Theorem 25.17 of [START_REF] Sato | Lvy processes and infinitely divisible distributions[END_REF] we have.

Proposition 3.9. Let D 0 = c ∈ R | [0,T ]×{|x|>1} e cx ν(dt, dx) < ∞ . Then, 1. D 0 is convex and contains the origin. 2. D 0 = D ∩ R. 3. If z ∈ C such that Re(z) ∈ D 0 , i.e. z ∈ D, then κ t (z) = zb t + z 2 2 c t + [0,t]×R (e zx -1 -zh(x))ν(ds, dx) . (3.6)
Proof.

1. is a consequence of Hlder inequality similarly as i) in Theorem 25.17 of [START_REF] Sato | Lvy processes and infinitely divisible distributions[END_REF] .

2. The characteristic function of the law of X t is given by (3.5). According to Theorem II.8.1 (iii) of Sato [START_REF] Sato | Lvy processes and infinitely divisible distributions[END_REF], there is an infinitely divisible distribution with characteristics (b t , c t , F t (dx)), fulfilling F t ({0}) = 0 and (1 ∧ x 2 )F t (dx) < ∞ and c t ≥ 0. By uniqueness of the characteristic function, that law is precisely the law of X t . By Corollary II.11.6, in [START_REF] Sato | Lvy processes and infinitely divisible distributions[END_REF], there is a Lévy process (L t s , 0 ≤ s ≤ 1) such that L t 1 and X t are identically distributed. We define

C t 0 = {c ∈ R | {|x|>1} e cx F t (dx) < ∞} and C t = {z ∈ C | E exp(Re(zL t 1 ) < ∞} . Remark 3.8 says that C T = D, moreover clearly C T 0 = D 0 . Theorem V.
25.17 of [START_REF] Sato | Lvy processes and infinitely divisible distributions[END_REF] implies D 0 = D∩R, i.e. point 2. is established.

3. Let t ∈ [0, T ] be fixed; let w ∈ D. We apply point (iii) of Theorem V.25.17 of [START_REF] Sato | Lvy processes and infinitely divisible distributions[END_REF] to the Lévy process L t .

Proposition 3.10. Let X be a semimartingale with independent increments. For all z ∈ D, t → κ t (z) has bounded variation and

κ dt (z) ≪ da t . (3.7)
Proof. Using (3.6), it remains to prove that

t → [0,T ]×R (e zx -1 -zh(x))ν(ds, dx)
is absolutely continuous with respect to (da t ). We can conclude

κ t (z) = t 0 db s da s da s + z 2 2 t 0 dc s da s da s + t 0 da s R (e zx -1 -zh(x)) F s (dx) , if we show that T 0 da s R |e zx -1 -zh(x)| F s (dx) < ∞ . (3.8)
Without restriction of generality we can suppose h(x) = x1 |x|≤1 . (3.8) can be bounded by the sum I 1 +I 2 +I 3

where

I 1 = T 0 da s |x|>1 |e zx | F s (dx) , I 2 = T 0 da s |x|>1 F s (dx) , and
I 3 = T 0 da s |x|≤1 |e zx -1-zx| F s (dx) .
Using Proposition 3.4, we have

I 1 = T 0 da s |x|>1 |e zx | F s (dx) = T 0 da s |x|>1 |e Re(z)x | F s (dx) = [0,T ]×|x|>1
|e Re(z)x |ν(ds, dx);

this quantity is finite because Re(z) ∈ D 0 taking into account Proposition 3.9. Concerning I 2 we have

I 2 = T 0 da s |x|>1 F s (dx) = T 0 da s |x|>1 (1 ∧ |x 2 |) F s (dx)
≤ a T because of (3.2). As far as I 3 is concerned, we have

I 3 ≤ e Re(z) z 2 2 [0,T ]×|x|≤1 da s (x 2 ∧ 1) F s (dx) = e Re(z) z 2 2 a T
again because of (3.2). This concludes the proof the Proposition.

The converse of the first part of previous corollary also holds. For this purpose we formulate first a simple remark.

Remark 3.11. For every z ∈ D, (exp(zX tκ t (z))) is a martingale. In fact, for all 0 ≤ s ≤ t ≤ T , we have

E[exp(z(X t -X s ))] = exp(κ t (z) -κ s (z)) . (3.9) Proposition 3.12. Let X be a PII. Let z ∈ D ∩ R ⋆ . (X t ) t∈[0,T ] is a semimartingale iff t → κ t (z) has bounded variation.
Proof. It remains to prove the converse implication. If t → κ t (z) has bounded variation then t → e κt(z) ) has the same property. Remark 3.11 says that e zXt = M t e κt(z) where (M t ) is a martingale. Finally, (e zXt ) is a semimartingale and taking the logarithm (zX t ) has the same property.

Remark 3.13. Let z ∈ D. If (X t ) is a semimartingale with independent increments then (e zXt ) is necessarily a special semimartingale since it is the product of a martingale and a bounded variation continuous deterministic function, by use of integration by parts.

Lemma 3.14. Suppose that (X t ) is a semimartingale with independent increments. Then for every z ∈ Int(D), t → κ t (z) is continuous.

Remark 3.15. The conclusion remains true for any process which is continuous in probability, whenever

t → κ t (z) is (locally) bounded. Proof of Lemma 3.14. Since z ∈ Int(D), there is γ > 1 such that γz ∈ D; so E[exp(zγX t )] = exp(κ t (γz)) ≤ exp(sup t≤T (κ t (γz))) , because t → κ t (γz) is bounded, being of bounded variation. This implies that (exp(zX t )) t∈[0,T ] is uniformly integrable. Since (X t ) is continuous in probability, then (exp(zX t )) is continuous in L 1 .
The result easily follows.

Proposition 3.16. The function (t, z) → κ t (z) is continuous. In particular, (t, z) → κ t (z), t ∈ [0, T ], z belonging to a compact real subset, is bounded.

Proof.

• Proposition 3.9 implies that z → κ t (z) is continuous uniformly with respect to t ∈ [0, T ].

• By Lemma 3.14, for z ∈ IntD, t → κ t (z) is continuous.

• To conclude it is enough to show that t → κ t (z) is continuous for every z ∈ D. Since D = IntD, there is a sequence (z n ) in the interior of D converging to z. Since a uniform limit of continuous functions on [0, T ] converges to a continuous function, the result follows.

Structure condition for PII (which are semimartingales)

Let X = (X t ) t∈[0,T ] be a real-valued semimartingale with independent increments and X 0 = 0. We assume

that E[|X t | 2 ] < ∞. We denote by ϕ t (u) = E[exp(iuX t )
] the characteristic function of X t and by u → Ψ t (u) its log-characteristic function introduced in Proposition 3.6. We recall that ϕ t (u) = exp(Ψ t (u)).

X has the property of independent increments; therefore

exp(iuX t )/E[exp(iuX t )] = exp(iuX t )/ exp(Ψ t (u)) , (3.10)
is a martingale.

Remark 3.17. Notice that the two first order moments of X are related to the log-characterisctic function of X, as follows

E[X t ] = -iΨ ′ t (0) , E[X t -X s ] = -i(Ψ ′ t (0) -Ψ ′ s (0)), (3.11)
V ar

(X t ) = -Ψ ′′ t (0) , V ar(X t -X s ) = -[Ψ ′′ t (0) -Ψ ′′ s (0)] . (3.12)
Proposition 3.18. Let X = (X t ) t∈[0,T ] be a real-valued semimartingale with independent increments.

1. X is a special semimartingale with decomposition X = M + A with the following properties:

M t = -Ψ ′′ t (0) and A t = -iΨ ′ t (0) . (3.13)
In particular t → -Ψ ′′ t (0) is increasing and therefore of bounded variation.

X satisfies condition (SC) of Definition 2.11 if and only if

Ψ ′ t (0) ≪ Ψ ′′ t (0) and T 0 d t Ψ ′ s d t Ψ ′′ s (0) 2 |dΨ ′′ s (0)| < ∞ . (3.14)
In that case

A t = t 0 α s d M s with α t = i d t Ψ ′ t (0) d t Ψ ′′ t (0) for all t ∈ [0, T ] . (3.15)
3. Under condition (3.14), FS decomposition exists (and it is unique) for every square integrable random variable.

In the sequel, we will provide an explicit decomposition for a class of contingent claims, under condition (3.14).

Proof.

1. Let us first determine A and M in terms of the log-characteristic function of X. Using (3.11) of Remark 3.17, we get

E[X t |F s ] = E[X t -X s + X s | F s ] , = E[X t -X s ] + X s , = -iΨ ′ t (0) + iΨ ′ s (0) + X s , then , E[X t + iΨ ′ t (0)|F s ] = X s + iΨ ′ s (0) .
Hence, (X t + iΨ ′ t (0)) is a martingale and the canonical decomposition of X follows

X t = X t + iΨ ′ t (0) Mt -iΨ ′ t (0) At ,
where M is a local martingale and A is a locally bounded variation process thanks to the semimartingale property of X. Let us now determine M , in terms of the log-characteristic function of X.

M 2 t = [X t + iΨ ′ t (0)] 2 , E[M 2 t |F s ] = E[(X t + iΨ ′ t (0)) 2 |F s ] , = E[(X s + iΨ ′ s (0) + X t -X s + iΨ ′ t (0) -iΨ ′ s (0)) 2 |F s ] , = E[(M s + X t -X s + i(Ψ ′ t (0) -Ψ ′ s (0))) 2 |F s ] ,
Using (3.11) and (3.12) of Remark 3.17, yields

M 2 t = E[(M s -E[X t -X s ] + X t -X s ) 2 ] , = M 2 s + V ar(X t -X s ) = M 2 s -Ψ ′′ t (0) + Ψ ′′ s (0) .
Hence, (M 2 t + Ψ ′′ t (0)) is a (F t )-martingale, and point 1. is established.

A t = t 0 α s d M s with α t = i d t Ψ ′ t (0) d t Ψ ′′ t (0) for all t ∈ [0, T ] .
2. is a consequence of point 1. and of Definition 2.11.

3. follows from Theorem 2.22. In fact

K t = - T 0 d t Ψ ′ s d t Ψ ′′ s (0) 2 dΨ ′′ s (
0) is deterministic and so Assumption 1 is fulfilled.

Examples

A continuous process example

Let ψ : [0, T ] → R be a continuous strictly increasing function, γ : [0, T ] → R be a bounded variation function such that dγ ≪ dψ. We set X t = W ψ(t) + γ(t), where W is the standard Brownian motion on R. Clearly,

X t = M t + γ(t), where M t = W ψ(t) , defines a continuous martingale, such that M t = [M ] t = ψ(t). Since X t ∼ N (γ(t), ψ(t)) for all u ∈ R and t ∈ [0, T ], we have Ψ t (u) = iγ(t)u - u 2 ψ(t) 2 ,
which yields 1. X is adapted to the filtration (F t ) t∈[0,T ] and has cdlg trajectories.

Ψ ′ t (0) = iγ(t)
2. X 0 = 0.

The distribution of X

t -X s depends only on t -s for 0 ≤ s ≤ t ≤ T . 4. X t -X s is independent of F s for 0 ≤ s ≤ t ≤ T .

X is continuous in probability.

For details on Lvy processes, we refer the reader to [START_REF] Protter | Stochastic integration and differential equations[END_REF], [START_REF] Sato | Lvy processes and infinitely divisible distributions[END_REF] and [START_REF] Jacod | Limit theorems for stochastic processes[END_REF].

Let X = (X t ) t∈[0,T ] be a real-valued Lvy process, with X 0 = 0. We assume that E[|X t | 2 ] < ∞ and we do not consider the trivial case where L 1 is deterministic.

Remark 3.20. 1. Since X = (X t ) t∈[0,T ] is a Lvy process then Ψ t (u) = tΨ 1 (u).
In the sequel, we will use the shortened notation Ψ := Ψ 1 .

2. Ψ is a function of class C 2 and Ψ ′′ (0) = V ar(X 1 ) which is strictly positive if X has no stationary increments.

Cumulative and characteristic functionals in some particular cases

We recall some cumulant and log-caracteristic functions of some typical Lvy processes.

Remark 3.21.

1. Poisson Case: If X is a Poisson process with intensity λ, we have that κ Λ (z) = λ(e z -1). Moreover, in this case the set D = C. [START_REF] Barndorff-Nielsen | Infinite divisibility of the hyperbolic and generalized inverse gaussian distributions[END_REF]. Then X is a Lvy process with

Concerning the log-characteristic function we have

Ψ(u) = λ(e iu -1) , Ψ ′ (0) = iλ and Ψ ′′ (0) = -λ, u ∈ R.

NIG Case: This process was introduced by Barndorff-Nielsen in

X 1 ∼ N IG(α, β, δ, µ), with α > |β| > 0, δ > 0 and µ ∈ R. We have κ Λ (z) = µz + δ(γ 0 -γ z ) and γ z = α 2 -(β + z) 2 , D =] -α -β, α -β[+iR . Therefore Ψ(u) = µiu + δ(γ 0 -γ iu ) , where γ iu = α 2 -(β + iu) 2 .
20

By derivation, one gets

Ψ ′ (0) = iµ + δ iβ γ 0 and Ψ ′′ (0) = -δ( 1 γ 0 + β 2 γ 3 0 ), Which yields α = i Ψ ′ (0) Ψ ′′ (0) = γ 2 0 (γ 0 µ + δβ) δ(γ 2 0 + β) . 3. Variance Gamma case: Let α, β > 0, δ = 0. If X is a Variance Gamma process with X 1 ∼ V G(α, β, δ, µ) with κ Λ (z) = µz + δLog α α-βz-z 2 2
, where Log is again the principal value complex logarithm defined in Section 2. The expression of κ Λ (z) can be found in [START_REF] Hubalek | Variance-optimal hedging for processes with stationary independent increments[END_REF][START_REF] Madan | The variance gamma process and option pricing[END_REF] or also [START_REF] Cont | Financial modelling with jump processes[END_REF], table IV.4.5 in the particular case µ = 0. In particular an easy calculation shows that we need z ∈ C such that Re

(z) ∈] -β -β 2 + 2α, -β + β 2 + 2α[ so that κ Λ (z) is well defined so that D =] -β -β 2 + 2α, -β + β 2 + 2α[+iR.
Finally we obtain

Ψ(u) = µiu + δLog α α -βiu + u 2 2 .
After derivation it follows

Ψ ′ (0) = i(µ -δβ), Ψ ′′ (0) = δ α (α 2 -β 2 ).

Structure condition in the Lévy case

By application of Proposition 3.18 and Remark 3.20, we get the following result.

Corollary 3.22. Let X = M + A be the canonical decomposition of X, then for all t ∈ [0, T ],

M t = -tΨ ′′ (0) and A t = -itΨ ′ (0) . (3.16)
Moreover X satisfies condition (SC) of Definition 2.11 with

A t = t 0 αd M s with α = i Ψ ′ (0) Ψ ′′ (0) for all t ∈ [0, T ] . (3.17)
Hence, FS decomposition exists for every square integrable random variable.

Remark 3.23. We have the following in previous three examples of subsubsection 3.4

1. Poisson case: α = 1.

NIG process

: α = γ 2 0 (γ 0 µ + δβ) δ(γ 2 0 + β) . 3. VG process: α = µ -δβ α 2 -β 2 α δ .

Wiener integrals of Lvy processes

We take X t = t 0 γ s dΛ s , where Λ is a square integrable Lvy process as in Section 3.3.2. Then, T 0 γ s dΛ s is well-defined for at least γ ∈ L ∞ ([0, T ]). It is then possible to calculate the characteristic function and the cumulative function of 1. The log-characteristic function of X t is such that for all u ∈ R,

Ψ Xt (u) = t 0 Ψ Λ (uγ s )ds , where E[exp(iuX t )] = exp Ψ Xt (u) ; 2. Let D Λ be the domain related to κ Λ in the sense of Definition 2.2. The cumulant generating function of X t is such that for all z ∈ {z | Rezγ t ∈ D Λ for all t ∈ [0, T ]}, κ Xt (z) = t 0 κ Λ (zγ s )ds.
Proof. We only prove 1. since 2. follows similarly. Suppose first γ to be continuous, then T 0 γ s dΛ s is the limit in probability of p-1 j=0 γ tj (Λ tj+1 -Λ tj ) where 0 = t 0 < t 1 < ... < t p = T is a subdivision of [0, T ] whose mesh converges to zero. Using the independence of the increments, we have

E   exp{i p-1 j=0 γ tj (Λ tj+1 -Λ tj )}   = p-1 j=0 E exp{iγ tj (Λ tj+1 -Λ tj )} , = p-1 j=0 exp{Ψ Λ (γ tj )(t j+1 -t j )} , = exp{ p-1 j=0 (t j+1 -t j )Ψ Λ (γ tj )} .
This converges to exp T 0 Ψ Λ (γ s )ds , when the mesh of the subdivision goes to zero. Suppose now that γ is only bounded and consider, using convolution, a sequence γ n of continuous functions, such that γ n → γ a.e. and sup t∈[0,T ] |γ n (t)| ≤ sup t∈[0,T ] |γ(t)|. We have proved that

E exp i T 0 γ n (s)dΛ s = exp T 0 Ψ Λ (γ n (s))ds (3.18)
Now, Ψ Λ is continuous therefore bounded, so Lebesgue dominated convergence and continuity of stochastic integral imply statement 1.

Remark 3.25. 1. Previous proof, which is left to the reader, also applies for statement 2. This statement in a slight different form is proved in [START_REF] Benth | The normal inverse Gaussian distribution and spot price modelling in energy markets[END_REF] 2. We prefer to formulate a direct proof. In particular statement 1. holds with the same proof even if Λ has no moment condition and γ is a continuous function with bounded variation. Stochastic integrals are then defined using integration by parts.

We suppose now that Λ is a Lvy process such that Λ 1 is not deterministic. In particular V ar(Λ 1 ) = 0 and so Ψ ′′ Λ = 0. In this case

Ψ ′ t (u) = t 0 Ψ ′ Λ (uγ s )γ s ds and Ψ ′′ t (u) = t 0 Ψ ′′ Λ (uγ s )γ 2 s ds . So Ψ ′ t (0) = Ψ ′ Λ (0) t 0 γ s ds and Ψ ′′ t (0) = Ψ ′′ Λ (0) t 0 γ 2 s ds . Condition (SC) is verified since dΨ ′ t (0) ≪ dΨ ′′ t (0) with α t = i dΨ ′ t (0) dΨ ′′ t (0) = Ψ ′ Λ (0) Ψ ′′ Λ (0) i γ t 1 {γt =0} and T 0 α 2 s |Ψ ′′ s (0)|γ 2 s ds = T |Ψ ′ Λ (0)| 2 |Ψ ′′ Λ (0)| < ∞ .
3.6 Explicit Fllmer-Schweizer decomposition in the PII case

Preliminaries

Let X = (X t ) t∈[0,T ] be a semimartingale with independent increments with log-characteristic function (t, u) → Ψ t (u). We assume that (X t ) t∈[0,T ] is square integrable and satisfies Assumption 2.

Remark 3.26.

1. u → Ψ t (u) is of class C 2 , for any t ∈ [0, T ] because X t is square integrable. 2. t → Ψ ′′ t (0) and t → Ψ ′ t (
0) have bounded variation because of Proposition 3.18. Therefore, they are bounded.

3. t → Ψ ′ t (u) is continuous for every u ∈ R. In fact, first t → X t is continuous in probability. Since M t = X t -Ψ ′ t (
P {|M 2 t | > N } ≤ Var(X t ) N ≤ Var(X T ) N ,
and so the family (M 2 t ) is again uniformly integrable.

We suppose the following.

Assumption 3.

1. t → Ψ ′ t (
u) is absolutely continuous with respect to dΨ ′′ t (0).

2.

For every u ∈ R, we suppose that the following quantity

K(u) := T 0 dΨ ′ t (u) dΨ ′′ t (0) 2 d(-Ψ ′′ s (0)) (3.19)
is finite.

Remark 3.27. If u = 0, the previous quantity (3.19) is finite because of the (SC) condition.

We consider a contingent claim which is given as a Fourier transform of X T ,

H = f (X T ) with f (x) = R e iux µ(du) , for all x ∈ R , (3.20)
for some finite signed measure µ.

Assumption 4. R K(u)d|µ(u)| < ∞.
Remark 3.28. We observe that the function

(u, t) → exp(Ψ T (u) -Ψ t (u))
is uniformly bounded because the characteristic function is bounded.

We will first evaluate an explicit Kunita-Watanabe decomposition of H w.r.t. the martingale part M of X. Later, we will finally obain the decomposition with respect to X.

Explicit elementary Kunita-Watanabe decomposition

By Propostion 3.18, X admits the following semimartingale decomposition, X t = A t + M t , where

A t = -iΨ ′ t (0) and M t = -Ψ ′′ t (0) . (3.21) Proposition 3.29. Let H = f (X T )
where f is of the form (3.20). We suppose that the PII X satisfies Assumptions 2, 3 and 4. Then, H admits the decomposition

V t = V 0 + t 0 Z s dM s + O t V T = H , (3.22)
with the following properties.

1. For all t ∈ [0, T ],

Z t = i R e iuXt- d Ψ ′ t (u) -Ψ ′ t (0) dΨ ′′ t (0) exp {Ψ T (u) -Ψ t (u)} µ(du) ; (3.23) 2.
O is a square integrable (F t )-martingale such that O, M = 0 ;

3. H = V T where (V t ) t∈[0,T ] is the (F t )-martingale defined by

V t = E[H|F t ] = R e iuXt exp {Ψ T (u) -Ψ t (u)} µ(du) . (3.24)
Remark 3.30. In particular,

1. V 0 = E[H] ; 2. E T 0 Z 2 s d M s < ∞ .
Proof. We start with the case µ = δ u (dx) for some u ∈ R so that f (x) = e iux . We consider the (F t )martingale

V t = E[f (X T )|F t ] = E[e iuXT |F t ]. 1. Clearly V 0 = E[e iuXT ] .
2. We calculate explicitely V t , which gives

V t = E[e iuXT |F t ] = e iuXt E[e iu(XT -Xt) ] = exp(iuX t -Ψ t (u)) exp(Ψ T (u)) = V t exp(Ψ T (u)) ,
where V t = exp(iuX t -Ψ t (u)) defines an (F t )-martingale.

3. We evaluate V, M .

Lemma 3.31. V, M t = -i t 0 V s (Ψ ′ ds (u) -Ψ ′ ds (0)) . Proof. We evaluate E[ V t M t |F s ].
Since V and M are (F t )-martingales and using the property of independent increments we get

E[ V t M t |F s ] = E[ V t M s |F s ] + E[ V t (M t -M s )|F s ] , = M s V s + V s E[exp{iu(X t -X s ) -(Ψ t (u) -Ψ s (u))}(M t -M s )] , = M s V s + V s e -(Ψt(u)-Ψs(u)) E[e iu(Xt-Xs) (M t -M s )] .
Previous expectation gives

E[e iu(Xt-Xs) (M t -M s )] = E[e iu(Xt-Xs) (X t -X s )] + E[e iu(Xt-Xs) i(Ψ ′ t (0) -Ψ ′ s (0))] , = -i ∂ ∂u E[e iu(Xt-Xs) ] + i(Ψ ′ t (0) -Ψ ′ s (0))E[e iu(Xt-Xs) ] , = -ie Ψt(u)-Ψs(u) (Ψ ′ t (u) -Ψ ′ s (u)) + i(Ψ ′ t (0) -Ψ ′ s (0))e Ψt(u)-Ψs(u) .
Consequently,

E[ V t M t |F s ] = M s V s -i V s (Ψ ′ t (u) -Ψ ′ s (u)) + i V s (Ψ ′ t (0) -Ψ ′ s (0)) = M s V s -i V s Ψ ′ t (u) -Ψ ′ t (0) -(Ψ ′ s (u) -Ψ ′ s (0)) . This implies that V t M t + i V t (Ψ ′ t (u) -Ψ ′ t (0))
t is an (F t )-martingale. Then by integration by parts,

V t (Ψ ′ t (u) -Ψ ′ t (0)) = t 0 V s (Ψ ′ ds (u) -Ψ ′ ds (0)) + t 0 (Ψ ′ s (u) -Ψ ′ s (0))d V s .
The second integral term of the right-hand side being a martingale, it follows that

V , M t = -i t 0 V s (Ψ ′ ds (u) -Ψ ′ ds (0)) .
and so

V, M t = -i t 0 V s (Ψ ′ ds (u) -Ψ ′ ds (0)) . (3.25)
4. We continue the proof of the Proposition 3.29. For given (Z t ) we have

t 0 ZdM, M t = t 0 Z s-d M s = - t 0 Z s Ψ ′′ ds (0) .

We want to identify

- t 0 Z s Ψ ′′ ds (0) = -i t 0 V s (Ψ ′ ds (u) -Ψ ′ ds (0)) .
This naturally leads to

Z s = i d(Ψ ′ s (u) -Ψ ′ s (0)) dΨ ′′ s (0) V s-. (3.26)
6. Finally, we obtain the general case, for general finite signed measure µ, similarly to the proof of Theorem 4.23 (in the sequel) in the case of exponential of PII processes. The use of Fubini's theorem is essential.

Example 3.32. We take X = M = W the classical Wiener process. We have

Ψ s (u) = -u 2 s 2 so that Ψ ′ s (u) = -us and Ψ ′′ s (u) = -s. So Z s = iuV s . We recall that V s = E[exp(iuW T )|F s ] = exp(iuW s ) exp -u 2 T -s 2 .
In particular, V 0 = exp(-u 2 T 2 ) and so

exp(iuW T ) = i T 0 u exp(iuW s ) exp -u 2 T -s 2 dW s + exp(- u 2 T 2 ).
In fact that expression is classical and it can be derived from Clark-Ocone formula.

Explicit Fllmer-Schweizer decomposition

We introduce a quantity which will be useful in the sequel. For t ∈ [0, T ], u ∈ R we set

η(u, t) = t 0 d(Ψ ′ s (u) -Ψ s ′ (0)) d(Ψ ′′ s (0)) Ψ ′ ds (0) . (3.27) Remark 3.33. 1. η is defined unambiguously since d Ψ ′ t (u) -Ψ ′ t (
0) is absolutely continuous with respect to dΨ ′′ t (0) .

η is well-defined, because for any

u ∈ R, η(u, t) = t 0 d(Ψ ′ s (u) -Ψ s ′ (0)) d(Ψ ′′ s (0)) d(Ψ s ′ (0)) d(Ψ ′′ s (0)) dΨ ′′ ds (0)
is bounded by Cauchy-Schwarz, taking into account Assumption 3 point 2.

We are now able to evaluate the FS decomposition of H = f (X T ) where f is given by (4.27).

We introduce now a supplementary hypothesis. (Re(η(u, t)) < ∞ .

Theorem 3.34. Under the assumptions of Proposition 3.29 and Assumption 5, the FS decomposition of H is the following

H t = H 0 + t 0 ξ s dX s + L t with H T = H (3.28)
and

H t = R H(u) t µ(du) , (3.29)
ξ t = R ξ(u) t µ(du) ,
where

ξ(u) t = i d(Ψ ′ t (u) -Ψ ′ t (0)) dΨ ′′ t (0) H(u) t-, (3.30) H(u) t = exp {η(u, T ) -η(u, t) + Ψ T (u) -Ψ t (u)} e iuXt .
Proof. Using Fubini's theorem, we reduce the problem to show that

H(u) t = H(u) 0 + t 0 ξ(u) s dX s + L(u) t with H(u) T = exp(iuX T ) ,
for fixed u ∈ R where L(u) is a square integrable martingale and L(u), M = 0, where M is the martingale part of the special semimartingale X. Notice that by equation (3.30),

H(u) t = H(u) 0 + e R T t η(u,ds) V (u) t with V (u) t = exp(iuX t + Ψ T (u) -Ψ t (u)) .
Integrating by parts, gives

H(u) t = H(u) 0 - t 0 e R T r η(u,ds) V (u) r η(u, dr) + t 0 e R T r η(u,ds) dV (u) r .
We denote again by Z(u) the expression provided by (3.26). We recall that

dV (u) r = Z(u) r dM r + dO(u) r = Z(u) r (dX r -dA r ) + dO(u) r ,
where A is given by (3.21) and O is a square integrable martingale strongly orthogonal to M (i.e. M, O . = 0).

H(u) t = H(u) 0 +L(u) t + t 0 e R T r η(u,ds) Z(u) r dX r - t 0 e R T r η(u,ds) Z(u) r (-iΨ ′ dr (0))- t 0 e R T r η(u,ds) V (u) r η(u, dr) ,
where

L(u) t = t 0 e R T r η(u,ds) dO(u) r ,
is a martingale strongly orthogonal to M . To conclude, we need to choose η so that

t 0 Z(u) r e R T r η(u,ds) (-iΨ ′ dr (0)) = t 0 e R T r η(u,ds) V (u) r η(u, dr)) .
This requires

η(u, dr) = d(Ψ ′ r (u) -Ψ r ′ (0)) d(Ψ ′′ r (0)) Ψ ′ dr (0) .
So we define η as in (3.27).

The Lvy case

Let X be a square integrable Lvy process, with characteristic function exp(Ψ(u)t). In particular, Ψ is of class C 2 (R). We have

dΨ ′ t (u) dΨ ′′ t (0) = Ψ ′ (u) Ψ ′′ (0) and η(u, t) = t Ψ ′ (u) -Ψ ′ (0) Ψ ′′ (0) Ψ ′ (0) .
We remark that Assumptions 2 is verified. Concerning Assumption 3, point 1. is trivial; point 2. is verified because

K(u) = Ψ ′ (u) Ψ ′′ (u) 2 (-T Ψ ′′ (u))
. On the other hand Assumption 5 is verified if

Re Ψ ′ (u)Ψ ′ (0) Ψ ′′ (0) < ∞ . (3.31) Since Ψ ′ (0) = iE[X 1 ] and Ψ ′′ (0) < 0, (3.31) is fulfilled if E[X 1 ]Im(Ψ ′ (u)) > -∞ . (3.32) Assumption 4 is verified if R Ψ ′ (u) 2 d|µ(u)| < ∞ . ( 3

Poisson case

We have Ψ ′ (u) = iλe iu .

NIG case

We have

Ψ ′ (u) = iµ + iδ (β + iu) α 2 -(β + iu) 2 -1 2 . Now |Ψ ′ (u)| ≤ 2 |µ| 2 + 2δ β 2 + u 2 (α 2 -β 2 + u 2 ) 2 + 4u 2 β 2 . Since |α| > |β|, u → |Ψ ′ (u)| is bounded.

Variance Gamma case

We have

Ψ ′ (u) = iµ -u-iβ α-iuβ+ u 2 2
Clearly |Ψ ′ (u)| is again bounded.

In conclusion, we can apply Theorem 3.34 and we obtain

V (u) t = exp(iuX t + (T -t)Ψ(u)) , H(u) t = exp ((T -t)Ψ(u) + η(u, T ) -η(u, t)) e iuXt , ξ(u) t = H t (u)i Ψ ′ (u) -Ψ ′ (0)
Ψ ′′ (0) .

Representation of some contingent claims by Fourier transforms

In general, it is not possible to find a Fourier representation, of the form (3.20), for a given payoff function which is not necessarily bounded or integrable. Hence, it can be more convenient to use the bilateral Laplace transform that allows an extended domain of definition including non integrable functions. We refer to [START_REF] Cramer | On the representation of a function by certain Fourier integrals[END_REF], [START_REF] Raible | Lvy processes in finance: theory, numerics and empirical facts[END_REF] and more recently [START_REF] Eberlein | Analysis of Fourier transform valuation formula and applications[END_REF] for such characterizations of payoff functions. This will be done in the next section. However, to illustrate the results of this section restricted to payoff functions represented as classical Fourier transforms, we give here two simple examples of such representation extracted from [START_REF] Eberlein | Analysis of Fourier transform valuation formula and applications[END_REF]:

1. A variant of the digital option is the so-called asset-or-nothing digital, where the option holder receives one unit of the asset, instead of currency, depending on wether the underlying reaches some barrier or not. Hence, the payoff of the asset-or-nothing digital put with barrier is

f (x) = e x 1 e x <B and f (u) = R e iux f (x) dx = B 1+iu 1 + iu .
2. The payoff of a self quanto put option with strike K is

f (x) = e x (K -e x ) + and f (u) = R e iux f (x) dx = K 2+iu (1 + iu)(2 + iu) .
In both cases the measure µ is finite.

Fllmer Schweizer decomposition for exponential of PII processes

In this section, we consisder the case of exponential of PII corresponding to geometric models (such as the Black-Scholes model) much more used in finance than arithmetic models (such as the Bachelier model).

The aim of this section is to generalize the results of [START_REF] Hubalek | Variance-optimal hedging for processes with stationary independent increments[END_REF] to the case of PII with possibly non stationary increments. Here again, this generalization is motivated by applications to energy derivatives where forward prices show a volatility term structure that requires the use of models with non stationary increments.

A reference variance measure

We come back to the main optimization problem which was formulated in Section 2. We assume that the process S is the discounted price of the non-dividend paying stock which is supposed to be of the form,

S t = s 0 exp(X t ) , for all t ∈ [0, T ] ,
where s 0 is a strictly positive constant and X is a semimartingale process with independent increments (PII), in the sense of Definition 3.1, but not necessarily with stationary increments.

Remark 4.1. Let γ ∈ R * , 1. E[exp(γ(X t -X s ))] > 0, since X t -X s > -∞ a.s. 2. exp(γ(X t -X s )) has a strictly positive variance if (X t -X s ) is non-deterministic.
We introduce a new function that will be useful in the sequel.

Definition 4.2. For any t ∈ [0, T ], let ρ t denote the complex valued function such that for all z, y ∈ D ρ t (z, y) = κ t (z + y)κ t (z)κ t (y) . For all z ∈ D, then z ∈ D and ρ t (z, z) is well defined. To shorten notations ρ t will also denote the real valued function defined on D such that,

ρ t (z) = ρ t (z, z) = κ t (2Re(z)) -2Re(κ t (z)) . (4.2)
Notice that the last equality results from Remark 2.3.

An important technical lemma follows below.

Lemma 4.3. Let z ∈ D, with z = 0, then, t → ρ t (z) is strictly increasing if and only if X has no deterministic increments.

Proof. It is enough to show that X has no deterministic increment if and only if for any 0 ≤ s < t ≤ T , the following quantity is positive,

ρ t (z) -ρ s (z) = κ t 2Re(z) -κ s 2Re(z) -2Re κ t (z) -κ s (z) . (4.3)
By Remark 3.11, for all z ∈ D, we have exp[κ t (z)κ s (z)] = E[exp(z∆ t s )] , where ∆ t s = X t -X s .

Applying this property and Remark 2.3 1., to the exponential of the first term on the right-hand side of (4.3) yields

exp κ t 2Re(z) -κ s 2Re(z) = E[exp(2Re(z)∆ t s )] = E[exp((z + z)∆ t s )] = E[ exp(z∆ t s ) 2 ] .
Similarly, for the exponential of the second term on the right-hand side difference of (4.3), one gets

exp 2Re κ t (z) -κ s (z) = exp κ t (z) -κ s (z) + κ t (z) -κ s (z) = E[exp(z∆ t s )] 2 .
Hence taking the exponential of

ρ t (z) -ρ s (z) yields exp[ρ t (z) -ρ s (z)] -1 = E[|exp(z∆ t s )| 2 ] |E[exp(z∆ t s )]| 2 -1 , = E[|Γ t s (z)| 2 ] |E[Γ t s (z)]| 2 -1 , where Γ t s (z) = exp(z∆ t s ) , = V ar Re Γ t s (z) + V ar Im Γ t s (z) |E[Γ t s (z)]| 2 . (4.4)
• If X has a deterministic increment ∆ t s = X t -X s , then Γ t s (z) is again deterministic and (4.4) vanishes and hence t → ρ t (z) is not strictly increasing.

• If X has never deterministic increments, then the nominator is never zero, otherwise Re Γ t s (z) , Im Γ t s (z) and therefore Γ t s (z) would be deterministic.

From now on, we will always suppose the following assumption.

Assumption 6.

1. (X t ) has no deterministic increments.

2. 2 ∈ D.

Remark 4.4.

1. In particular for γ ∈ D, γ = 0, the function t → ρ t (γ) is strictly increasing.

2. If z = 1, (4.4) equals V ar exp(∆ t s ) E[exp(∆ t s )]
2 , which is a mean-variance quantity.

We continue with a simple observation.

Lemma 4.5. Let I be a compact real interval included in D.

sup γ∈I sup t≤T E[S γ t ] < ∞ .
Proof. Let t ∈ [0, T ] and γ ∈ I, we have

E[S γ t ] = s γ 0 exp{κ t (γ)} ≤ max(1, s sup I 0 ) exp( sup t≤T,γ∈I |κ t (γ)|) .
since κ is continuous.

We state now a result that will help us to show that κ dt (z) is absolutely continuous with respect to ρ dt (1) = κ dt (2) -2κ dt (1). Lemma 4.6. We consider two positive finite non-atomic Borel measures on E ⊂ R n , µ and ν. We suppose the following:

1. µ ≪ ν ;

µ(I) = 0 for every open ball of E.

Then dµ dν := h = 0 ν a.e. In particular µ and ν are equivalent.

Proof. We consider the Borel set

B = {x ∈ E|h(x) = 0} .
We want to prove that ν(B) = 0. So we suppose that there exists a constant c > 0 such that ν(B) = c > 0 and another constant ǫ such that 0 < ǫ < c. Since ν is a Radon measure, there are compact subsets K ǫ and K ǫ 2 of E such that

K ǫ ⊂ K ǫ 2 ⊂ B and ν(B -K ǫ ) < ǫ , ν(B -K ǫ 2 ) < ǫ 2 .
Setting ǫ = c 2 , we have

ν(K ǫ ) > c 2 and ν(K ǫ 2 ) > 3c 4 .
By Urysohn lemma, there is a continuous function ϕ : E → R such that, 0 ≤ ϕ ≤ 1 with

ϕ = 1 on K ǫ and ϕ = 0 on K c ǫ 2 . Now E ϕ(x)ν(dx) ≥ ν(K ǫ ) > c 2 > 0 .
By continuity of ϕ there is an open set O ⊂ E with ϕ(x) > 0 for x ∈ O. Clearly O ⊂ K ǫ 2 ⊂ B; since O is relatively compact, it is a countable union of balls, and so B contains a ball I. The fact that h = 0 on I implies µ(I) = 0 and this contradicts Hypothesis 2. of the statement. Hence the result follows.

Remark 4.7. From now on, in this section, dρ t = ρ dt will denote the measure

dρ t = ρ dt (1) = d(κ t (2) -2κ t (1)) . ( 4
S z t = M (z) t + A(z) t satisfies A(z) t = t 0 S z u-κ du (z) , M (y), M (z) t = t 0 S y+z u-ρ du (z, y) , M (z) 0 = s z 0 , (4.7)
where dρ u (z) is defined by equation (4.2). In particular we have the following:

1. M (z), M t = t 0 S z+1 u-ρ du (z, 1) 2. M (z), M (z) t = t 0 S 2Re(z) u- ρ du (z) .
Proof. For simplicity, we will only treat the case y = 1 in (4.7), i.e. statement 1. The general case will follow similarly. By Remark 3.11, N (z) t := e -κt(z) S z t is a martingale. Integration by parts yields

S z t = e κt(z) N (z) t = M (z) t + A(z) t with M 0 (z) = s z 0 , A(z) t = t 0 S z u-κ du (z) and [M (z), M ] t = [S z , S] t , = S z t S 1 t -S z 0 S 1 0 - t 0 S z s-dS 1 s t 0 S 1 s-dS z s , = S z t S 1 t -S z 0 S 1 0 - t 0 S z s-dM s - t 0 S z s-dA s - t 0 S 1 s-dM (z) s - t 0 S 1 s-dA(z) s , = S z+1 t -S z+1 0 - t 0 S z s-dM s - t 0 S 1 s-dM (z) s - t 0 S z+1 s-κ ds (1) - t 0 S z+1 s-κ ds (z) , = M (z + 1) t - t 0 S z s-dM s - t 0 S s-dM (z) s + t 0 (κ ds (z + 1) -κ ds (z) -κ ds (1))S z+1 s-.
Note that the first three terms on the right-hand side are local martingales. Since M (z), M t is the predictable part of finite variation of the special semimartingale M (z)M , equation (1) follows.

On the Structure Condition

If we apply Proposition 4.11 with y = z = 1, we obtain S = M + A where M is a martingale and

A t = t 0 S u-κ du (1) , (4.8)
and

M, M t = t 0 S 2 u-(κ du (2) -2κ du (1)) = t 0 S 2 u-ρ du . (4.9)
At this point, the aim is to exhibit a predictable R-valued process α such that

1. A t = t 0 α s d M s ; 2. K T = T 0 α 2 s d M s is bounded.
In that case, according Theorem 2.22, there will exist a unique FS decomposition for any H ∈ L 2 and so the minimization problem (2.1) will have a unique solution, by Theorem 2.25.

Proposition 4.13. Under Assumption 6, we have

A t = t 0 α s d M s , (4.10)
where α is given by

α u := λ u S u- with λ u := dκ u (1) dρ u , for all u ∈ [0, T ] . (4.11)
Moreover the MVT process is given by

K t = t 0 d(κ u (1)) dρ u 2 dρ u . (4.12)
Corollary 4.14. Under Assumption 6, the structure condition (SC) is verified if and only if

K T = T 0 d(κ u (1)) dρ u 2 dρ u < ∞ .
In particular, (K t ) is deterministic therefore bounded.

T 0 v 2 t d M, M t ] < ∞. Since M, M t = t 0 S 2 s-dρ s ,
the assertion follows.

Explicit Fllmer-Schweizer decomposition

We denote by D the set of z ∈ D such that

T 0 dκ u (z) dρ u 2 dρ u < ∞. (4.13)
From now on, we formulate another assumption which will be in force for the whole section.

Assumption 7 implies that K is uniformly bounded.

The proposition below will constitute an important step for determining the FS decomposition of the contingent claim H = f (S T ) for a significant class of functions f , see Section 4.5.

Proposition 4.17. Let z ∈ D with z + 1 ∈ D and 2Re(z) ∈ D.

S z

T ∈ L 2 (Ω, F T ).

We suppose

L(z) t := H(z) t -H(z) 0 - t 0 ξ(z) u dS u . (4.18)
Proof.

1. is a consequence of Lemma 4.5.

γ(z,

•) is square integrable because Assumption 7 and z, z + 1 ∈ D. Moreover η is well-defined since

T 0 |γ(z, s)| dκ s (1) dρ s ρ ds 2 ≤ T 0 |γ(z, s)| 2 ρ ds T 0 | dκ s (1) dρ s | 2 ρ ds . (4.19)
3. In order to prove that (4.16),(4.17

H(z) t = H(z) 0 + t 0 e R T u η(z,ds) dS z u + t 0 S z u d(e R T u η(z,ds) ) = H(z) 0 + t 0 e R T u η(z,ds) dM (z) u + t 0 e R T u η(z,ds) dA(z) u + t 0 S z u d(e R T u η(z,ds) ) = H(z) 0 + t 0 e R T u η(z,ds) dM (z) u + t 0 e R T u η(z,ds) dA(z) u - t 0 e R T u η(z,ds) S z u η(z, du) = H(z) 0 + t 0 e R T u η(z,ds) dM (z) u - t 0 e R T u η(z,ds) S z u η(z, du) + t 0 e R T u η(z,ds) S z u-κ du (z) .
On the other hand

t 0 ξ(z) u dS u = t 0 ξ(z) u dM u + t 0 ξ(z) u dA u , = t 0 ξ(z) u dM u + t 0 ξ(z) u S u-κ du (1) , = t 0 ξ(z) u dM u + t 0 γ(z, u)e R T u η(z,ds) S z u-κ du (1) .
Hence,

L(z) t = H(z) t -H(z) 0 - t 0 ξ(z) u dS u , = t 0 e R T u η(z,ds) dM (z) u - t 0 e R T u η(z,ds) S z u η(z, du) + t 0 e R T u η(z,ds) S z u-κ du (z) - t 0 ξ(z) u dM u - t 0 γ(z, u)e R T u η(z,ds) S z u-κ du (1) , = t 0 e R T u η(z,ds) dM (z) u - t 0 ξ(z) u dM u + t 0 e R T u η(z,ds) S z u-[κ du (z) -η(z, du) -γ(z, u)κ du (1)].
Then, by definition of η in (4.15), η(z, du) = κ du (z)γ(z, u)κ du (1) , hence,

L(z) t = t 0 e R T u η(z,ds) dM (z) u - t 0 ξ(z) u dM u , (4.20)
which implies that L(z) is a local martingale.

From point 1 of Proposition 4.11, it follows that

L(z), M t = t 0 e R T u η(z,ds) d M (z), M u - t 0 ξ(z) u d M, M u , = t 0 e R T u η(z,ds) S z+1 u-ρ du (z, 1) - t 0 ξ(z) u S 2 u-ρ du , = t 0 e R T u η(z,ds) S z+1 u-ρ du (z, 1) - t 0 γ(z, u)e R T u η(z,ds) S z+1 u-ρ du .
Consequently,

L(z), M t = t 0 e R T u η(z,ds) S z+1 u-[ρ du (z, 1) -γ(z, u)ρ du ] .
Then by definition of γ in (4.14), ρ dt (z, 1) = γ(z, t)ρ dt , which yields,

L(z), M t = 0 . (4.21)
Consequently, point (b) follows. To continue the proof of this proposition we need the lemma below.

Lemma 4.18. For all z ∈ C as in Proposition 4.17, dρ t a.e. we have

1. γ(z, t) = γ(z, t) ; 2. η(z, t) = η(z, t) .
Proof. Using Remark 2.3 1) we observe z, z + 1 ∈ D.

1. By definition of γ in (4.14), γ(z, t)ρ dt = ρ dt (z, 1) . Then, taking the complex conjugate of the integral from 0 to t and using Remark 2.3.1 yields,

t 0 γ(z, s)ρ ds = t 0 γ(z, s)ρ ds , = ρ t (z, 1) = κ t (z + 1) -κ t (z) -κ t (1) , = = κ t (z + 1) -κ t (z) -κ t (1) = ρ t (z, 1) , = t 0 γ(z, s)ρ ds .
2. By definition of η in (4.15), η(z, t) = κ t (z) -t 0 γ(z, u)κ du (1) , so taking the complex conjugate,

η(z, t) = κ t (z) - t 0 γ(z, s)κ ds (1) , = κ t (z) - t 0 γ(z, s)κ ds (1) , = η(z, t) .
We continue with the proof of point 3. of Proposition 4.17

L(z) t = L(z) t , (4.22)
hence,

L(z), L(z) t = L(z), L(z) t , = L(z), . 0 e R T s η(z,du) dM s (z) t , (4.23) = t 0 e R T s η(z,du) e R T s η(z,du) d M (z), M (z) s - t 0 ξ(z) s e R T s η(z,du) d M, M (z) s .
By Proposition 4.11 we have

L(z), L(z) t = t 0 e R T s η(z,du) e R T s η(z,du) S 2Re(z) s- ρ ds (z) - t 0 ξ(z) s e R T s η(z,du) S 1+z s-ρ ds (z, 1) .
Using Lemma 4.18 and expressions (4.14), and (4.17) of γ(z, s) and ξ(z) s , we have

L(z), L(z) t = t 0 e R T s 2Re(η(z,du)) S 2Re(z) s- ρ ds (z) - t 0 ξ(z) s e R T s η(z,du) S 1+z s-γ(z, s)ρ ds (1) , = t 0 e R T s 2Re(η(z,du)) S 2Re(z) s- ρ ds (z) - t 0 γ(z, s)e R T s 2Re(η(z,du)) S 2Re(z) s- γ(z, s)ρ ds , = t 0 e R T s 2Re(η(z,du)) S 2Re(z) s- ρ ds (z) - t 0 e R T s 2Re(η(z,du)) S 2Re(z) s- |γ(z, s)| 2 ρ ds .
Consequently

L(z), L(z) t = t 0 e R T s 2Re(η(z,du) )S 2Re(z) s- [ρ ds (z) -|γ(z, s)| 2 ρ ds ] . (4.24)
Then, point 2. implies This concludes the proof of Proposition 4.17.

T 0 |ξ(z) s | 2 S 2 s-ρ ds = T 0 ξ(z) s ξ(z) s S 2 s-ρ ds = T 0 γ(z, s)e R T s η(z,du) S z-1 s-γ(z, s)e R T s η(z,du) S z-1 s-S 2 s-ρ ds , (4.25)

FS decomposition of special contingent claims

Now, we will proceed to the FS decomposition of more general contingent claims. We consider now options of the type

H = f (S T ) with f (s) = C s z Π(dz) , (4.27)
where Π is a (finite) complex measure in the sense of Rudin [START_REF] Rudin | Real and complex analysis[END_REF], Section 6.1. An integral representation of some basic European calls can be found later.

We need now the new following assumption.

Assumption 8. Let I 0 = suppΠ ∩ R. We denote We need now to obtain upper bounds on z for the quantity (4.26). We will first need the following lemma.

I := [inf I 0 ∧ 2 inf I 0 , 2 sup I 0 ∨ sup I 0 + 1] . 1. ∀z ∈ suppΠ, z, z + 1 ∈ D.
Lemma 4.20. There are positive constants c 1 , c 2 , c 3 such that dρ s a.e.

1.

sup z∈suppΠ dRe(η(z, s)) dρ s ≤ c 1 .

For any

z ∈ suppΠ |γ(z, s)| 2 ≤ dρ s (z) dρ s ≤ c 2 -c 3 dRe(η(z, s)) dρ s 3.
- To prove (4.32) it is enough to show that

sup
Re(η(z, T ) -η(z, t)) ≤ c 0 (ρ T -ρ t ) + 1 2 Re(κ T (z) -κ t (z)), ∀t ∈ [0, T ]. (4.34)
Again Assumption 8 point 2. implies that

T t γ(z, s)dκ s (1) ≤ c 12 T t |γ(z, s)|dρ s , (4.35)
where c 12 = dκs(1) dρs ∞ . Using (4.30), and Assumption 8 it follows

|γ(z, s)| 2 ≤ dρ s (z) dρ s = dκ(2Re(z)) dρ s - 2dRe(κ s (z)) dρ s , (4.36)
≤ c 11 - 2dRe(κ s (z)) dρ s .
This implies that

c 2 12 |γ(z, s)| 2 ≤ c 2 13 + 1 4 dRe(κ s (z)) dρ s 2 ,
where c 13 > 0 is chosen such that c 2 13 ≥ 4c

(η(z, T ) -η(z, t)) ≤ T t dρ s c 0 + 1 2 dRe(κ s (z)) dρ s , (4.37)
where c 0 = c11 2 + c 13 . This concludes the proof of point 1. In order to prove point 2. we first observe that (4.32) implies

- dRe(κ s (z)) dρ s ≤ 2 c 0 - dRe(η(z, s)) dρ s (4.38)
dρ s a.e. (4.36) implies

|γ(z, s)| 2 ≤ c 21 -4 dRe(η(z, s)) dρ s , (4.39)
where c 21 = c 11 + 4c 0 . Point 2. is now established with c 2 = c 21 and c 3 = 4.

We continue with the proof of point 3. We decompose

Re(η(z, t)) = A + (z, t) -A -(z, t) ,
where

A + (z, t) = t 0 dRe(η(z, s)) dρ s + dρ s , and A -(z, t) = t 0 dRe(η(z, s)) dρ s - dρ s .
A + (z, .) and A -(z, .) are increasing non negative functions. Moreover point 1. implies

A + (z, t) ≤ c 1 ρ t .
At this points for z ∈ I + iR

- T 0 Re(η(z, dt))e R T t 2Re(η(z,ds)) = T 0 A -(z, dt) -A + (z, dt) e 2 R T t Re(η(z,ds)) ≤ T 0 A -(z, dt)e 2(A + (z,T )-A + (z,t)) e -2(A -(z,T )-A -(z,t)) ≤ e 2c1ρT T 0 e -2(A -(z,T )-A -(z,t)) A -(z, dt) = e 2c1ρT 2 1 -e -2A -(z,T ) ≤ e 2c1ρT 2 ,
which concludes the proof of point 3 of Lemma 4.20.

Let γ = sup z∈I (2Re(z)), by Lemma 4.5, it follows sup z∈I,s≤T

E S 2Re(z) s < ∞ . ( 4
• H t = H(z) t Π(dz) , • ξ t = ξ(z) t Π(dz) , • L t = L(z) t Π(dz) ,
where for z ∈ supp(Π), H(z), ξ(z) and L(z) are the same as those introduced in Proposition 4.17 and we convene that they vanish if z / ∈ supp(Π).

Previous decomposition is real-valued if f is real-valued.

Remark 4.24. Taking Π = δ z0 (dz), z 0 ∈ C, Assumption 8 is equivalent to the assumptions of Proposition 4.17.

Proof. a) f (S T ) ∈ L 2 since by Jensen,

E C Π(dz)S z T 2 ≤ C |Π|(dz)E|S 2Rez T ||Π|(C) ≤ sup x∈I0 E(S 2x T )|Π|(C) 2 ,
where |Π| denotes the total variation of the finite measure Π. Previous quantity is bounded because of Lemma 4.17.

We go on with the FS decomposition. We would like to prove first that H and L are well defined square-integrable processes and E( T

0 |ξ s | 2 d M s ) < ∞. We denote K = supp(Π)
. By Jensen's inequality, we have

E C L(z) t Π(dz) 2 ] ≤ E C |Π|(dz)|L t (z)| 2 t |Π(C)| = C |Π|(dz)E[|L t (z)| 2 t ]|Π| Similar calculations allow to show that E[ξ 2 t ] ≤ |Π|(C) C |Π|dz)E[|ξ t (z)| 2 ] and E[L 2 t ] ≤ |Π(C)| C |Π|(dz)E[|L t (z)| 2 ] .
We will show now that

• (A1): sup t≤T,z∈I+iR E[|H t (z)| 2 ] < ∞ ; • (A2): C |Π|(dz)E[|L t (z)| 2 t ] < ∞; • (A3): E T 0 dρ t S 2 t C |ξ t (z)| 2 |Π|(dz) < ∞ . (A1): Since H(z) t = e R T t η(z,ds) S z t , we have |H(z) t | 2 = H(z) t H(z) t = e R T t 2Re(η(z,ds)) S 2Re(z) t , so E[|H(z) t | 2 ] = e R T t 2Re(η(z,ds)) E[S 2Re(z) t ] ≤ e R T t 2Re(η(z,ds)) sup t≤T E[S γ t ] ,
with γ = sup z∈I 2Re(z). Inequality (4.40) and Lemma 4.20 imply (A1). Therefore (H t ) is a well-defined square-integrable process.

(A2): E[|L t (z)| 2 ] ≤ E[|L T (z)| 2 ] = E[ L(z), L(z) T ] ,
where the first inequality is due to the fact that

|L t (z)| 2 is a submartingale. E L(z), L(z) T = E T 0 e R T s 2Re(η(z,du) S 2Re(z) s- dρ s (z) -|γ(z, s)| 2 dρ s .
By Fubini, Lemma 4.5 and (4.24), we have

E L(z), L(z) T = T 0 e R T s 2Re(η(z,du) E[S 2Re(z) s- ] dρ s (z) dρ s -|γ(z, s)| 2 dρ s ≤ T 0 e R T s 2Re(η(z,du) dρ s (z) dρ s -|γ(z, s)| 2 E[S 2Re(z) s- ]dρ s ≤ c 4 T 0 e R T s 2Re(η(z,du) dρ s (z) dρ s dρ s ,
where

c 4 = sup s≤T E[S 2Re(z) s ] .
According to Lemma 4.20 point 2, previous expression is bounded by c 4 I(z), where

I(z) := T 0 dρ t exp T t 2Re(η(z, ds)) c 2 -c 3 dRe(η(z, t)) dρ t (4.41) = c 2 I 1 (z) + c 3 I 2 (z) ,
where

I 1 (z) = T 0 dρ t exp T t 2Re(η(z, ds)) and I 2 (z) = T 0 exp T t 2Re(η(z, ds)) dRe(η(z, ds)) .
Using Lemma 4.20, we obtain

sup z∈I+iR |I 1 (z)| ≤ ρ T exp (2c 1 ρ T ) and sup z∈I+iR |I 2 (z)| < ∞ , (4.42)
and so

sup z∈I+iR E L(z), L(z) T < ∞ . (4.43)
This concludes (A2).

We verify now the validity of (A3). This requires to control

E T 0 ρ dt S 2 t C |Π|(dz)|ξ(z) t | 2 ≤ E   T 0 ρ dt S 2 t   C |Π|(dz) γ(z, t) exp T t Re(η(z, ds)) S z-1 t 2     .
Using Jensen inequality, this is smaller or equal than

|Π(C)| C |Π|(dz) T 0 ρ dt E S 2Re(z) t |γ(z, t)| 2 exp 2 T t
Re(η(z, ds)) .

Lemma 4.20 gives the upper bound

|Π|(C) sup t≤T,γ∈I E S 2Re(z) t C |Π|(dz)I(z) ,
where I(z) was defined in (4.42). Since Π is finite and because of (4.43), (A3) is now established.

We show now that (L t ) is an (F t )-martingale. Let 0 ≤ s ≤ t ≤ T , B ∈ F s . By Proposition 4.17, since (L(z) t ) is a martingale, we obtain

E[(L t -L s )1 B ] = E[ C (L(z) t -L(z) s )Π(dz)1 B ] .
By Fubini's theorem we conclude that

E[(L t -L s )1 B ] = C E[(L(z) t -L(z) s )1 B ]Π(dz) ,
and

E[(L(z) t -L(z) s )1 B ] = 0. So E[(L t -L s )1 B ] = 0 .
Hence, L is a square-integrable martingale.

Similarly, it can be shown that E[(M t L t -M s L s )1 B ] = 0 and so ML is a square-integrable martingale as well. Hence L is orthogonal to M. By Fubini's theorem for stochastic integrals, cf. [START_REF] Protter | Stochastic integration and differential equations[END_REF], Theorem IV.46, we have

t 0 ξ(z) s dS s Π(dz) = t 0 ξ(z) s Π(dz)dS s = t 0 ξ s dS s .
Consequently, (H 0 , ξ, L) provide a (possibly complexe) FS decomposition of H.

b) It remains to prove that the decomposition is real-valued. Let (H 0 , ξ, L) and (H 0 , ξ, L) be two FS decomposition of H. Consequently, since H and (S t ) are real-valued, we have

0 = H -H = (H 0 -H 0 ) + T 0 (ξ s -ξ s )dS s + (L T -L T ) , which implies that 0 = Im(H 0 ) + T 0 Im(ξ s )dS s + Im(L T )
. By Theorem 2.22, the uniqueness of the real-valued Fllmer-Schweizer decomposition yields that the processes (H t ),(ξ t ) and (L t ) are real-valued.

Representation of some typical contingent claims

We used some integral representations of payoffs of the form (4.27). We refer to [START_REF] Cramer | On the representation of a function by certain Fourier integrals[END_REF], [START_REF] Raible | Lvy processes in finance: theory, numerics and empirical facts[END_REF] and more recently [START_REF] Eberlein | Analysis of Fourier transform valuation formula and applications[END_REF], for some characterizations of classes of functions which admit this kind of representation. In order to apply the results of this paper, we need explicit formulae for the complex measure Π in some example of contingent claims.

Call

The first example is the European Call option H = (S T -K) + . We have two representations of the form (4.27) which result from the following lemma. 1. For arbitrary R > 1, s > 0, we have

(s -K) + = 1 2πi R+i∞ R-i∞ s z K 1-z z(z -1) dz . (4.44)
2. For arbitrary 0 < R < 1, s > 0, we have

(s -K) + -s = 1 2πi R+i∞ R-i∞ s z K 1-z z(z -1) dz . (4.45) 4.6.2 Put
Lemma 4.26. Let K > 0, the European Put option H = (K -S T ) + gives for an arbitrary R < 0, s > 0

(K -s) + = 1 2πi R+i∞ R-i∞ s z K 1-z z(z -1) dz . (4.46)
5 The solution to the minimization problem 5.1 Mean-Variance Hedging FS decomposition will help to provide the solution to the global minimization problem. Next theorem deals with the case where the underlying process is a PII.

Theorem 5.1. Let X = (X t ) t∈[0,T ] be a process with independent increments with log-characteristic function

Ψ t . Let H = f (X T )
where f is of the form (3.20). We suppose that the PII, X, satisfies Assumptions 2, 3, 4 and 5. Then, the variance-optimal capital V 0 and the variance-optimal hedging strategy ϕ, solution of the minimization problem (2.1), are given by

V 0 = H 0 , (5.1)
and the implicit expression

ϕ t = ξ t + λ t S t- (H t--V 0 - t 0 ϕ s dS s ) , (5.2)
where the processes (H t ),(ξ t ) and (λ t ) are defined by

H t = R H(u) t µ(du) , ξ t = R i d(Ψ ′ t (u) -Ψ ′ t (0)) dΨ ′′ t (0) H(u) t µ(du) and λ t = i dΨ ′ t (0) dΨ ′′ t (0) , (5.3)
and H(u) t = e η(u,T )-η(u,t)+ΨT (u)-Ψt(u) e iuXt-with η(u, t)

= i t 0 dΨ ′ t (0) dΨ ′′ t (0) d Ψ ′ s (u) -Ψ ′ s (0) .
(5.4)

The optimal initial capital is unique. The optimal hedging strategy ϕ t (ω) is unique up to some (P (dω) ⊗ dt)null set.

Proof. Since K is deterministic, the optimality follows from Theorem 3.34, Theorem 2.25 and Corollary 2.26. Uniqueness follows from Theorem 2.24.

Next theorem deals with the case where the payoff to hedge is given as a bilateral Laplace transform of the exponential of a PII. It is an extension of Theorem 3.3 of [START_REF] Hubalek | Variance-optimal hedging for processes with stationary independent increments[END_REF] to PII with no stationary increments.

Theorem 5.2. Let X = (X t ) t∈[0,T ] be a process with independent increments with cumulant generating function κ. Let H = f (e XT ) where f is of the form (4.27). We assume the validity of Assumptions 6, 7, 8.

The variance-optimal capital V 0 and the variance-optimal hedging strategy ϕ, solution of the minimization problem (2.1), are given by

V 0 = H 0 (5.5)
and the implicit expression

ϕ t = ξ t + λ t S t- (H t--V 0 - t 0 ϕ s dS s ) , (5.6)
where the processes (H t ), (ξ t ) and (λ t ) are defined by γ(z, t) := dρ t (z, 1) dρ t with ρ t (z, y) = κ t (z + y)κ t (z)κ t (y) , (

η(z, dt) := κ dt (z)γ(z, t)κ dt (1) , (5.8)

λ t := d(κ t (1)) dρ t , (5.9)
H t := C e R T t η(z,ds) S z t Π(dz) , (5.10)
ξ t := C γ(z, t)e R T t η(z,ds) S z-1 t-Π(dz) .
(5.11)

The optimal initial capital is unique. The optimal hedging strategy ϕ t (ω) is unique up to some (P (dω) ⊗ dt)null set.

Remark 5.3. The mean variance tradeoff process can be expressed as follows, see (4.12):

K t = t 0 dκ u (1) dρ u κ du (1) .
Proof of Theorem 5.2. Since K is deterministic, the optimality follows from Theorem 4.23, Theorem 2.25 and Corollary 2.26. Uniqueness follows from Theorem 2.24.

The quadratic error

Again, ρ dt denotes the measure κ dt (2) -2κ dt [START_REF] Ansel | Lois de martingale, densits et dcomposition de Fllmer-Schweizer[END_REF]. Let V, ϕ and H appearing in Theorem 5.2. The quantity

E[(V 0 + G T (ϕ) -H) 2
] will be called the variance of the hedging error. Remark 5.5. We have

α(y, z, t) = (η(z, T ) -η(z, t)) -(η(y, T ) -η(y, t)) -(K T -K t ) ,
where K is the MVT process.

Proof. The quadratic error can be calculated using Corollary 2.29 and Corollary 2.26. It gives

E T 0 exp {-(K T -K s )} d L s , (5.12)
where L is the remainder martingale in the FS decomposition of H. We proceed now to the evaluation of L .

Using is a well-defined, continuous, predictable, with bounded variation complex-valued process. We recall that L t = L(z) t Π(dz) so

L 2 t = L(y) t L(z) t Π(dy)Π(dz) .
An application of Fubini's theorem yields that We come back to (5.12). Recalling Remark 5. which equals J 0 (y, z).

L 2 t - L(y), L(z) t Π(dy)Π(dz) ,
Another application of Fubini's theorem gives Corollaries 2.29 and 2.26 imply that the left-hand side of the previous equation provides the variance of the hedging error.

The exponential Lvy case

In this section, we specify rapidly the results concerning FS decomposition and the minimization problem when (X t ) is a Lvy process (Λ t ). Using the fact that (Λ t ) is a process with independent stationary increments it is not difficult to show that

κ t (z) = tκ Λ (z) , (5.14)
where κ Λ (z) = κ 1 (z), κ Λ : D → C. Since for every z ∈ D, t → κ t (z) has bounded variation then X = Λ is a semimartingale and Proposition 3.16 implies that (t, z) → κ t (z) is continuous.

We make the following hypothesis.

Assumption 9.

1. 2 ∈ D ;

2. κ Λ (2) -2κ Λ (1) = 0 . Remark 5.6. 1. ρ dt = κ Λ (2) -2κ Λ (1) dt ; 2. dκ t dρ t (z) = 1 κ Λ (2) -2κ Λ (1)
κ Λ (z) for any t ∈ [0, T ], z ∈ D ; so D = D.

3. Assumptions 6, and 7 are verified.

Again we denote the process S as

S t = s 0 exp(X t ) = s 0 exp(Λ t ) .
It remains to verify Assumption 8 which of course depends on the contingent claim.

Example 5.7.

1. H = (S T -K) + . We choose the second representation for the call. So, for 0 < R < 1,

I 0 = supp(Π) ∩ R = {R, 1} .
In this case Assumption 8.1 becomes I = [R, R + 1] ⊂ D. This is always satisfied since D ⊃ [0, 2] and it is convex. Assumption 8.2 is always verified because I is compact and κ Λ is continuous.

2. H = (K -S T ) + . We recall that R < 0 and so

I 0 = supp(Π) ∩ R = {R}.
In this case, Assumption 8.1, gives again I = [2R, 1] ⊂ D. Since [0, 2] is always included in D, we need to suppose here that 2R (which is a negative value) belongs to D. This is not a restriction provided that D contains some negative values since we have the degree of freedom for choosing R.

In this subsection, we reobtain results obtain in [START_REF] Hubalek | Variance-optimal hedging for processes with stationary independent increments[END_REF]. From Proposition 4.13, we obtain the following.

Remark 5.12. If X is a Poisson process with parameter λ > 0 then the quadratic error is zero. In fact, the quantities

κ Λ (z) = λ(exp(z) -1)) ρ t (y, z) = λt(exp(y) -1)(exp(z) -1) γ(z, t) = exp(z) -1 e -1
imply that β(y, z, t) = 0 for every y, z ∈ C, t ∈ [0, T ]. Therefore J 0 (y, z, t) ≡ 0. In particular all the options of type (4.27) are perfectly hedgeable.

Exponential of a Wiener integral driven by a Lvy process

Let Λ be a Lvy process. The cumulant function of Λ t equals κ Λ t (z) = tκ Λ 1 (z) for κ Λ 1 = κ Λ : D Λ → C. We formulate the following hypothesis: Assumption 10.

1. There is r > 0 such that r ∈ D Λ .

2. Λ has no deterministic increments.

κ Λ (2l s ) -2κ Λ (l s ) > 0 .
In fact, Λ has no deterministic increments. This shows Assumption 6.

2. For z ∈ D ε,r , by Remark 5.14 1. we have

dκ t (z) dρ t = κ Λ (zl t ) κ Λ (2l t ) -2κ Λ (l t ) ≤ sup x∈[ε,r] |κ Λ (xz)| inf x∈[ε,r/2] (κ Λ (2x) -2κ Λ (x))
.

Previuous supremum and infimum exist since x → κ Λ (zx) is continuous and it attains a maximum and a minimum on a compact interval. So, D ε,r ⊂ D and Assumption 7 is verified because of Remark 5.15 We only have to require that D Λ contains some negative values, which is the case for the three examples introduced at Section 3.4. Selecting R in a proper way, (5.17)is fulfilled.

We provide now the solution to the minimization problem under Assumption 10. . By Theorem 5.2, we have

λ(s) = κ Λ (l s ) κ Λ (2l s ) -2κ Λ (ls) , γ(z, s) = κ Λ ((z + 1)l s ) -κ Λ (zl s ) -κ Λ (l s ) κ Λ (2l s ) -2κ Λ (l s ) , η(z, s) = κ Λ (zl s ) - κ Λ (l s ) κ Λ (2l s ) -2κ Λ (l s ) κ Λ ((z + 1)l s ) -κ Λ (zl s ) -κ Λ (l s ) , hence η(z, s) = κ Λ (zl s ) -λ(s) κ Λ ((z + 1)l s ) -κ Λ (zl s ) -κ Λ (l s ) ,
We obtain finally the optimal hedging

ϕ t = ξ t + λ t S t- (H t--V 0 - t 0 ϕ s dS s ) ,
where the processes (H t ),(ξ t ) are defined by

H t = C e R T t η(z,ds) S z t Π(dz) , ξ t = C γ(z, t)e R T t η(z,ds) S z-1 t-Π(dz) .
given future period, typically a month, a quarter or a year. Hence, the corresponding term contracts are in fact swaps (i.e. forward contracts with delivery over a period) but are improperly named forward. However, the strong assumption that there are tradable forward contracts for all future time points T d ≥ 0 is usual and will be assumed here.

Because of non-storability of electricity, no dynamic hedging strategy can be performed on the spot market. Hedging instruments for electricity derivatives are then futures or forward contracts. The value of a forward contract offering the fixed price F T d 0 at time 0 for delivery of 1MWh at time T d is by definition of the forward price, S 0,T d 0 = 0. Indeed, there is no cost to enter at time 0 the forward contract with the current market forward price F T d 0 . Then, the value of the same forward contract S 0,T d at time t ∈ [0, T d ] is deduced by an argument of Absence of (static) Arbitrage as S 0,T d t = e -r(T -t) (F T d t -F T d 0 ). Hence, the dynamic of the hedging instrument (S 0,T d t ) 0≤t≤T d is directly related (for deterministic interest rates) to the dynamic of forward prices (F T d t ) 0≤t≤T d . Consequently, in the sequel we will focus on the dynamic of forward prices.

Electricity price models for pricing and hedging application

Observing market data, one can notice two main stylised features of electricity spot and forward prices:

• Volatility term structure of forward prices: the volatility increases when the time to maturity decreases;

• Non-Gaussianity of log-returns: log-returns can be considered as Gaussian for long-term contracts but they are clearly leptokurtic for short-term contratcs with huge spikes on the Spot market.

Hence, a challenge is to be able to describe with a single model, both the spikes on the short term and the volatility term structure of the forward curve. One reasonable attempt to do so is to consider the exponential Lvy factor model, proposed by Benth and Benth [START_REF] Benth | The normal inverse Gaussian distribution and spot price modelling in energy markets[END_REF], or [START_REF] Collet | Some non-Gaussian models for electricity spot prices[END_REF]. The forward price given at time t for delivery at time T d ≥ t, denoted F T d t is then modeled by a p-factors model, such that

F T d t = F T d 0 exp(m T d t + p k=1 X k,T d t
) , for all t ∈ [0, T d ] , where (6.18)

• (m T d t ) 0≤t≤T d is a real deterministic trend;

• For any k = 1, • • • p, (X k,T d t ) 0≤t≤T d is such that X k,T d t = t 0 σ k e -λ k (T d -s) dΛ k s , where Λ = (Λ 1 , • • • , Λ p ) is a Lvy process on R d , with E[Λ k 1 ] = 0 and V ar[Λ k 1 ] = 1;
• σ k > 0 , λ k ≥ 0 , are called respectively the volatilities and the mean-reverting rates.

Hence, forward prices are given as exponentials of PII with non-stationary increments. Then, the spot model is derived by setting S T d = F T d T d and reduces to the exponential of a sum of possibly non-Gaussian Ornstein-Uhlenbeck processes. In practice, we consider the case of a one or a two factors model (p = 1 or 2), where the first factor X 1 is a non-Gaussian PII and the second factor X 2 is a Brownian motion with σ 1 ≫ σ 2 . Notice that this kind of model was originally developed and studied in details for interest rates in [START_REF] Raible | Lvy processes in finance: theory, numerics and empirical facts[END_REF], as an extension of the Heath-Jarrow-Morton model where the Brownian motion has been replaced by a general Lvy process. Recent contributions in the subject are [START_REF] Filipovic | Existence of Lvy term structure models[END_REF][START_REF] Rdiger | Stability results for term structure models driven by Lvy processes[END_REF]. Of course, this modeling procedure (6.18), implies incompleteness of the market. Hence, if we aim at pricing and hedging a European call on a forward with maturity T ≤ T d , it won't be possible, in general, to hedge perfectly the payoff (F T d T -K) + with a hedging portfolio of forward contracts. Then, a natural approach could consist in looking for the variance optimal price and hedging portfolio. In this framework, the results of Section 4 generalizing the results of Hubalek & al in [START_REF] Hubalek | Variance-optimal hedging for processes with stationary independent increments[END_REF] to the case of non stationary PII can be useful. Similarly, some arithmetic models proposed in [START_REF] Benth | A non-Gaussian Ornstein-Uhlenbeck process for electricity spot price modeling and derivatives pricing[END_REF] for electricity prices, consists of replacing the right-hand side of (6.18) by its logarithm. Hence, with this kind of models the results of Section 3.6 can also be useful.

The non Gaussian two factors model

To simplify let us forget the upperscript T d denoting the delivery period (since we will consider a fixed delivery period). We suppose that the forward price F follows the two factors model

F t = F 0 exp(m t + X 1 t + X 2 t ) , for all t ∈ [0, T d ] , where (6.19)
• m is a real deterministic trend starting at 0. It is supposed to be absolutely continuous with respect to Lebesgue;

• X 1 t = t 0 σ s e -λ(T d -u) dΛ u ,
where Λ is a Lvy process on R with Λ following a Normal Inverse Gaussian (NIG) distribution or a Variance Gamma (VG) distribution. Moreover, we will assume that E[Λ 1 ] = 0 and V ar[Λ 1 ] = 1;

• X 2 = σ l W where W is a standard Brownian motion on R;

• Λ and W are independent.

• σ s and σ l standing respectively for the short-term volatilty and long-term volatility.

Verification of the assumptions

The result below helps to extend Theorem 5.2 to the case where X is a finite sum of independent PII semimartingales, each one verifying Assumptions 6, 7 and 8 for a given payoff H = f (s 0 e XT ). Lemma 6.1. Let X 1 , X 2 be two independent PII semimartingales with cumulant generating functions κ i and related domains D i , D i , i = 1, 2 characterized in Remark 3.8 and (4.13). Let f : C → C of the form (4.27). For X = X 1 + X 2 with related domains D, D and cumulant generating function κ, we have the following.

1. D = D 1 ∩ D 2 . 2. D 1 ∩ D 2 ⊂ D.
3. If X 1 , X 2 verify Assumptions 6, 7 and 8, then X has the same property.

Proof. Since X 1 , X 2 are independent and taking into account Remark 3.8 we obtain 1. and

κ t (z) = κ 1 t (z) + κ 2 (z), ∀z ∈ D.
We denote by ρ i , i = 1, 2, the reference variance measures defined in Remark 4.7. Clearly ρ = ρ 1 + ρ 2 and

dρ i ≪ dρ with dρ i dρ ∞ ≤ 1. If z ∈ D 1 ∩ D 2 , we can write T 0 dκ t (z) dρ t 2 dρ t ≤ 2 T 0 dκ 1 t (z) dρ 1 t dρ 1 t dρ t 2 dρ t + 2 T 0 dκ 2 t (z) dρ 2 t dρ 2 t dρ t 2 dρ t = 2 T 0 dκ 1 t (z) dρ 1 t 2 dρ 1 t dρ t dρ 1 t + 2 T 0 dκ 2 t (z) dρ 2 t 2 dρ 2 t dρ t dρ 2 t ≤ 2 T 0 dκ 1 t (z) dρ 1 t 2 dρ 1 t + T 0 dκ 2 t (z) dρ 2 t 2 dρ 2 t . λ t = m t + σ 2 l 2 + κ Λ ( z) σ 2 l + κ Λ (2 z) -2κ Λ ( z) , H t = C e R T t η(z,ds) S z t Π(dz), ξ t = C γ(z, t)e R T t η(z,ds) S z-1 t-Π(dz) .
The optimal initial capital is unique. The optimal hedging strategy ϕ t (ω) is unique up to some (P (dω) ⊗ dt)null set.

Remark 6.5. Previous formulae are practically exploitable numerically. The last condition to be checked is 7 Simulations

2σ s ∈ D Λ . ( 6

Exponential Lvy

We consider the problem of pricing a European call, with payoff (S T -K) + , where the underlying process S is given as the exponential of a NIG Lvy process i.e. for all t ∈ [0, T ],

S t = e Xt , where X is a Lvy process with X 1 ∼ N IG(α, β, δ, µ) . produced by the VO and the BS strategies as functions of the strike, for three different sets of parameters C = 0.08 , C = 1 , C = 2. We consider N = 12 trading dates, which corresponds to operational practices on electricity markets, for an option expirying in three months. One can observe that BS results are very similar to VO results for C ≥ 1 which corresponds to almost Gaussian returns. However, for small values of C, for C = 0.08, corresponding to highly non Gaussian returns, BS approach under-estimates out-of-themoney options and over-estimates at-the-money options. For instance, on Figure 3, one can observe that for K = 99 Euros the Black-Scholes Initial Capital (IC BS ) represents 122% of the variance optimal Initial Capital (IC V O ), while for K = 150 it represents only 57% of the variance optimal price. Moreover, the hedging strategy differs sensibly for C = 0.08, while it is quite similar to BS's ratio for C ≥ 1. (on the right graph) of the hedging error have been estimated by Monte Carlo method on 5000 runs. Note that we could have used the formula stated in Theorem 5.4 to compute the variance of the error, but this would have give us the limiting error which does not take into account the additional error due to the finite number of trading dates.

In terms of standard deviation, the VO strategy seems to outperform sensibly the BS strategy, for small values of C. For instance, one can observe on Figure 5, for C = 0.08 that the VO strategy allows to reduce 10% of the standard deviation of the error. As expected, one can observe that the VO error converges to the BS error when C increases. This is due to the convergence of NIG log-returns to Gaussian log-returns when C increases (recall that the simulated log-returns are almost symmetric). One can distinguish two sources of incompleteness, the rebalancing error due to the dicrete rebalancing strategy and the intrinsic error due to the model incompleteness. On Figure 4, the hedging error (both for BS and VO) decreases with the number of trading dates and seems to converge to a limiting error corresponding to the intrinsic error. For C = 1 and for a small number of trading dates N ≤ 5, the rebalancing error represents the most part of the hedging error, then it seems to vanish over N = 30 trading dates, where the intrinsic error is predominant. For small values of C ≤ 0.2, even for small numbers of trading dates, the intrinsic error seems to be predominant. For C ≤ 0.2 and N ≥ 12 trading dates, it seems useless to increase the number of trading dates. Moreover, one can observe that for a small number of trading dates N ≤ 12 and for large values of C ≥ 1, BS seems to outperform the VO strategy, in terms of standard deviation. This can be interpreted as a consequence of the central limit theorem. Indeed, when the time between two trading dates increases the corresponding increments of the Lvy process converge to a Gaussian variable. Hence, the model error comitted by the BS approach decreases when the number of trading dates decreases.

In term of bias, the over-estimation of at-the-money options (observed for C = 0.08, on Figures 2,3) seems to induce a positive bias for the BS error (see Figure 4), whereas the Bias of the VO error is negligeable (as expected from the theory). However, one can observe on Figure 5, that the difference between VO and BS bias error is smaller than the difference between the Initial Capitals, therefore one can conclude that, in our simulations, the BS hedging strategy induces more losses in average than the VO strategy. However, to be more relevant in our analysis, we have compared on Figure 7, the performances of the BS hedging portfolio with the VO hedging portfolio starting with the same Initial Capital as the BS hedging portfolio. One can observe on Figure 5 that this approach allows to reduce the standard deviation of the VO hedging error (increasing the bias and of course the global quadratic error w.r.t. the VO strategy with optimal Initial capital).

It is interesting to notice that, in terms of skewness and kurtosis, the VO strategy seems to outperform sensibly the BS strategy for small values of C. Figure 6 shows that for C = 0.08, the skewness of the BS hedging error is strongly negative (3 times greater than the VO error using the same Initial Capital) and the kurtosis is high (14 times greater than the VO error). Hence, in our simulations, BS strategy seems to imply more extreme losses than the VO strategy.

In conclusion, the VO approach provides initial capital and hedging strategies which are not significantly different from the BS approach except for log-returns with high excess kurtosis (with small values of parameter α in the NIG case). Similarly, we can observe (though the figures are not reported here) the same behaviour w.r.t. to the asymmetry of the distribution: the VO approach allows to outperform significantly the BS approach for strongly asymmetric log-returns (with high (absolute) values of parameter β in the NIG case). On the other hand, in more standard cases, the VO strategy seems to be comparable with the BS strategy in terms of quadratic error and to have the significant and unexpected advantage to limit extreme losses (skewness and kurtosis) compared to the BS strategy.

Exponential PII

We consider the problem of hedging and pricing a European call on an electricity forward, with a maturity T = 0.25 of three month. The maturity is equal to the delivery date of the forward contract T = T d .

As stated in Section 6, the natural hedging instrument is the corresponding forward contract with value S 0 t = e -r(T -t) (F T t -F T 0 ) for all t ∈ [0, T ], where F T = F is supposed to follow the NIG one factor model: Those parameters correspond to a standard and centered NIG distribution with a skewness of -0.019. The estimated annual short-term volatility and mean-reverting rate are σ s = 57.47% and λ = 3. The other sets of parameters considered in simulations are obtained by multiplying parameter α by a coefficient C, (β, δ, µ being such that the first three moments are unchanged). Table 1 shows how the excess kurtosis is modified with C. Figure 9 shows the Bias and Standard deviation of the hedging error as a function of the number of trading dates estimated by Monte Calo method on 5000 runs. The results are comparable to those obtained in the case of the Lvy process, on Figure 9. However, one can notice that the BS strategy does no more outperform the VO strategy for small numbers of trading dates as observed in the Lvy case. This is due to the fact that X t is no more a sum of i.i.d. variables.

F t = e Xt

ACKNOWLEDGEMENTS:

The first named author was partially founded by Banca Intesa San Paolo. The second named author was supported by FiME, Laboratoire de Finance des Marchs de l'Energie (Dauphine, CREST, EDF R&D) www.fime-lab.org.

All the authors are grateful to F. Hubalek for useful advices to improve the numerical computations of Laplace transforms performed in simulations.

Theorem 2 . 23 .

 223 Under Assumption 1, the subspaces G T (Θ) and {L 2 (F 0 ) + G T (Θ)} are closed subspaces of L 2 .

• 0 γ

 0 s dΛ s . Let (t, z) → tΨ Λ (z), (resp. (t, z) → tκ Λ (z)) denoting the log-characteristic function (resp. the cumulant generating function) of Λ. Lemma 3.24. Let γ : [0, T ] → R be a Borel bounded function.

Assumption 5 .

 5 The quantity sup u∈suppµ,t∈[0,T ]

. 33 ) 3 . 35 .

 33335 Example We start with the toy model X t = σW t + mt, σ, m ∈ R. We haveΨ(u) = -u 2 2 σ 2 + imu so Ψ ′ (u) = -uσ 2 +im and Im(Ψ ′ (u)) = m. Condition (3.32) is always verified and Condition (3.33) is verified if R u 2 dµ(u) < ∞ . (3.34) (3.34) is verified for instance in the example of the beginning of subsection 3.7 since c -∞ u 2 e u du < ∞ for c > 0. Remark 3.36. In the examples introduced in Remark 3.21, we can show that u → Ψ ′ (u) is bounded and so (3.32) and (3.33) are always verified for the following reasons.

Remark 4 . 12 .

 412 Lemma 4.5 implies that E [| M (y), M (z) |] < ∞ and so M (z) is a square integrable martingale for any z ∈ D such that 2Re(z), Re(z) + 1 ∈ D.

  ) and (4.18) constitute the FS decomposition of H(z), taking into account Remark 2.19 we need to show that (a) H(z) 0 is F 0 -measurable, (b) L(z), M = 0, (c) ξ(z) ∈ Θ, (d) L(z) is a square integrable martingale. Point (a) is obvious. Partial integration and point 1 of Proposition 4.11 yield

  . It remains to prove that L(z) is a squareintegrable martingale for all z ∈ D and that Re(ξ(z)) and Im(ξ(z)) are in Θ. (4.20) says that L(z) t = z,du) dM s (z) -t 0 ξ(z) s dM s . By Proposition 4.11 and Lemma 4.18, it follows

  Taking the expectation in (4.25), using again point 2., (4.14), (4.15) and Lemma 4.5, we obtainE L(z), L(z) T < ∞ . (4.26)Therefore, L is a square-integrable martingale. Similarly, (4.25) yields that Re(ξ(z)) ∈ Θ and Im(ξ(z)) ∈ Θ.

Lemma 4 . 25 .

 425 Let K > 0, the European Call option H = (S T -K) + has two representations of the form (4.27):

Theorem 5 . 4 .s y+z 0 T

 540 Under the assumptions of Theorem 5.2, the variance of the hedging error equalsJ 0 := C C J 0 (y, z)Π(dy)Π(dz) ,where J 0 (y, z) := 0 β(y, z, dt)e κt(y+z)+α(y,z,t) dt : y, z ∈ suppΠ 0 : otherwise. and α(y, z, t) := η(z, T )η(z, t) -(η(y, T )η(y, t))z, t) := ρ t (y, z) -t 0 γ(z, s)ρ ds (y, 1) .

  (4.22), (4.23), Remark 2.4 and the bilinearity of the covariation giveRe ( L(y), L(z) ) = 1 2 L(y), L(z) + L(y), L(z) ) + L(z), L(y) + L(z) -L(y), L(y) -L(z), L(z) ,andL(y) + L(z), L(y) + L(z) ≤ L(y) + L(z), L(y) + L(z) + L(y) -L(z), L(y) -L(z) , =2L(y), L(y) + 2 L(z), L(z) .

( 4 .

 4 43) in the proof of Theorem 4.23, and considerations above allow to prove that sup y,z∈I+iR |Re ( L(y), L(z) )| < ∞. Similarly we can bound Im( L(y), L(z) t ), writing Im ( L(y), L(z) ) = 1 2 L(y) -L(z), L(y) -L(z) -L(y), L(y) -L(z), L(z) , so that we obtain Im ( L(y), L(z) ) ≤ L(y), L(y) + L(z), L(z) and sup y,z∈I+iR |Im ( L(y), L(z) )| < ∞ . Therefore L(y), L(z) t Π(dy)Π(dz)

  is a martingale. This implies L, L t = L(y), L(z) t Π(dy)Π(dz) , by definition of oblique bracket. It remains to evaluate L(y), L(z) for y, z ∈ supp(Π).We know by Proposition 4.11 that for all y, z, y+ z ∈ D, M (y), M (z) t = t 0 S y+z u-ρ du (y, z) .Using the same terminology of Proposition 4.17, (4.21) says L(z), M t = 0 and (4.20) implyL(y), L(z) t = z,du)+η(y,du)) d M (y), M (z) s -y,du) d M, M (y) s , z,du)+η(y,du)) d M (y), M (z) s -z,du)+η(y,du)) S z-1 s-d M, M(y) s , z,du)+η(y,du)) S y+z s-ρ ds (y, z) -z,du)+η(y,du)) S z-1 s-S y+1 s-ρ ds (y, 1) , (z,du)+η(y,du)) S y+z s-[ρ ds (y, z)γ(z, s)ρ ds (y, 1)] . z,du)+η(y,du)) S y+z s-β(y, z, ds) .

3 we have T 0 e 0 e 0 e

 000 -(KT -Kt) d L(y), L(z) t = T 0 e -(KT -Kt)+ R T t (η(z,du)+η(y,du)) S y+z t-β(y, z, dt) ,=T α(y,z,t) S y+z t-β(y, z, dt) .Since E[S y+z t-] = s y+z 0 e κt(y+z) , an application of Fubini's theorem yieldsE T -(KT -Kt) d L(y), L(z) t = ET 0 e α(y,z,t) S y+z t-β(y, z, dt) , y,z,t)+κt(y+z) β(y, z, dt) .

  KT -Kt) d L(y), L(z) t Π(dy)Π(dz) , KT -Kt) d L(y), L(z) t Π(dy)Π(dz) , = C CJ 0 (y, z)Π(dy)Π(dz) .

2 .

 2 H = (K -S T ) + . Again, here R < 0, I = [2R, R + 1].

  The time unit is the year and the interest rate is zero in all our simulations. The initial value of the underlying is S 0 = 100 Euros. The maturity of the option is T = 0.25 i.e. three months from now. Five different sets of parameters for the NIG distribution have been considered, going from the case of almost Gaussian returns corresponding to standard equities, to the case of highly non Gaussian returns. The standard set of parameters is estimated on the Month-ahead base forward prices of the French Power market in 2007: α = 38.46 , β = -3.85 , δ = 6.40 , µ = 0.64 . (7.25) Those parameters imply a zero mean, a standard deviation of 41%, a skewness (measuring the asymmetry) of -0.02 and an excess kurtosis (measuring the fatness of the tails) of 0.01. The other sets of parameters are obtained by multiplying parameter α by a coefficient C, (β, δ, µ) being such that the first three moments are unchanged. Note that when C grows to infinity the tails of the NIG distribution get closer to the tails of the Gaussian distribution. For instance,

Figure 1 :

 1 Figure 1: Excess kurtosis of X1 for different values of α, (β, δ, µ) insuring the same three first moments.

Figure 2

 2 Figure2shows the Initial Capital (on the left graph) and the initial hedge ratio (on the right graph)

Figure 2 : 40 ICFigure 3 :

 2403 Figure 2: Initial Capital (on the left) and hedge ratio (on the right) w.r.t. the strike, for C = 0.08 , C = 1 , C = 2.

7. 1 . 2 Figure 4

 124 Figure 4 considers the hedging error (the difference between the terminal value of the hedging portfolio and the payoff) as a function of the number of trading dates, for a strike K = 99 Euros (at the money) and for five different sets of parameters C described on Figure 1. The bias (on the left graph) and standard deviation

Figure 4 :

 4 Figure 4: Hedging Error w.r.t. the number of trading dates for different values of C and for K = 99 Euros (Bias, on the left and standard deviation, on the right).

  CoefficientC = 0.08 C = 0.14 C = 0.

Figure 6 :

 6 Figure 6: Empirical moments of the hedging error for C = 0.08, N = 12 and K = 99 Euros (averaged values for different number of trading dates).

, where X t = t 0 σFigure 7 :

 07 Figure 7: Hedging Error of BS strategy v.s. the VO strategy with the same initial capital as BS w.r.t. the number of trading dates for different values of C and for K = 99 Euros (Bias, on the left and standard deviation, on the right).

Figure 8 :

 8 Figure 8: Excess kurtosis of Λ1 for different values of α (β, δ, µ) insuring the same three first moments

Figure 9 :

 9 Figure 9: Hedging Error w.r.t. the number of trading dates for C = 0.08 and C = 1, for K = 99 Euros (Bias, on the left and standard deviation, on the right).

Figure 10 :

 10 Figure 10: Empirical moments of the hedging error for C = 0.08, N = 10 and K = 99 Euros.

  .6)Proof. See the proof of Corollary 9 of[START_REF] Schweizer | Approximating random variables by stochastic integrals[END_REF] with Remark 2.16.

	Corollary 2.29. If M, M is continuous

  In fact, again it is enough to prove t → ϕ ′′ t (0) is continuous. This follows if we prove that (M t ) is continuous in L 2 . This is true because M is continuous in probability and for any N > 0, t ∈ [0, T ], Chebyshev implies that

0) is a square integrable martingale and t → Ψ ′ t (0) is bounded, then the family (E(X 2 t )) is bounded and so (X t ) is uniformly integrable. So t → ϕ ′ t (u) is continuous and the result follows by Assumption 2 4. t → Ψ ′′ t (0) is continuous.

  Proof of Proposition 4.13. By Proposition 4.9, dκ t (1) is absolutely continuous with respect to dρ t . Setting α u as in(4.11), relation (4.12) follows from Proposition 4.11, expressing K t =

			t 0 α 2 u d M u .
	0	T	v 2 t S 2 t-dρ

Lemma 4.15. The space Θ is constituted by all predictable processes v such that

E[ t ] < ∞ .

Proof. According to Proposition 2.14, the fact that K is bounded and S satisfies (SC), then v ∈ Θ holds if and only if v is predictable and E[

  Remark 4.19. 1. Point 2. of Assumption 8 implies sup z∈I+iR κ dt (Re(z)) T < ∞ . 2. Under Assumption 8, H = f (S T ) is square integrable. In particular it admits an FS decomposition. 3. Because of (4.6) in Proposition 4.9, the Radon-Nykodim derivative at Point 2. of Assumption 8, always exists.

	2. I ⊂ D and sup x∈I∪{1}	d(κt(x)) dρt	∞	< ∞.

  .40) Theorem 4.23. Let Π be a finite complex-valued Borel measure on C. Suppose Assumptions 6, 7, 8. Any complex-valued contingent claim H = f (S T ), where f is of the form (4.27), and H ∈ L 2 , admits a unique FS decomposition H = H 0 +

	T 0 ξ t dS t + L T with the following properties.
	1. H ∈ L 2 and

  Remark 5.13. According toLemma 4.3 for every γ > 0, such that γ ∈ D, κ Λ (2γ) -2κ Λ (γ) > 0 .(5.16)We consider the PII process X t = t 0 l s dΛ s where l : [0, T ] → [ε, r/2] is a (deterministic continuous) function and ε, r > 0 such that 2ε ≤ r.Remark 5.14.1. Lemma 3.24 says that D contains D ε,r := x ∈ R | εx, rx 2 ∈ D Λ + iR , and κ t (z) = (2l s ) -2κ Λ (l s ) ds .

	t 0 κ Λ (zl s )ds .
	2. ρ t =	t 0 κ Λ (2l s ) -2κ Λ (l s ) ds ;
	3. 2 ∈ D ; X is a PII semimartingale since t → κ t (2) has bounded variation, see Lemma 3.14.
	4. 1 ∈ D ε,r since 0, r ∈ D Λ .
	Proposition 5.15. Assumptions 6 and 7 are verified. Moreover D ε,r ⊂ D.
	Proof. increasing. Now 1. Using Lemma 4.3, Assumption 6 is verified if we show that t → ρ t (1) = κ t (2) -2κ t (1) is strictly
	κ t (2) -2κ t (1) = κ Λ Inequality (5.16) and Lemma 4.3 imply that ∀s ∈ [0, T ] t 0

4 .

 4 Remark 5.16.1. Point 1. of Assumption 8 is also verified if we show that I ⊂ D ε,r ; in fact D ε,r ⊂ D and I 0 ∪ (I 0 + 1) ⊂ I. T -K) + . Similarly to the case where X is a Lvy process, we take the second representation of the European Call. In this case I = [R, R + 1] and (5.17) is verified.

	2. From previous proof it follows that
		dκ t (z) dρ t	=	κ Λ (zl t ) κ Λ (2l t ) -2κ Λ (l t )	.
	dρt 3. Since I is compact and t → dκt(z) all cases provided that I ⊂ D ε,r .	is continuous, point 2. of Assumption 8 would be verified again for
	It remains to verify Assumption 8 for the same class of options as in previous subsections. The only point
	to establish will be to show	I ⊂ {x|εx,	rx 2	∈ D Λ }.	(5.17)
	Example 5.17.	1. H = (S

  .24) Λ 1 is a Variance Gamma random variable then (6.24) is verified. if for instance σ s ≤

	In our classical examples, this is always verified.
	1. Λ 1 is a Normal Inverse Gaussian random variable. If σ s ≤ α-β 2 2. -β+ then (6.24) is verified.	√ β 2 +2α 2	.

Table 1

 1 shows how the excess kurtosis (which is zero for a Gaussian distribution) is modified with the five values of C chosen in our simulations.We have compared on simulations the Variance Optimal strategy (VO) using the real NIG incomplete market model with the real values of parameters to the Black-Scholes strategy (BS) assuming Gaussian returns with the real values of mean and variance. Of course, the VO strategy is by definition theoritically optimal in continuous time, w.r.t. the quadratic norm. However, both strategies are implemented in discrete time, hence the performances observed in our simulations are spoiled w.r.t. the theoritical continuous

	rebalancing framework.
	57

  Variance optimal hedging error vs Black-Scholes hedging error for different values of C and for K = 99 Euros (averaged values for different numbers of trading dates).

				2	C = 1	C = 2
	Std V O /Std BS	91.19%	95.88%	97.63% 107.52% 109.39%
	Bias BS -Bias V O IC BS -IC V O	1.20 1.55	0.57 0.7	0.32 0.39	0.022 0.01	0.019 0
	Figure 5: Moments	Mean	Standard deviation Skewness Kurtosis
	VO BS VO with IC V O = IC BS	-0.049 1.27 1.39		6.59 7.25 6.47	-3.50 -7.65 -2.37	31.51 152.09 10.70

Corollary 5.8. Under Assumption 9, the process (S t ) can be written as

where

The mean-variance tradeoff process equals

t .

(5.15)

From Theorem 4.23 and Theorem 5.2, we obtain the following result.

Theorem 5.9. We suppose the validity of Assumption 9. We consider an option H of the type (4.27). The following properties hold true.

The FS decomposition is given by

;

According to the notations of Lemma 4.17, we have

2. The solution of the minimization problem is given by a pair (V 0 , ϕ) where

.

Remark 5.10. Lemma 2.14 implies that Θ is the linear space of predictable processes v such that

Remark 5.11. We come back to the examples introduced in Remark 3.21. In all the three cases, Assumption 9 is verified if 2 ∈ D. This is happens in the following situations:

A toy example

Let (W t ) be a standard Brownian motion, we consider X t = W ψ(t) , where ψ : R + → R + is a strictly increasing function, including the pathological case where ψ ′ t = 0 a.e. We have

and the MVT process verifies

All the conditions to apply Theorem 5.2 are satisfied so the function γ(z, t) is equal to the Radon-Nykodim

Hence we can compute the variance-optimal hedging strategy ϕ and the variance-optimal initial capital V 0 in this case

and

Remark 5.18. Calculating β(y, z, t) of the quadratic error section, we find β ≡ 0. Therefore here also the quadratic error is zero. This confirms the fact that the market is complete, at least for the considered class of options.

6 Application to Electricity

Hedging electricity derivatives with forward contacts

Electricity markets are composed by the Spot market setting prices for each delivery hour of the next day and the forward or futures market setting prices for more distant delivery periods. For simplicity, we will assume that interest rates are deterministic and zero so that futures prices are equivalent to forward prices.

Forward prices given by the market correspond to a fixed price of one MWh of electricity for delivery in a

This concludes the proof of D 1 ∩ D 2 ⊂ D and therefore of the of Point 2. Finally Point 3. follows then by inspection.

With the two factors model, the forward price F is then given as the exponential of a PII, X, such that for all t ∈ [0, T d ],

For this model, we formulate the following assumption.

Assumption 11.

2. If σ l = 0, we require Λ not to have deterministic increments.

3. We define ε = σ s e -λT d , r = 2σ s . f : C → C is of the type (4.27) fulfilling (5.17).

Proposition 6.2.

1. The cumulant generating function of X defined by (6.20), κ : [0,

In particular for fixed z ∈ D ε,r , t → κ t (z) is absolutely continuous with respect to Lebesgue measure.

2. Assumptions 6, 7 and 8 are verified.

Proof. We set X2 = m + X 2 . We observe that

We recall that Λ and W are independent so that X2 and X 1 are independent. X 1 is a process of the type studied at Section 5.4. According to Proposition 5.15, Remark 5.16 and (5.17) it follows that Assumptions 6, 7 and 8 are verified for X 1 .

Both statements 1. and 2. are now a consequence of Lemma 6.1.

Remark 6.3. For examples of f fulfilling (5.17), we refer to Example 5.17.

The solution to the mean-variance problem is provided by Theorem 5.2.

Theorem 6.4. The variance-optimal capital V 0 and the variance-optimal hedging strategy ϕ, solution of the minimization problem (2.1), are given by V 0 = H 0 (6.22) and the implicit expression

where the processes (H t ),(ξ t ) and (λ t ) are defined as follows: