Sebastien Darses
email: darses@cmi.univ-mrs.fr

Emmanuel Denis
email: emmanuel.denis@ceremade.dauphine.fr

LIMIT THEOREM FOR A MODIFIED LELAND HEDGING STRATEGY UNDER CONSTANT TRANSACTION COSTS RATE

Keywords:

, we identify the rate of convergence and the associated limit distribution. This rate turns out to be improved using the modified strategy and non periodic revision dates.

Introduction

The present paper is concerned with the study of asymptotic hedging in the presence of transaction costs. The asymptotic replication of a given payoff is performed via a modified Leland's strategy recently introduced in [START_REF] Denis | Modified Leland's strategy for constant transaction costs rate[END_REF].

Let us briefly recall the history and the main known results about Leland's strategy. In 1985 Leland suggested an approach to price contingent claims under proportional transaction costs. His main idea was to use the classical Black-Scholes formula with a suitably adjusted volatility for a periodically revised portfolio whose terminal value approximates the payoff. The intuition behind this practical method is to compensate for transaction cost by increasing the volatility in the following way:

(1.1)

σ 2 t = σ 2 + σ √ nk n 8/π f ′ (t),
where n is the number of the portfolio revision dates and k n = k 0 n -α , α ∈ [0, 1 2 ] is the transaction costs coefficient generally depending of n; f is an increasing and smooth function whose inverse g := f -1 defines the revision dates

t n i := g( i n ), 1 ≤ i ≤ n.
The principal results on convergence for models with transaction costs can be described as follows. First consider the case of approximate hedging of the European call option using the strategy with periodic portfolio revisions (i.e g(t) = t). We know the following results with T = 1:

(a) For α = 1 2 , Lott gave the first rigorous result on the convergence of the approximating portfolio value V n 1 to the payoff V 1 = (S 1 -K) + . The sequence V n 1 -V 1 tends to zero in probability [START_REF] Lott | Ein verfahren zur replikation von optionen unter transaktionkosten in stetiger zeit[END_REF], and a stronger result holds: n E (V n 1 -V 1 ) 2 converges to a constant A 1 > 0 [START_REF] Gamys | Mean square error for the Leland-Lott hedging strategy[END_REF];

(b) For α ∈ (0, 1 2 ), the sequence V n 1 -V 1 tends to zero in probability (see [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF]), and it is shown in [START_REF] Denis | Marchés avec coûts de transaction: approximation de Leland et arbitrage[END_REF] that n pα E (V n 1 -V 1 ) 2 → 0 as n → ∞, with p α < α. (c) For α = 0, the terminal values of portfolios do not converge to the European call as shown by Kabanov and Safarian [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF]. Namely, there is a negative σ{S 1 }-measurable random variable ξ such that V n 1 -V 1 → ξ in probability. Pergamenshchikov [START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF] then analyzed the rate of convergence and proved a limit theorem: the sequence n 1 4 (V n 1 -V 1 -ξ) converges in law to a mixture of Gaussian distributions [START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF]. He noticed that one can increase the modified volatility to obtain the asymptotic replication. To do so, he utilizes the explicit form of the systematic hedging error for the European call option. For related results see also [START_REF] Granditz | Leland's approach to option pricing: The evolution of discontinuity[END_REF] and [START_REF] Sekine | Hedging errors of Leland's strategies with time-inhomogeneous rebalancing[END_REF].

For models including uniform and non-uniform revision intervals one needs to impose certain conditions on the scale transform g. Generalizations of some of the above results to this more technical case as well as extensions to contingent claims of the form h(S 1 ) can be found in [START_REF] Sekine | Hedging errors of Leland's strategies with time-inhomogeneous rebalancing[END_REF][START_REF] Gamys | Mean square error for the Leland-Lott hedging strategy[END_REF][START_REF] Denis | Marchés avec coûts de transaction: approximation de Leland et arbitrage[END_REF]. In particular, n 1/2+α E (V n 1 -V 1 ) 2 converges to a constant in the case α > 0. Moreover, for α = 1 2 , the distributions of the process Y n t := n 1 2 (V n t -V t ) * in the Skorohod space D[0, 1] converges weakly to the distribution of a two-dimensional Markov diffusion process component (see [START_REF] Denis | Mean square error for the Leland-Lott hedging strategy: convex payoffs[END_REF]). Notice that the asymptotic replication still does not hold for α = 0 in this more general setting. For more details we refer to [START_REF] Denis | Marchés avec coûts de transaction: approximation de Leland et arbitrage[END_REF][START_REF] Denis | Approximate hedging of contingent claims under transaction costs[END_REF][START_REF] Denis | Mean square error for the Leland-Lott hedging strategy: convex payoffs[END_REF] and references therein.

We solve the case α = 0 for a large class of payoff and with specific non uniform revision dates by means of the modified strategy introduced in [START_REF] Denis | Modified Leland's strategy for constant transaction costs rate[END_REF]. This one makes the portfolio's terminal value converge to the contingent claim as n tends to infinity, that is the approximation error vanishes. The analysis we performed here suggests that it might be difficult to obtain a better convergence rate regarding uniform revision dates. In the framework of the non uniform grid we use, concentrating the revision dates near the maturity T = 1 accelerates the convergence rate. We leave the issue concerning periodic revision dates as an open problem.

The asymptotic behavior of the hedging error is a practical important issue. Since traders obviously prefer gains than losses, measuring the L 2 -norm of hedging errors is strongly criticized. Of course, the limiting distribution of the hedging error is much more informative. Our present work also aims at tackling this issue: we prove that

n 1 4 +p (V n 1 -h(S 1 )) d ----→ n→∞ Z,
where the law of Z is explicitly identified and p > 0 depends on the choosen grid.

The paper is organized as follows. In Section 2, we introduce the basic notations, models and assumptions of our study; In particular we recall the modified Leland's strategy defined in [START_REF] Denis | Modified Leland's strategy for constant transaction costs rate[END_REF]. In Section 3, we state our main result: a limit theorem for the renormalized asymptotic hedging error. In Section 4, we establish two lemmas concerning, on one hand, random variables constructed from the geometric Brownian motion, and on the other hand, some change of variables for the revision dates. These auxiliary results will be used repeatedly throughout the paper. In Section 5, we prove the main result. An appendix finally recalls all the known technical results we need for the various proofs.

Notations and Models

2.1. Black-Scholes model and hedging strategy. We are given a filtered probability space (Ω, F, (F t ) [0,1] , P) on which a standard one-dimensional (F t )-adapted Brownian motion W is defined. As usual, we denote by L 2 (Ω) the space of square integrable F 1 -measurable random variables endowed with its norm X 2 := √ E X 2 . We consider the classical Black-Scholes model composed of two assets without transaction costs, i.e. k 0 = 0 and σ = σ. The first one is riskless (bond) with the interest rate r = 0 and the second asset is S = (S t ), t ∈ [0, 1], a geometric Brownian motion that is

S t = S 0 e σWt-1 2 σ 2 t . (2.2)
It satisfies the SDE

dS t = σS t dW t ,
with positive constants S 0 , σ. It means that the risky asset is seen under the martingale measure.

The well-known Black and Scholes problem without transaction costs is to hedge a payoff h(S 1 ), h being a continuous function of polynomial growth. The pricing function solves the terminal valued Cauchy problem

(2.3)      C t (t, x) + σ 2 t 2 x 2 C xx (t, x) = 0, t ∈ [0, 1], x > 0, C(1, x) = h(x).
Its solution can be written as

C(t, x) = ∞ -∞ h xe ρty-ρ 2 t 2 ϕ(y)dy (2.4)
where ρ 2 t = (1 -t)σ 2 and ϕ is the standard Gaussian density. Without transaction costs (σ = σ) the self-financed portfolio process reads

V t = C(0, S 0 ) + t 0 C x (u, S u )dS u . (2.5)
In the Itô formula for C(t, S t ) the integral over dt vanishes and, therefore, V t = C(t, S t ) for all t ∈ [0, 1]. In particular, V 1 = h(S 1 ): At maturity the portfolio V replicates the terminal payoff of the option. Modeling assumptions of the above formulation include frictionless market and continuous trading for instance.

However, an investor revises the portfolio at a finite set of dates

T n = {t i ∈ [0, 1], i = 0, • • • , n}
and keeps C x (t i , S ti ) units of the stock until the next revision date t i+1 . It is well known that this discretized model converges to the Black-Scholes one in the sense that the corresponding portfolio terminal value converges to the payoff as the number of revision dates tends to infinity.

2.2.

Reminder about Leland's strategy. We are now concerned with transaction costs. We directly work in a discrete time setting. Leland suggested to replace σ in the Cauchy problem above by a suitable modified volatility σ. In the case where σ does not depend on t, the solution C satisfies C(t, x) = C(t, x, σ), (2.6) i.e. practitioners do not need to rectify their algorithms to compute the strategy. Leland obtained an explicit expression of σ by equalizing the transaction costs of the portfolio and the drift term generated by the additional term σ -σ 2 > 0 in the Ito expansion of the payoff h(S 1 ) = C(1, S 1 ). In the general case, the pricing function can be written as

C(t, x) = ∞ -∞ h xe ρ n t y-(ρ n t ) 2 /2 ϕ(y)dy (2.7)
where

(ρ n t ) 2 := 1 t σ 2 s ds, (2.8) σ 2 t := σ 2 + σ √ nk n 8/π f ′ (t), (2.9)
ϕ is the Gaussian density and g = f -1 is the revision date function.

2.3.

A possible modification of Leland's strategy. In the model with proportional transaction costs and a finite number of revision dates the current value of the portfolio process at time t is described as (2.10)

V n t := V n 0 + t 0 D n u dS u - ti<t k 0 S ti |D n i+1 -D n i |
where D n is a piecewise-constant process with D n = D n i on the interval (t i-1 , t i ], t i = t n i , i ≤ n, are the revision dates, and D n i are F ti-1 -measurable random variables.

Recall that the transaction costs coefficient is a constant k 0 > 0 (that is α = 0 in the Leland model) and the dates t i are defined by a function g, namely t i = g( i n ). Let us denote by f the inverse of g. Set for all i 0 < n

J n i0 (t) = {i ≥ i 0 , t i ≤ t, t i ∈ T n } and let us define the dates t - n (t) = t (n-1)∧max J n 0 (t) t + n (t) = t 1+(n-1)∧max J n 0 (t) .
The "enlarged volatility", depending on n, is given by the formula (2.9). We modify the usual Leland strategy (see [START_REF] Denis | Modified Leland's strategy for constant transaction costs rate[END_REF]) by considering the process D n with

D n i := C x (t i-1 , S ti-1 ) - i-1 j=1 tj tj-1 C xt (u, S tj-1 )du. (2.11)
Moreover, let us define

K n t := i∈J n 1 (t) ∆K n t n i
where ∆K n t n 0 := 0 and for i ≥ 1,

∆K n t n i := - ti ti-1 C xt (u, S u )du. (2.12)
In the same way, we set

L n t := i∈J n 1 (t) ∆L n t n i
where ∆L n t n 0 := 0 and for i ≥ 1,

∆L n t n i := - ti ti-1 C xt (u, S ti-1 )du. (2.13)
2.4. Assumptions and notational conventions. Throughout the paper, we adopt the following rules:

(i) we will often omit the indexes n and the variable t (especially in the appendix) when there is no ambiguity; (ii) the constants C appearing in the various inequalities is independent of n and may change from one line to the next; (iii) we use the classical Landau notations O and o. These quantities will be always deterministic. We use the abbreviations

δ t := C x (t, S t ), (2.14)
γ t := C xx (t, S t ). (2.15)
We denote by (δ n t ) t the process equal to δ t n i on the interval [t n i , t n i+1 ) and (γ n t ) t is defined similarly. For an arbitrary process H, we set ∆ H ti = H ti -H ti-1 .

We will work under the following assumptions: (A1) The function g has the following form:

(2.16)

g(t) = 1 -(1 -t) µ , µ ∈ 1, 3 + √ 57 8 .
(A2) h is a convex and continuous function on [0, ∞) which is twice differentiable except the points K 1 < • • • < K p where h ′ and h ′′ admit right and left limits;

|h ′′ (x)| ≤ M x -β for x ≥ K p where β ≥ 3/2.
Assumption (A1) is not too restrictive. A trader can in particular choose µ sufficiently close to 1 to balance its portfolio quasi periodically. However, as we will see, it is more preferable to increase µ to obtain a better rate of convergence.

Note that f (t) = 1 -(1 -t) 1/µ , hence the derivative f ′ for µ > 1 explodes at the maturity date and so does the enlarged volatility. We define the increasing function

p := p(µ) := µ -1 4(1 + µ) . (2.17)
Under Assumption (A1), we have 0 < p < 1/16.

In the sequel, the quantity

Q(µ) = µ 1/2-2p (1 + µ) 4p 2 4p 8/π 4p+1
(2.18) will frequently appear.

0 i-1 i i+1 i+2 i+3 1 0 t(i-1) t(i) t(i+1) t(i+2) t(i+3) 1 Revision dates f(t, 1/mu) Figure 1. Revision dates with f (t) = 1 -(1 -t)

Main Result

In [START_REF] Denis | Modified Leland's strategy for constant transaction costs rate[END_REF], it is proven that V n 1 converges in probability to h(S 1 ). Our main result here provides the rate of convergence and identifies the associated limit distribution of the deviation: Theorem 3.1. Consider the portfolio V n defined by (2.10) and (2.11) under Assumptions (A1) and (A2). The following convergence then holds:

n 1 4 +p (V n 1 -h(S 1 )) d ----→ n→∞ Z, (3.19)
where the law of Z is given by the characteristic function φ

Z (s) = E e -s 2 2 η 2 with η 2 := Q(µ)(k 0 σ) 1-4p S 2 1 × ∞ 0 x 4p ∞ x J(y, S 1 )dy 2 + 1 - 2 π J(x, S 1 ) 2 dx, and
J(x, S 1 ) := 1 2x ∞ -∞ h ′ (S u e √ xy+x/2 )(-y 2 - √ xy + 1)ϕ(y)dy (3.20) J(x, S 1 ) := 1 √ x ∞ -∞ h ′ (S u e √ xy+x/2 )yϕ(y)dy. (3.21) Moreover n 1 2 +2p E (V n 1 -h(S 1 )) 2 ----→ n→∞ E η 2 .

Auxiliary results

4.1. Geometric Brownian motion and related quantities. In the sequel, we shall use the decomposition given by Ito formula

(4.22) C x (t, S t ) = C x (0, S 0 ) + M n t + A n t
where

M n t := t 0 σ u S u C xx (u, S u )dW u , A n t := t 0 C xt (u, S u ) + 1 2 σ 2 u S 2 u C xxx (u, S u ) du.
The process M n is a square integrable martingale on [0, 1] by virtue of [START_REF] Denis | Modified Leland's strategy for constant transaction costs rate[END_REF].

We set for u < v

E v u = S v S u -1,
and

[E v u ] c = E (|E v u |) -|E v u | . {E v u } 2 s := (E v u ) 2 sgn E v u .
In the sequel, we will use several times the following basic results. Lemma 4.1. For all i the following inequalities and expansions hold:

E (E v u ) 2m ≤ C m (v -u) m , u ≤ v (4.23) E E ti ti-1 2 = σ 2 ∆t i (1 + o(1)) (4.24) E E ti ti-1 2 c = 1 - 2 π σ 2 ∆t i (1 + o(1)) (4.25) E E ti ti-1 2 c sgn E v u = 1 - 2 π σ 2 (∆t i ) 3 2 (1 + o(1)) (4.26) E {E ti ti-1 } 2 s = k(∆t i ) 3/2 1 + o(n -1/4 ) . (4.27)
Proof. We refer to [START_REF] Denis | Marchés avec coûts de transaction: approximation de Leland et arbitrage[END_REF] or [START_REF] Denis | Mean square error for the Leland-Lott hedging strategy: convex payoffs[END_REF]. For the sake of completeness we recall the proof of the last one. Let us notice the equality in law

{E ti ti-1 } 2 s d = exp σ ∆t j ξ -σ 2 ∆t j /2 -1 2 1 ξ≥σ √ ∆tj /2 -1 ξ≤σ √ ∆tj /2
, where ξ is the standard Gaussian variable. Since ξ and -ξ has the same law, this yields

E {E ti ti-1 } 2 s = E e uξ-u 2 /2 -1 2 -e -uξ-u 2 /2 -1 2 1 ξ≥u/2 -E e -uξ-u/2 -1 2 1 |ξ|≤u/2 ,
where u = σ ∆t j . Moreover, we have the inequality

E e -uξ-u 2 /2 -1 2 1 |ξ|≤u/2 ≤ u 4 .
From [START_REF] Gamys | Mean square error for the Leland-Lott hedging strategy[END_REF], we recall that

E e uξ-u 2 /2 -1 2 -e -uξ-u 2 /2 -1 2 1 ξ≥u/2 = 2 √ 2π u 3 + O(u 4 ).
We then conclude.

4.2.

Basic results concerning the revision dates. The function ρ t decreases from ρ 0 to 0. The following useful bounds are obvious:

ρ 2 t ≥ (σ 2 + cn 1 2 )(1 -t) (4.28) ρ 2 t ≤ σ 2 (1 -t) + σk 0 n 1 2 8/π(1 -t) 1 2 (1 -f (t)) 1 2 . (4.29)
Moreover, it is straightforward that

ρ 2 t ≥ cn 1 2 f ′ (t)(1 -t), (4.30) provided that f ′ is no decreasing.
Note that there is a constant C independent of n such that for all i ≤ n -1,

1 -t i-1 1 -t i ≤ C. (4.31) From there we deduce ∆t i 1 -t i ≤ C. (4.32)
We shall often use the inequality

n-1 i=1 ∆t i 1 -t i ≤ C log(n)
where C is a constant independent of n.

Lemma 4.2. Fix x > 0 and t := t(n, x) such that x = ρ 2 t . Set

x i-1 = ρ 2 ti-1
and

x i = ρ 2 ti where t i-1 , t i are such that t ∈ [t i-1 , t i ). Then, x ∈ (x i , x i-1 ] with |x i-1 -x i | ≤ c n -1/2 , c is a constant. There exists a constant C > 0 such that ∆t i n 1/2+2p x i-1 -x i ≤ C (x + 1). (4.33) Moreover, for a given x, (1 -t) ≤ cn -1/2 x → 0 as n → ∞ and ∆t i n 1/2+2p x i-1 -x i ----→ n→∞ Q(µ)x 4p (σk 0 ) 4p+1 . (4.34) Proof. Let us write ∆t i n 1/2+2p x i-1 -x i = n 2p σ 2 n -1/2 + σk 0 8/π 1 ∆ti ti ti-1 f ′ (u)du ∼ n→∞ n 2p σk 0 8/π f ′ ( ti ) where ti ∈ [t i-1 , t i ]. Moreover x = ρ 2 t = σ 2 (1 -t) + σk 0 8/πn 1/2 1 t f ′ (u)du = σ 2 (1 -t) + σk 0 8/πn 1/2 2µ 1/2 1 + µ (1 -t) 1+µ 2µ
and

1 -t = x -σ 2 (1 -t) σk 0 8/πn 1/2 1 + µ 2µ 1/2 2µ 1+µ . Note that x ≥ cn 1/2 (1 -t) so that (1 -t) ≤ cn -1/2 x → 0.
In a similar way, we have

x i-1 -x i = ρ 2 ti-1 -ρ 2 ti = σ 2 ∆t i + cn 1/2 f ′ ( ťi )∆t i where ťi ∈ [t i-1 , t i ]. We deduce that x i-1 -x i = σ 2 ∆t i + cn 1/2 f ′ ( ťi )g ′ (θ i )n -1 where θ i ∈ [(i -1)/n, i/n]. Moreover, f ′ ( ťi )g ′ (θ i ) = g ′ (θ i ) g ′ (f ( ťi )) is bounded since f ( ťi ) ∈ [(i -1)/n, i/n], i ≤ n -1. Hence there is a constant c satisfying x i-1 -x i ≤ cn -1/2 . Since ρ is decreasing, x ∈ [x i , x i-1 ]. Eventually, ti ∈ [t i-1 , t i ] is such that xi = ρ 2 ti ∈ [x i , x i-1 ] and xi → x. Similarly we have 1 -ti = xi -σ 2 (1 -ti ) σk 0 8/πn 1/2 1 + µ 2µ 1/2 2µ 1+µ (4.35)
which yields

f ′ ( ti ) = µ -1/2 xi -σ 2 (1 -ti ) σk 0 8/πn 1/2 1 + µ 2µ 1/2 1-µ 1+µ (4.36)
and

∆t i n 1/2+2p x i-1 -x i ∼ n→∞ n 2p σk 0 8/π f ′ ( ti ) ∼ n→∞ n 2p σk 0 8/π µ 1/2 σk 0 8/πn 1/2 xi -σ 2 (1 -ti ) 2µ 1/2 1 + µ 1-µ 1+µ ∼ n→∞ 1 σk 0 8/π µ 1/2 σk 0 8/π xi -σ 2 (1 -ti ) 2µ 1/2 1 + µ 1-µ 1+µ .
Since xi → x and ti → 0, we deduce that

∆t i n 1/2+2p x i-1 -x i ----→ n→∞ 1 σk 0 8/π µ 1/2 σk 0 8/π x 2µ 1/2 1 + µ 1-µ 1+µ . Since 0 < (µ -1)/(1 + µ) < 1,
we also find a constant c such that

∆t i n 1/2+2p x i-1 -x i ≤ c xi -σ 2 (1 -ti ) µ-1 1+µ ≤ c (x + 1),
which concludes the proof.

We now stress an important remark regarding a slight abuse of notation repeatedly used along the paper. Remark 4.3. Throughout the sequel, we shall often use the change of variable x = ρ 2 t with dx = -σ 2 t dt. For ease of notation, we will use the abuse of notation t instead of t(n, x) := (ρ 2 ) -1 (x) when applying this change of variable in an integral.

Similarly, a direct computation yields the following lemma.

:= v(n, y) such that y = ρ 2 v . There exists a constant C > 0 such that (1 -v)n 1/2+2p y ≤ C y.
Moreover, for a given y,

(1 -v) ≤ cn -1/2 y → 0 as n → ∞ and (1 -v)n 1/2+2p y ----→ n→∞ µ -1/2-2p (1 + µ) 4p+1 y 4p
2 4p σk 0 8/π 4p+1 .

Proof of the limit theorem

The proof is divided into three parts. In Step 1 we split the hedging error into a martingale part M and a residual part ε. In Step 2 we show that the residual terms tend to 0 in L 2 (Ω) with rate n 1 4 +p as n tends to infinity. We identify in Step 3 the asymptotic distribution of the martingale n 1 4 +p M n and we conclude the proof of the main result.

5.1.

Step 1: Splitting of the hedging error. Comparing Expression (2.10) with the Ito expansion of h(S 1 ) = C(1, S 1 ) yields the following decompositions. The hedging error reads

V n 1 -h(S 1 ) = M n 1 + ε n 1 (5.37)
where for all n ∈ N, M n is a martingale of terminal value

M n 1 := k 0 i≤n-1 γ ti-1 S 2 ti-1 E ti ti-1 c + 1 0 K n u dS u . (5.38)
The residual term can be splited as

ε n t = R n 0 (t) + R n 1 (t) + R n 2 (t) + R n 3 (t) (5.39) where R n 0 (t) := k 0 i∈J n 1 (t) γ ti-1 S 2 ti-1 σ 2 π nf ′ (t i-1 )∆t i -E E ti ti-1 (5.40) R n 1 (t) := t 0 (δ n u -δ u )dS u , (5.41) R n 2 (t) := k 0 i∈J n 1 (t) |∆ δ n ti + ∆ K n ti | -|∆ δ n ti + ∆ L n ti | S ti , (5.42) R n 3 (t) := t 0 (L n u -K n u )dS u . (5.43) 5.2.
Step 2: The mean square residue tends to 0 with rate n 1 2 +2p . The most technical part of this paper is the following. The deviation of the approximating portfolio from the payoff has been written in an integral form by virtue of the Ito formula. The "real world" portfolio may be interpreted as a discretetime approximation of the theoretical portfolio C(t, S t ) yielding the residual terms above. Consequently, the following analysis is mainly based on Taylor approximations involving the successive derivatives of C and so heavily utilizes estimates of the appendix. Standard tools from stochastic calculus are also frequently used.

Theorem 5.1. The following convergence holds:

n 1 2 +2p E (ε n t ) 2 ----→ n→∞ 0. (5.44)
To prove this theorem, we show the suitable convergence to 0 concerning the R j , 0 ≤ j ≤ 3.

Lemma 5.2.

n 1 2 +2p E (R n 0 ) 2 ----→ n→∞ 0. (5.45) Proof. We have E E ti ti-1 = 4Φ σ √ ∆t i 2 -2 = σ 2 π ∆t i + (∆t i )o(1), σ 2 π n 1 2 f ′ (t i-1 )∆t i = σ 2 π ∆t i ε i
where

ε i = n 1 2 √ ∆t i f ′ (t i-1 ) verifies |ε i -1| ≤ c∆t i
1 -t i by virtue of Lemma 6.12. Hence, there is a constant C > 0 such that:

sup t |R n 0 (t)| ≤ Ck 0 n-1 i=1 γ ti-1 S 2 ti-1 (∆t i ) 3 2 1 -t i .
From Corollary 6.5 and Inequalities (4.28-4.32), we deduce the following

n 1 4 +p E sup t |R n 0 (t)| 2 ≤ Cn 1 8 +p n-1 i=1 (∆t i ) 3 2 (1 -t i ) 5/4 (5.46) ≤ C n 1 8 +p n 1 4 log n ----→ n→∞ 0. (5.47)
A Taylor formula suggests to write the following splitting:

R n 1 = σ (R n 10 -R n 11 -R n 12 -R n 13 + 2R n 14 ) , (5.48) where R n 10 (t) := i≤n γ ti-1 S 2 ti-1 ti∧t ti-1∧t E u ti-1 S u S ti-1 dW u R n 11 (t) := n-1 i=1 ti∧t ti-1∧t C xt (t i-1 , S ti-1 )(u -t i-1 )S u dW u , R n 12 (t) := 1 2 n-1 i=1 S 3 ti-1 ti∧t ti-1∧t C xxx ( t i-1 , S ti-1 ) E u ti-1 2 S u S ti-1 dW u , R n 13 (t) := 1 2 n-1 i=1 S ti-1 ti∧t ti-1∧t C xtt ( t i-1 , S ti-1 )(u -t i-1 ) 2 S u S ti-1 dW u , R n 14 (t) := - 1 2 n-1 i=1 S 2 ti-1 ti∧t ti-1∧t C xxt ( t i-1 , S ti-1 )E u ti-1 (u -t i-1 ) S u S ti-1 dW u . Lemma 5.3. n 3 2 +2p E sup t∈[0,1] R n 10 (t) 2 ----→ n→∞ 0. (5.49)
Proof. The Doob inequality yields

n 3 2 +2p E sup t (R n 10 (t)) 2 ≤ 4n 3 2 +2p E (R n 10 (1)) 2
where the r.h.s tends to 0 as shown below. Indeed, by the independence of the increments of the Wiener process, we write:

E (R n 10 (1)) 2 = σ 2 n i=1 Λ ti-1 ti ti-1 E E u ti-1 2 S 2 u S 2 ti-1 du
where

Λ t := E C 2 xx (t, S t )S 4 t . (5.50)
It is easy to check the following asymptotic

E E u ti-1 2 S 2 u S 2 ti-1 = σ 2 (u -t i-1 ) + (u -t i-1 )O(n -1 ). Therefore E (R n 10 (1)) 2 = σ 4 2 i≤n Λ ti-1 (∆t i ) 2 (1 + O(n -1 ))
where

∆t i = g ′ (θ i )/n with θ i ∈ [(i -1)/n, i/n]. We then deduce n 3 2 +2p E (R n 10 (1)) 2 = σ 4 (1 + O(n -1 )) 2 i≤n Λ ti-1 (∆t i n) ∆t i n 1 2 +2p x i-1 -x i (x i-1 -x i )
where x i = ρ 2 ti . So, we have:

n 3 2 +2p E (R n 10 (1)) 2 = σ 4 (1 + O(n -1 )) 2 ρ 2 0 0 f n (x)dx
where

f n (x) = n i=1 Λ ti-1 (∆t i n) ∆t i n 1 2 +2p x i-1 -x i 1 (xi,xi-1] (x).
Let us remark the abuse of notations ρ 2 0 = ρ 2 0,n and t i = t(n, x i ) as previously mentioned.

First, let us show that f n satisfies the dominated convergence bound condition. If x ∈ (x i , x i-1 ] then from Corollary 6.5, we have

0 ≤ Λ ti-1 ≤ C √ x i-1 e -xi-1/4 ≤ C √ x e -x/4 .
Thus, from (4.33) we obtain

f n (x) ≤ C √ x e -x/4 (1 + x). (5.51) Regarding the pointwise convergence of f n , for a given x ∈ (x i , x i-1 ], there exists u ∈ [t i-1 , t i ) such that x = ρ 2 u ≥ cn 1 2 (1 -u). It follows that not only u → 1 but also t i , t i-1 → 1. Recall that ∆t i = g ′ (θ i )n -1 where θ i ∈ [(i -1)/n, i/n]. Thus g(θ i ) → 1 and θ i → 1 since f is continuous. Therefore ∆t i n → g ′ (1) = 0. Moreover, note that Λ ti-1 = 1 x i-1 ∞ -∞ e 2σ √ ti-1z-σ 2 ti-1 Υ i (z)ϕ(z)dz where Υ i (z) = ∞ -∞ h ′ e σ √ ti-1z-σ 2 t i-1 2 + √ xi-1y+ x i-1 2 yϕ(y)dy 2 .
Applying the Lebesgue theorem, we deduce that Λ ti-1 converges to

Λ(x) := 1 x ∞ -∞ e 2σz-σ 2 ∞ -∞ h ′ e σz-σ 2 2 + √ xy+ x 2 yϕ(y)dy 2 ϕ(z)dz.
Finally, together with (4.34), f n ----→ n→∞ 0 a.e. We then apply the Lebesgue theorem to conclude the following limit

σ 4 (1 + O(n -1 )) 2 ρ 2 0 0 f n (x)dx ----→ n→∞ 0. Lemma 5.4. n 1 2 +2p E (sup t R n 11 (t)) 2 ----→ n→∞ 0.
Proof. Using the Doob inequality, we obtain that

E (sup t R n 11 (t)) 2 ≤ 4E (R n 11 (1)) 2
. By independence of the increments of the Wiener process, we deduce that

n 1 2 +2p E (R n 11 (1)) 2 = n 1 2 +2p n-1 i=1 E C 2 xt (t i-1 , S ti-1 )S 2 ti-1 ti ti-1 (u-t i-1 ) 2 E S u S ti-1 2 du.
It follows that

n 1 2 +2p E (R n 11 (1)) 2 ≤ cn 1 2 +2p n-1 i=1 E C 2 xt (t i-1 , S ti-1 )S 2 ti-1 (∆t i ) 3 ≤ cn -1 4 +2p log n,
since Corollary 6.14 gives

E C 2 xt (t i-1 , S ti-1 )S 2 ti-1 ≤ c n 1 4 f ′ (t i-1 ) (1 -t i-1 ) 3 2
where nf ′ (t i-1 )∆t i is bounded. We then conclude.

Lemma 5.5. n 1 2 +2p E (sup t R n 12 (t)) 2 ----→ n→∞ 0.
Proof. As previously, we have the Doob inequality E (sup t R n 12 (t)) 2 ≤ 4E (R n 12 (1)) 2 and the equality

4E (R n 12 (1)) 2 = n-1 i=1 ti ti-1 E C 2 xxx ( t i-1 , S ti-1 )S 6 ti-1 1 - S t S ti-1 4 S 2 t S 2 ti-1 dt.
From (6.125), there exists a constant C such that:

E C 4 xxx ( t i-1 , S ti-1 ) ≤ C ρ 8 ti . (5.52)
Using the Cauchy-Schwarz inequality and (4.23) with m = 8, we deduce that

n 1 2 +2p E (R n 12 (1)) 2 ≤ Cn 1 2 +2p n-1 i=1 (∆t i ) 3 n(1 -t i ) 2 ≤ C n 2p log n n
which proves the desired convergence to 0.

Lemma 5.6. n

1 2 +2p E (sup t R n 13 (t)) 2 ----→ n→∞ 0.
Proof. We still consider the Doob inequality E (sup t R n 13 (t)) 2 ≤ 4E (R n 13 (1)) 2 and

4E (R n 13 (1)) 2 ≤ n-1 i=1 ti ti-1 E C 2 xtt ( t i-1 , S ti-1 )(t -t i-1 ) 4 S 2 t dt.
Moreover, using Lemma 6.19 and the Cauchy-Schwarz inequality, we deduce that

E C 2 xtt ( t i-1 , S ti-1 )S 2 t ≤ c (1 -t i ) 4 .
Then, we obtain

n 1 2 +2p E (R n 13 (1)) 2 ≤ Cn 1 2 +2p n-1 i=1 (∆t i ) 5 (1 -t i ) 4 ≤ C n -1 2 +2p log n.
The conclusion follows.

Lemma 5.7. n

1 2 +2p E (sup t R n 14 (t)) 2 ----→ n→∞ 0.
Proof. We use the Doob inequality E (sup t R n 14 (t)) 2 ≤ 4E (R n 14 (1)) 2 and the equality

4E (R n 14 (1)) 2 = n-1 i=1 ti ti-1 E S 4 ti-1 C 2 xxt ( t i-1 , S ti-1 ) 1 - S t S ti-1 2 (t -t i-1 ) 2 S 2 t S 2
ti-1 dt.

From (6.126), we deduce that

E S 4 ti-1 C 2 xxt ( t i-1 , S ti-1 ) 1 - S t S ti-1 2 S 2 t S 2 ti-1 ≤ c t -t i-1 (1 -t i ) 3 .
Then,

n 1 2 +2p E (R n 14 (1)) 2 ≤ cn 1 2 +2p n-1 i=1 (t i -t i-1 ) 4 (1 -t i ) 3 ≤ c n -1 2 +2p log n.
Thus, we can conclude.

Let us now study the residual term R n 2 . Again, a Taylor formula suggests to write the following splitting:

R n 2 = R n 20 + • • • + R n 24 , (5.53) where R n 20 (t) := σk 0 2 π n 1 2 t t - n (t) S 2 u γ u f ′ (u)du, R n 21 (t) := σk 0 n 1 2 2 π i∈J n 1 (t) ti ti-1 S 2 u γ u f ′ (u) -S 2 ti-1 γ ti-1 f ′ (t i-1 ) du, R n 22 (t) := k n i∈J n 1 (t) γ ti-1 |S ti -S ti-1 |(S ti-1 -S ti ), R n 23 (t) := k 0 i∈J n 1 (t) Θ i (S ti -S ti-1 ), R n 24 (t) := k 0 i∈J n 1 (t) Θ i S ti-1 , Θ i := γ ti-1 |S ti -S ti-1 | -| C x (t i , S ti ) -C x (t i-1 , S ti-1 ) + ∆ K n ti |. (5.54) Lemma 5.8. n 1 2 +2p E (R n 20 (1)) 2 ----→ n→∞ 0.
Proof. We have

n 1 2 +2p E (R n 20 (1)) 2 = c n 3 2 +2p E [tn-1,1] 2 S 2 u γ u S 2 v γ v f ′ (u) f ′ (v)dudv.
We use the Cauchy-Schwarz inequality, Inequalities (6.5) and (4.30). From the explicit formula of f ′ , we thus obtain

n 1 2 +2p E (R n 20 (1)) 2 ≤ c n 1+2p [tn-1,1] 2 dudv (1 -u) 5/8-3/(8µ) (1 -v) 5/8-3/(8µ) , ≤ c n 1+2p n 3/4+3/(4µ) . Since µ ∈ (1, 2], 3 4 + 3 4µ -(1 + 2p) = -3µ 2 + 5µ + 3 4µ(µ + 1)
> 0 so that we can conclude.

Lemma 5.9. n

1 2 +2p E (sup t R n 21 (t)) 2 ----→ n→∞ 0. Proof. Let us consider Ψ(t, x) := x 2 C xx (t, x) f ′ (t).
The Ito formula yields

Ψ(t, S t ) = Ψ(t i-1 , S ti-1 ) + t ti-1 ∂Ψ ∂x (u, S u )σS u dW u + t ti-1 ∂Ψ ∂t (u, S u )du + 1 2 t ti-1 ∂ 2 Ψ ∂x 2 (u, S u )σ 2 S 2 u du,
where

∂Ψ ∂t (t, x) = x 2 C xxt (t, x) f ′ (t) + C xx (t, x) f ′′ (t) 2 f ′ (t) , ∂Ψ ∂x (t, x) = 2x C xx (t, x) + x 2 C xxx (t, x) f ′ (t), ∂ 2 Ψ ∂x 2 (t, x) = 2 C xx (t, x) + 4x C xxx (t, x) + x 2 C xxxx (t, x) f ′ (t).
If we set

X t = S 2 t C xx (t, x) f ′ (t) then dX t = µ t dt + β t dW t where µ t = ∂Ψ ∂t (t, S t ) + 1 2 ∂ 2 Ψ ∂x 2 (t, S t )σ 2 S 2 t , β t = ∂Ψ ∂x (t, S t )σS t .
We write n

1 4 +p R n 21 (t) = A n t + B n t with A n t := σk 0 n 3 4 +p 2 π i∈J n 1 (t) ti ti-1 t ti-1 β u dW u dt, B n t := σk 0 n 3 4 +p 2 π i∈J n 1 (t) ti ti-1 t ti-1 µ u du dt.
From (6.120), there exists a constant C such that:

E β 2 t ≤ c E S 4 t δ 2 t + E S 6 t C 2 xxx (t, S t ) f ′ (t) ≤ Cf ′ (t) 1 4 n 3 4 (1 -t) 3 2
.

Using A1, we claim that there exists a constant c such that

|f ′′ (t)| f ′ (t) = c (1 -t) 3 2 -1/(2/µ) .
Thus, using (6.119-6.124), we obtain some constant C such that the following inequality holds:

E µ 2 t ≤ c (1 -t) 3/(4µ) n 1 4 (1 -t) 13/4 + c (1 -t) 3/(4µ) n 3/4 (1 -t) 7/4 + c n 5/4 (1 -t) 9/4+1/(4µ) . (5.55)
By means of the stochastic Fubini theorem, we obtain that

A n t = σk 0 n 3 4 +2p 2 π i∈J n 1 (t) ti ti-1 (t i -u)β u dW u .
Since the Doob inequality

E (sup t A n t ) 2 ≤ 4E (A n 1 )
2 holds, it suffices to estimate

E (A n 1 ) 2 .
From the boundedness of (t i -u)/(1 -u) and f ′ (u)(t i -u)n on u ∈

[t i-1 , t i ), we deduce the following estimates:

E (A n 1 ) 2 ≤ cn 3 2 +2p n-1 i=1 ti ti-1 (t i -u) 2 E β 2 u du, ≤ cn 3 2 +2p n-1 i=1 ti ti-1 (t i -u) 2 f ′ (u) 1 4 n 3/4 (1 -u) 3 2 du, ≤ c n 2p n 1 4 n-1 i=1 ti ti-1 (t i -u) (1 -u) 3 2 du ≤ c n 2p log n n 3/4 ----→ n→∞ 0.
Then, we conclude that

E (sup t A n t ) 2 ----→ n→∞ 0.
Secondly, we write:

B n t = cn 3/4+p i∈J n 1 (t) ti ti-1 µ u ti ti-1 1 t≥u dt du, = cn 3/4+p i∈J n 1 (t) ti ti-1 (t i -u)µ u du.
Then,

sup t |B n t | ≤ cn 3/4+p n-1 i=1 ti ti-1 (t i -u)|µ u |du.
Thus there exists a constant c such that E sup t |B n t | 2 ≤ c n 3 2 +2p Υ n where

Υ n = E 1 0 n-1 i=1 (t i -u)|µ u |1 (ti-1,ti] (u)du 2 , = E 1 0 1 0 n-1 i, j=1 (t i -u)(t j -v)|µ u ||µ v |1 (ti-1,ti] (u)1 (tj-1,tj ] (v)du dv.
Using the Cauchy-Schwarz inequality and (5.55), we can then bound Υ n :

Υ n ≤ 1 0 1 0 n-1 i, j=1 (t i -u)(t j -v) E µ 2 u 1 2 E µ 2 v 1 2 1 (ti-1,ti] (u)1 (tj-1,tj ] (v)du dv, ≤ 1 0 n-1 i=1 (t i -u) E µ 2 u 1 2 1 (ti-1,ti] (u)du 2 , ≤ c (Υ 1n + Υ 2n + Υ 3n )
where

Υ 1n ≤   i≤n-1 (∆t i ) 2 (1 -t i ) 1 n 1/8 (1 -t) 5/8-3/(8µ)   2 ≤ c log n n 1+3/(4µ) . (5.56)
In a same way, we obtain the following inequalities

Υ 2n ≤   i≤n-1 (∆t i ) 2 n 3/8 (1 -t i ) 7/8   2 ≤ C n 5 4
(5.57)

Υ 3n ≤   i≤n-1 (∆t i ) 2 n 5/8 (1 -t i ) 1+(1/8+1/(8µ))   2 ≤
Ct log n n 7/2+1/(4µ) .

(5.58) Then, from inequalities (5.56), (5.57) and (5.58) we deduce that

E sup t |B n t | 2 ≤ C n 3 2 +2p log n n 1+3/(4µ) ≤ C log n n 3/(4µ)-1 2 -2p where 3/(4µ) - 1 2 -2p = -4µ 2 + 3µ + 3 4µ(µ + 1)
.

Assumption (A1) yields -4µ 2 + 3µ + 3 > 0. Hence the result follows.

Lemma 5.10. n

1 2 +2p E (sup t R n 22 (t)) 2 ----→ n→∞ 0.
Proof. We write

-R n 22 (t) = k n i∈J n 1 (t) γ ti-1 S 2 ti-1 {E ti ti-1 } 2 s = U n (t) + V n (t)
where U n is a martingale defined as

U n (t) := k 0 i∈J n 1 (t) γ ti-1 S 2 ti-1 {E ti ti-1 } 2 s -E {E ti ti-1 } 2 s ,
and

V n (t) := k 0 i∈J n 1 (t) γ ti-1 S 2 ti-1 E {E ti ti-1 } 2 s .
Recall that from Lemma 4.1 4 ) .

E {E ti ti-1 } 2 s = k(∆t j ) 3 2 1 + o(n - 1
We deduce that for n large enough, 0

≤ E {E ti ti-1 } 2 s ≤ c(∆t i ) 3 2 . Using the Doob inequality E (sup t U n (t)) 2 ≤ 4E (U n (1)) 2 , it suffices to estimate E (U n (1)) 2 .
The independence of the increments of the Brownian motion implies the equality

E (U n (1)) 2 = k 2 0 n-1 i=1 E C 2 xx (t i-1 , S ti-1 )S 4 ti-1 E {E ti ti-1 } 2 s -E {E ti ti-1 } 2 s 2 .
Then, there exists a constant C such that

n 1 2 +2p E (U n (1)) 2 ≤ C n 2p n 1 4 ----→ n→∞ 0.
At last, for n large enough, E {E ti ti-1 } 2 s ≥ 0. Hence, 0 ≤ sup t V n (t) ≤ N n (1). In order to prove that n

1 2 +2p E V n (1) 2 ----→ n→∞ 0, we first analyze the following sum n 1 2 +2p k 2 0 n-1 i=1 E C 2 xx (t i-1 , S ti-1 )S 4 ti-1 (E {E ti ti-1 } 2 s ) 2 ≤ c n 2p n 7/4 ----→ n→∞ 0.
Using the Cauchy-Schwarz inequality, we also have

n 1 2 +2p ti<tj ≤tn-1 E γ ti-1 S 2 ti-1 γ tj-1 S 2 tj-1 E {E ti ti-1 } 2 s E {E tj tj-1 } 2 s ≤ c n 2p n ----→ n→∞ 0.
We then deduce that n

1 2 +2p E V n (1) 2 ----→ n→∞ 0 and finally n 1 2 +2p E (sup t R n 22 (t)) 2 ----→ n→∞ 0. Lemma 5.11. n 1 2 +2p E (sup t R n 23 (t)) 2 ----→ n→∞ 0. Proof. We write R n 23 (t) = R n 231 (t) + R n 232 (t) where R n 231 (t) := k 0 i∈J n 1 (t) Θ 1 i (S ti -S ti-1 ) R n 232 (t) := k 0 i∈J n 1 (t) Θ i -Θ 1 i (S ti -S ti-1 )
with

Θ 1 i := γ ti-1 |S ti -S ti-1 | -| C x (t i , S ti ) -C x (t i-1 , S ti-1 )|. We note that sup t |R n 231 (t)| is bounded by k 0 n-1 i=1 C x (t i , S ti ) -C x (t i-1 , S ti-1 ) -γ ti-1 (S ti -S ti-1 ) |S ti -S ti-1 |.
Applying Taylor's formula to the difference C x (t i , S ti )-C x (t i-1 , S ti-1 ) it is sufficient to estimate the following sums (5.59), • • • ,(5.62). The first one satisfies

n 1 4 +p k 0 n-1 i=1 C xt (t i-1 , S ti-1 )(∆t i )(S ti -S ti-1 ) 2 ≤ C n p n 1/8 → 0. (5.59)
Indeed, from Corollary 6.14, we deduce that:

E C 2 xt (t i-1 , S ti-1 )(∆t i ) 2 (S ti -S ti-1 ) 2 ≤ C (∆t i ) 3 n 1 4 f ′ (t i-1 ) 1 4
(1

-t i ) 3 2
.

The second one verifies

n 1 4 +p k 0 n-1 i=1 C xxx ( t i-1 , S_t i-1 )(S ti -S ti-1 ) 3 2 ≤ C n p log n n 1 2
→ 0. (5.60) Thirdly, from (6.126), we deduce that

E C 2 xxt ( t i-1 , S ti-1 )(S ti -S ti-1 ) 4 (∆t i ) 2 ≤ C(∆t i ) 4 (1 -t i ) 3
and it follows that

n 1 4 +p k 0 n-1 i=1 C xxt ( t i-1 , S ti-1 )(S ti -S ti-1 ) 2 ∆t i 2 ≤ C n p log n n 1 4
----→ n→∞ 0. (5.61) Finally, from Lemma 6.19, we get that

E C 2 xtt ( t i-1 , S ti-1 )(S ti -S ti-1 ) 2 (∆t i ) 4 ≤ C(∆t i ) 5 (1 -t i ) 4
and

n 1 4 +p k 0 n-1 i=1 C xtt ( t i-1 , S ti-1 )(S ti -S ti-1 )(∆t i ) 2 2 ≤ C n p log n n 1 4 ----→ n→∞ 0. (5.62)
From above, we can conclude about that n

1 2 +2p E (sup t R n 231 (t)) 2 ----→ n→∞ 0.
As for R n 232 (t), we use the inequality

Θ i -Θ 1 i ≤ |∆ K n
ti | and we deduce from Definition (2.12) the bound

|R n 232 (t)| ≤ c sup i |∆S ti | tn-1 0 | C xt (t u , S u )|du, with | C xt (t u , S u )| ≤ c √ S u (1 -u) so that |R n 232 (t)| ≤ c log(n) sup t S -1 2 t sup i |∆S ti | .
Using the Cauchy-Schwarz inequality, the boundedness of E sup t S -2

t yields E sup t (R n 232 (t)) 2 ≤ C log 2 (n) E sup i (∆S ti ) 4 .
Moreover,

E sup i (∆S ti ) 4 ≤ n -3 2 + E sup i (∆S ti ) 4 1 sup i (∆St i ) 4 ≥n -3 2 , ≤ n -3 2 + C P sup i (∆S ti ) 4 ≥ n -3 2 .
By virtue of the Bienaymé-Tchebytchev inequality

P(|X| ≥ k) ≤ k -8 E X 8 , P sup i (∆S ti ) 4 ≥ n -3 2 ≤ n 12 i E (∆S ti ) 32 ≤ C n -3 .
We deduce that

E sup i (∆S ti ) 4 ≤ C n -3 2
and finally E sup t (R n 232 (t)) 2 ≤ Cn -3/4 log 2 (n) so that we can conclude the lemma.

Lemma 5.12. We have

n 1 2 +2p E (sup t R n 24 (t)) 2 ----→ n→∞ 0.
Proof. Let us notice that sup t |R n 24 (t)| is bounded by the random variable

k 0 n-1 i=1 C x (t i , S ti ) -C x (t i-1 , S ti-1 ) + ∆ K n ti -C xx (t i-1 , S ti-1 ) S ti -S ti-1 S ti-1 .
Using the Ito formula for the increments C x (t i , S ti ) -C x (t i-1 , S ti-1 ), we obtain

sup t |R n 24 (t)| ≤ k 0 n-1 i=1 S ti-1 ti ti-1 σS u C xx (u, S u ) -C xx (t i-1 , S ti-1 ) dW u + 1 2 ti ti-1 σ 2 S 2 u C xxx (u, S u )du . (5.63) Thus n 1 4 +p sup t R n 24 (t) 2 ≤ T 1 n + T 2 n
where

T 1 n = σk 0 n 1 4 +p n-1 i=1 ti ti-1 E S 2 ti-1 S 2 u γ u -γ ti-1 2 du 1 2
and

T 2 n = k 0 n 1 4 +p σ 4 4 n-1 i=1 (∆t i ) 1 2 ti ti-1 E S 2 ti-1 S 4 u C 2 xxx (u, S u )du 1 2
.

We first prove that T 1 n ----→ n→∞ 0. Using the Taylor Formula, we get that

γ u -γ ti-1 = γ u -C xx (u, S ti-1 ) + C xx (u, S ti-1 ) -γ ti-1 = C xxx (u, S ti-1 )(S u -S ti-1 ) + 1 2 C xxxx (u, S ti-1 )(S u -S ti-1 ) 2 + C xxt ( t i-1 , S ti-1 )(u -t i-1 ).
Using the suitable estimations from the Appendix, we then obtain

E S 2 ti-1 S 2 u γ u -γ ti-1 ) 2 ≤ c∆t i n 7 8 (1 -t i ) 7 4 + c(∆t i ) 2 n 3 2 (1 -t i ) 3 f ′ (t i ) 3 2 + c(∆t i ) 2 n 3/4 (1 -t i ) 1 1 4 .
The last estimate follows from Corollary 6.124. Indeed, the proof is the same since ρ e ti-1 ≤ ρ ti-1 . We can therefore deduce that T 1 n ----→ n→∞ 0.

We then prove that T 2 n ----→ n→∞ 0. We deduce from Appendix the following inequality:

E S 2 ti-1 S 4 u C 2 xxx (u, S u ) ≤ c n 7/8 (1 -t i ) 7/4 .

It suffices to obtain the convergence

n 1 4 +p n-1 i=1 ∆t i n 7/16 (1 -t i ) 7/8 ≤ c n p n 3/16 ----→ n→∞ 0
to conclude the lemma. This last lemma completes the proof of Theorem 5.1.

Step 3: Asymptotic distribution.

From the previous subsection, it turns out that the deviation between the "real world" terminal portfolio and the payoff h(S 1 ) is essentially composed of a martingale as n → ∞. To study the asymptotic distribution of n

1 4 +p M n
1 , we consider it as terminal values of the following sequence of martingales (N n j ) j=0,••• ,n with respect to the filtration F n = (F ti ) i :

N n j := n 1 4 +p M n j = j i=1 (χ i + ϑ i ),
where

χ i := k 0 n 1/4+p γ ti-1 S 2 ti-1 E ti ti-1 c
, (5.64)

ϑ i := k 0 n 1/4+p K n ti-1 (S ti -S ti-1
). (5.65) Theorem 5.13. The following convergence holds:

N n 1 d ----→ n→∞ Z, (5.66)
where the law of Z is given by the characteristic function φ Z (s) = E e -s 2 2 η 2 with

η 2 := Q(µ)(k 0 σ) 1-4p S 2 1 × ∞ 0 x 4p ∞ x J(y, S 1 )dy 2 + 1 - 2 π J(x, S 1 ) 2 dx, and
J(x, S 1 ) := 1 2x ∞ -∞ h ′ (S u e √ xy+x/2 )(-y 2 - √ xy + 1)ϕ(y)dy (5.67) J(x, S 1 ) := 1 √ x ∞ -∞ h ′ (S u e √ xy+x/2 )yϕ(y)dy. (5.68) Moreover E (N n 1 ) 2 ----→ n→∞ E η 2 .
We achieve the proof of this theorem by means of the result page 58 in [START_REF] Hall | Martingale limit theory and its application[END_REF] recalled by Theorem 6.1 in the Appendix. (Note that the completion of the σ-field generated by the collection (F n ) n coincides with F 1 .) We thus need to prove the following lemmas.

Lemma 5.14. The sequence of martingales

(N n i ) i=0,••• ,n satisfies for all ε > 0, i E (χ i + ϑ i ) 2 1 |χi+ϑi|>ε |F ti-1 P ----→ n→∞ 0. (5.69)
Proof. We use the inequality (χ i + ϑ i ) 2 ≤ 2χ 2 i + 2ϑ 2 i and we shall prove a stronger result, i.e. a convergence in L 1 .

First let us show that

E ϑ 2 i 1 |χi+ϑi|>ε ----→ n→∞ 0. (5.70)
By virtue of the Markov inequality, we obtain

E ϑ 2 i 1 |ϑi|>ε/2 ≤ E ϑ 4 i P(|ϑ i | > ε/2) ≤ Cε -6 E ϑ 4 i E ϑ 12 i .
By independance, we have

E ϑ 4 i = k 2 0 n 1+4p E (K n ti-1 ) 4 S 4 ti-1 E (E ti ti-1 ) 4 (5.71) E ϑ 12 i = k 2 0 n 3+12p E (K n ti-1 ) 12 S 12 ti-1 E (E ti ti-1 ) 12 (5.72) Recall that K n ti-1 = - ti-1 0 C xt (u, S u )du
and by virtue of Lemma 6.8 there exists a constant C such that

|K n ti-1 | 4 ≤ C sup 0≤u≤T S -2 u tn-1 0 du 1 -u 4 ≤ C sup 0≤u≤T S -2 u log 4 (n).
We deduce that

E ϑ 4 i ≤ C log 4 (n)n 4p-1 , E ϑ 12 i ≤ C log 12 (n)n 12p-3 . Since p < 1/8, we deduce i E ϑ 2 i 1 |ϑi|>ε/2 ≤ C ε -6 n 8p-1 log 8 (n) i≤n n -1 ≤ C ε -6 n 8p-1 log 8 (n) ----→ n→∞ 0. Let us study E ϑ 2 i 1 |χi|>ε/2 . Again, E ϑ 2 i 1 |χi|>ε/2 ≤ E ϑ 4 i P(|χ i | > ε/2) ≤ Cε -2 E ϑ 4 i E χ 4 i .
Once again by independence,

E χ 4 i = k 4 0 n 1+4p E γ 4 ti-1 S 8 ti-1 E E ti ti-1 4 c .
We easily deduce from Lemma 4.1 the inequality

E E ti ti-1 4 c ≤ C(∆t i ) 2 .
Using Corollary 6.119 we obtain

E χ 4 i ≤ C n 1+4p (∆t i ) 2 n 1/4 √ 1 -t i-1 3 ≤ C n 4p-1/4 . (5.73) Since p < 1 16 < 3 32 , then i E ϑ 2 i 1 |χi|>ε/2 ≤ C ε -2 n 4p log 2 (n) i ∆t i n 3/8 (1 -t i-1 ) 3/4 . ≤ C ε -2 n 4p-3/8 log 2 (n) i ∆t i 1 -t i-1 ≤ Cε -2 n -3/8+4p log 3 (n) ----→ n→∞ 0.
From the inequality

1 |χi+ϑi|>ε ≤ 1 |χi|>ε/2 + 1 |ϑi|>ε/2 we then deduce that i E ϑ 2 i 1 |χi+ϑi| > ε ----→ n→∞ 0.
Second let us show that

E χ 2 i 1 |χi+ϑi|>ε ----→ n→∞ 0. (5.74)
In the same way, we have

E χ 2 i 1 |ϑi|>ε/2 ≤ E χ 4 i P(|ϑ i | > ε/2) (5.75) ≤ Cε -6 E χ 4 i E ϑ 12 i . (5.76) From (5.73) we have E χ 4 i ≤ C n 4p-1/4 . Thus, using p < 1 16 < 5 64 , i E χ 2 i 1 |ϑi|>ε/2 ≤ Cε -6 n 8p-5/8 log 6 (n) ----→ n→∞ 0. (5.77) Let us now study E χ 2 i 1 |χi|>ε/2 . E χ 2 i 1 |χi|>ε/2 ≤ E χ 4 i P(|χ i | > ε/2) (5.78) ≤ Cε -2 E χ 4 i . (5.79)
Using the bound (5.73), we obtain

i E χ 2 i 1 |χi|>ε/2 ≤ Cn 4p-1/4 ----→ n→∞ 0. (5.80)
We finally conclude the lemma.

Lemma 5.15. The sequence of martingales (M n i ) i=0,••• ,n satisfies the following convergence

V 2 n := i E (χ i + ϑ i ) 2 |F ti-1 d ----→ n→∞ η 2 , (5.81) where η := Q(µ)(k 0 σ) 1-4p S 2 1 × ∞ 0 x 4p ∞ x J(y, S 1 )dy 2 + 1 - 2 π J(x, S 1 ) 2 dx, with J(x, S 1 ) := 1 2x ∞ -∞ h ′ (S u e √ xy+x/2 )(-y 2 - √ xy + 1)ϕ(y)dy (5.82) J(x, S 1 ) := 1 √ x ∞ -∞ h ′ (S u e √ xy+x/2
)yϕ(y)dy. (5.83) Proof. First, let us study the term

ξ ϑ n := i E ϑ 2 i |F ti-1 . (5.84)
By independence, we obtain

E ϑ 2 i |F ti-1 = k 2 0 n 1/2+2p K n ti-1 2 S 2 ti-1 E E ti ti-1 2 .
(5.85) Hence, using Lemma 4.1 and the change of variable y = ρ 2 u and

x i = ρ 2 ti , E ϑ 2 i |F ti-1 = k 2 0 n 1/2+2p K n2 ti-1 S 2 ti-1 σ 2 ∆t i (1 + O(n -1 )) = k 2 0 σ 2 n 1/2+2p S 2 ti-1 ti-1 0 C xt (u, S u )du 2 ∆t i (1 + O(n -1 )) = k 2 0 σ 2 S 2 ti-1 ti-1 0 C xt (u, S u )du 2 n 1/2+2p ∆t i x i-1 -x i ∆x i (1 + O(n -1 )) = k 2 0 σ 2 S 2 ti-1 ρ 2 0 xi-1 C xt (u, S u ) σ -2 u dy 2 n 1/2+2p ∆t i x i-1 -x i ∆x i (1 + O(n -1 )).
We then deduce that

ξ ϑ n = (1 + O(n -1 )) ∞ 0 z ϑ n (x)dx (5.86) where z ϑ n (x) := S 2 ti-1 k 2 0 σ 2 i ρ 2 0 xi-1 C xt (u, S u ) σ -2 u dx 2 n 1/2+2p ∆t i x i-1 -x i 1 (xi,xi-1] (x). Recall that | C xt (u, S u ) σ -2 u |du ≤ c G 1 (x, S u ), x = ρ 2 u where G 1 (x, y) = 1 x e -x/8   p j=1 | log(y/K j )| √ x exp - log 2 (y/K j ) 2x + √ x + x   .
In particular,

√ xG 1 (x, y) ≤ G(x) (5.87)
where G(x) = c x -1 2 e -x/16 , c > 0 is a constant. Hence, a.s.

ρ 2 0 xi-1 C xt (u, S u ) σ -2 u dy ≤ ∞ x G(x ′ )dx ′ ≤ ∞ 0 G(x ′ )dx ′ < +∞. (5.88)
Therefore, using (4.34),

|z ϑ n (x)| ≤ C(1 + x) ∞ x G(x ′ )dx ′ 2 sup u∈[0,1] S 2 u . (5.89) But, due to Hölder's inequality, ∞ 0 (1 + x) ∞ x G(x ′ )dx ′ 2 dx < +∞. (5.90)
We can thus apply Lebesgue's theorem using Corollary 6.22 and (4.33):

(5.91)

ξ ϑ n a.s. ----→ n→∞ Q(µ)(k 0 σ) 1-4p S 2 1 ∞ 0 x 4p ∞ x J(y, S 1 )dy 2 dx.
Second, let us study the term

ξ χ n = i E χ 2 i |F ti-1 . (5.92)
By independence, we obtain

E χ 2 i |F ti-1 = k 2 0 n 1/2+2p γ 2 ti-1 S 4 ti-1 E E ti ti-1 2 c . (5.93) Then E χ 2 i |F ti-1 = k 2 0 σ 2 n 1/2+2p γ 2 ti-1 S 4 ti-1 1 - 2 π ∆t i (1 + o(1)). (5.94) We then deduce that i E χ 2 i |F ti-1 = (1 + O(n -1 )) ∞ 0 z χ n (x)dx (5.95) where z χ n (x) := S 4 ti-1 k 2 0 σ 2 i γ 2 ti-1 n 1/2+2p ∆t i x i-1 -x i 1 (xi,xi-1] (x).
Let us obtain a suitable bound for z χ n (x), integrable in x. Recall that

γ ti-1 = C xx (t i-1 , S ti-1 ) (5.96) = 1 ρ ti-1 S ti-1 ∞ -∞ h ′ (S ti-1 e ρt i-1 y+ρ 2 t i-1 /2 )yϕ(y)dy (5.97) = 1 √ x i-1 S ti-1 ∞ -∞ h ′ (S ti-1 e √ xi-1y+xi-1/2
)yϕ(y)dy. (5.98) Due to Inequality (6.117), we claim that a.s.(ω) for n large enough, there is a constant c ω which does not depend on n such that

|γ ti-1 | ≤ C sup u≤1 S -3/2 u e -x/8 1 x≥1 + e -cω x √ x + 1 1 x≤1 . (5.99) Indeed, this is obvious for x ≥ 1. Otherwise, 1 ≥ x = ρ 2 u ≥ c n 1/2 (1-u n (x)
) implies that u = u n (x) is close to 1 uniformly in x ≤ 1 as soon as n is large enough. It then suffices to choose S 1 out of the null-set {S 1 = K 1 , • • • , K p } to obtain by continuity that S u n (x) is also far enough from the points

K 1 , • • • , K p if x ≥ 1.
We conclude that for all j, log 2 (K j /S u n (x) ) ≥ c ω,j for some constants c ω,j > 0. Therefore, (5.100)

|S ti-1 | 4 |γ ti-1 | 2 ≤ C sup 0≤u≤1 S 4 u sup 0≤u≤1 S -3 u e -x/4 1 x≥1 + e -cω x √ x + 1 2 1 x≤1 .
Thus, due to (4.33)

(5.101)

|z χ n (x)| ≤ C sup 0≤u≤1 S 4 u sup 0≤u≤1 S -3 u (1 + x)e -x/4 1 x≥1 + e -cω x √ x + 1 2 1 x≤1 .
We can then apply the dominated convergence theorem using the limit (4.34). We obtain (5.102)

ξ χ n a.s. ----→ n→∞ 1 - 2 π Q(µ)(k 0 σ) 1-4p S 2 1 ∞ 0 x 4p J(x, S 1 ) 2 dx.
Finally, let us study the term i E χ i ϑ i |F ti-1 . By independence, we have

E χ i ϑ i |F ti-1 = k 2 0 n 1/2+2p γ ti-1 S 2 ti-1 K n ti-1 S ti-1 E E ti ti-1 c E ti ti-1 .
But

E E ti ti-1 c E ti ti-1 = E E ti ti-1 2 c sgn E ti ti-1 = 1 - 2 π σ 2 (∆t i ) 3 2 (1 + o(1)).
Due to (4.33), we obtain

(∆t i ) 3 2 n 1/2+2p x i-1 -x i ----→ n→∞ 0.
From the bounds (4.34), (5.88), (5.99) and by applying again Lebesgue's theorem, we then deduce the following limit

(5.103) i E χ i ϑ i |F ti-1 a.s.
----→ n→∞ 0.

Lemma 5.16. We have

E (N n 1 ) 2 ----→ n→∞ E η 2 . (5.104)
Proof. Due to the independence of the increments of the Wiener process, we have E (χ i + ϑ i )(χ j + ϑ j ) = 0 whenever i = j. We thus obtain

E (N n 1 ) 2 = i E (χ i + ϑ i ) 2 (5.105) = E i E (χ i + ϑ i ) 2 |F ti-1 (5.106) But i E (χ i + ϑ i ) 2 |F ti-1 ≤ 2 i E χ 2 i + ϑ 2 i |F ti-1 = 2(ξ χ n + ξ ϑ n ). (5.107)
Let us show that ξ n := ξ χ n + ξ ϑ n is uniformly integrable. First let us note that ξ n is bounded in L 1 (Ω). Indeed, from Corollary 6.5, Inequalities (5.89) and (4.33), we obtain for all n

E |ξ n | ≤ C ∞ 0 (1 + x) (E S 2 1 ) e -x √ x + ∞ x G(x ′ )dx ′ < ∞. (5.108)
Now, using the Cauchy-Schwarz inequality and then the Markov inequality, we have

E ξ ϑ n 1 ξn≥k ≤ C ∞ 0 (1 + x) ∞ x G(x ′ )dx ′ 2 dx E S 4 1 P(ξ n ≥ k) ≤ C sup n E |ξ n | k ----→ k→∞ 0.
Recall that

z χ n (x)1 ξ n ≥M0 := k 2 0 σ 2 i S 4 ti-1 γ 2 ti-1 1 ξ n ≥M0 n 1/2+2p ∆t i x i-1 -x i 1 (xi,xi-1] (x).
Therefore, applying successively the Cauchy-Schwarz inequality, (4.33), Corollary 6.16, and the Markov inequality, we obtain

E ξ χ n 1 ξn≥k ≤ C i E S 5 ti-1 γ 5/2 ti-1 4/5 (P(ξ n ≥ k)) 1/5 n 1/2+2p ∆t i x i-1 -x i 1 (xi,xi-1] (x) ≤ C ∞ 0 (1 + x) e -5x/32 x 15/16 4/5 dx sup n E ξ n k 1/5 ----→ k→∞ 0.
Therefore, ξ n is uniformly integrable, and so is i E (χ i + ϑ i ) 2 |F ti-1 , which moreover converges to η a.s. This yields the conclusion of the Lemma. This last lemma completes the proof of Theorem 5.13.

E (V n 1 -h(S 1 )) 2 ----→ n→∞ E η 2 = E Z 2 .
The proof of the limit theorem is then complete.

Appendix

The following limit result combines Theorem 3.2 page 58 [START_REF] Hall | Martingale limit theory and its application[END_REF] with its corollary 3.1 and its remarks. Theorem 6.1 ([7] page 58). Let {M n i , F ti , 0 ≤ i ≤ n} be a zero-mean square integrable martingale with increments ∆M n i = X n i and let η 2 be an a.s. finite r.v. Suppose that

for all ε > 0, i E (X n i ) 2 1 |χ n i |>ε |F ti-1 P ----→ n→∞ 0, (6.112) V 2 n = i E (X n i ) 2 |F ti-1 P ----→ n→∞ η 2 . (6.113) Then M n n d ----→ n→∞ Y where the r.v. Y has the characteristic function E exp -1 2 η 2 t 2 .
6.1. A0. We recall from [START_REF] Denis | Approximate hedging of contingent claims under transaction costs[END_REF] the following expressions for the successive derivatives. They are based on direct computations using the integration by parts formula under suitable assumptions on the payoff function h. Lemma 6.2. Let C(t, x) is given by (2.7). Then

C x (t, x) = ∞ -∞ h ′ (xe ρy+ρ 2 /2 )ϕ(y)dy, C xx (t, x) = 1 ρx ∞ -∞ h ′ (xe ρy+ρ 2 /2 )yϕ(y)dy, C xxx (t, x) = 1 ρ 2 x 2 ∞ -∞ h ′ xe ρy+ρ 2 /2 )P 2 (y)ϕ(y)dy, C xxxx (t, x) = 1 ρ 3 x 3 ∞ -∞
h ′ (xe ρy+ρ 2 /2 )P 3 (y)ϕ(y)dy where P 2 (y) := y 2 -ρy -1, P 3 (y) := y 3 -3ρy 2 + (2ρ 2 -3)y + 3ρ.

In particular, | C x (t, x)| ≤ ||h ′ || ∞ .
Similarly, we obtain the following expressions for the successive derivatives in t: Lemma 6.3. Let C(t, x) is given by (2.7). Then

C t (t, x) = -σ 2 t x 2ρ ∞ -∞ h ′ (xe ρy+ρ 2 /2
)yϕ(y)dy, (6.114) To study the residual terms generated by the discretization of the theoretical C(T, S T ), we use Taylor approximations. We then need to estimate some bounds of the successive derivatives of C. Lemma 6.4. There is a constant C > 0 such that

C tx (t, x) = σ 2 t 2ρ 2 ∞ -∞ h ′ (xe ρy+ρ 2 /2 )Q 2 (y)ϕ(y)dy, (6.115) C xxt (t, x) = σ 2 t 2ρ 3 x ∞ -∞ h ′ (xe ρy+ρ 2 /2 )Q 3 (
(6.117) | C xx (t, x)| ≤ C e -ρ 2 /8 ρx 3/2 p j=1 exp - 1 2 log 2 (K j /x) ρ 2 + c e -ρ 2 /8
x 3/2 . Corollary 6.5. There exists a constant C such that for t ∈ [0, 1[

E S 4 t C 2 xx (t, S t ) ≤ C ρ e -ρ 2 /4 .
Similarly, we can deduce the following bounds: Corollary 6.6. There exists a constant c such that for t ∈ [0, 1[

E S 2 t C 2 xx (t, S t ) ≤ c   p j=1 1 ρ 2 √ 2u 2 + 1 exp - v 2 j 2u 2 + 1 + e -ρ 2 /4  
where c is a constant, u = α t /ρ and

v j := log(S 0 /K j ) -α 2 t /2 ρ + ρ 2 .
Corollary 6.7. There exists a constant c such that for t ∈ [

| C xxx (t, x)| ≤ ce -ρ 2 /8 ρ 2 x 5/2 (L(x, ρ) + ρ) , | C xxxx (t, x)| ≤ ce -ρ 2 /8 x -7/2 P 3 (ρ -1 ), | C tx (t, x)| ≤ c σ 2 e -ρ 2 8 x 1/2 ρ 2 L(x, ρ) + ρ + ρ 2 , | C xxt (t, x)| ≤ c σ 2 e -ρ 2 /8 x -3/2 (ρ -1 + ρ -3 ),
where P 3 is a polynomial of the third order and

L(x, ρ) := p j=1 | log(x/K j )| ρ exp - log 2 (x/K j ) 2ρ 2 .
Lemma 6.9. There exists a constant c and a polynomial Q of third order such that

E S m t C 2 tx (t, S t ) ≤ c σ 4 t Q(ρ -1 )e -ρ 2 /4 .

A.1.

We give here some necessary calculus and inequalities for the present work. Moreover, we recall some results from [START_REF] Denis | Modified Leland's strategy for constant transaction costs rate[END_REF] and [START_REF] Denis | Approximate hedging of contingent claims under transaction costs[END_REF]. In particular, we can show the following in a similar way: Lemma 6.10. We have:

C xxt (t, x) = σ 2 t 2ρ 3 t x ∞ -∞ h ′ (xe ρty+ρ 2 t /2 )P 1 (ρ t , y)ϕ(y)dy, C xtt (t, x) = - ρ ′′ t ρ t ∞ -∞ h ′ (
| C xxt (t, x)| ≤ c e -ρ 2 t /8 x 3/2 σ 2 t ρ 3 t   p j=1 ̺ j (x) 2 + ρ 2 t /4 + 1 e -̺j (x) 2 /2 + ρ t + ρ 3 t   , | C xtt (t, x)| ≤ X 1 (t, x) + X 2 (t, x)
where

X 1 (t, x) := c e -ρ 2 t /8 √ x |ρ ′′ t | ρ t   p j=1 ̺ j (x)e -̺j (x) 2 /2 + ρ t + ρ 2 t   , X 2 (t, x) := c e -ρ 2 t /8 √ x σ 4 t ρ 4 t   p j=1 ̺ j (x) 3 + ̺ j (x) e -̺j (x) 2 /2 + 4 j=1 ρ j t   and ̺ j (x) := |log(K j /x)| /ρ t .
Lemma 6.12. Assume that Assumption A1 holds. Then there exists a constant c such that ε

i := n 1/2 √ ∆t i f ′ (t i-1 ), i ≤ n -1 satisfies the inequality |ε i -1| ≤ c∆t i /(1 -t i ) for n large enough.
Proof. We have obviously

|ε i -1| ≤ |n∆t i f ′ (t i-1 ) -1|, where ∆t i = g ′ (θ i )n -1 and θ i ∈ [(i -1)/n, i/n]. Then, d i := g(θ i ) -t i-1 ∈ [0, ∆t i ].
We deduce that:

|ε i -1| ≤ f ′ (g(θ i ) -h i ) f ′ (g(θ i )) -1 ≤ c ∆t i 1 -t i .
Indeed, we use a first order Taylor expansion to estimate the difference f ′ (g(θ i )h i ) -f ′ (g(θ i )). We conclude by using the explicit expression of f , g but also the inequality (1

-t i-1 )/(1 -t i ) ≤ c for i ≤ n -1.
The following lemma is of first importance to get estimations of expectations we need in some of our proofs. Lemma 6.13. Suppose that t ≤ u < 1, m ∈ R, q ∈ 2N and K > 0. There exists a constant c = c(m, q) such that

E S m u log q S u K exp - log 2 (S u /K) ρ 2 t ≤ cP q (ρ t )
where

P 0 (ρ t ) := ρ t , P 2 (ρ t ) := ρ 3 t + ρ 5 t , P 4 (ρ t ) := ρ 5 t + ρ 7 t + ρ 9 t , P 2q (ρ t ) := ρ 2q+1 t + ρ 2q+3 t + • • • + ρ 4q+1 t . Proof. We set p = log S0 K -σ 2 u/2 , α = σ √ u and A(q) = E S m u log q S u K exp - log 2 (S u /K) ρ 2 t .
Then,

A(q) = S m 0 √ 2π ∞ -∞ (p + αy) q exp αmy -α 2 m/2 - 1 ρ 2 t (p + αy) 2 -y 2 /2 dy, A(q) = S m 0 e A1 √ 2π ∞ -∞ (p + αy) q exp - 1 2 1 + 2α 2 ρ 2 t y 2 + α m - 2p ρ 2 t y dy
where

A 1 = - α 2 m 2 - p 2 ρ 2 t . Let y = z/A 2 with A 2 = 1 + 2α 2 /ρ 2 t . Then A(q) = S m 0 e A4 √ 2πA 2 ∞ -∞ (p + αz A 2 ) q exp - 1 2 z 2 -2(A 3 /A 2 )z + A 2 3 /A 2 2 dz where A 3 = α m -2p/ρ 2 t and A 4 = A 1 + A 2 3 /(2A 2
2 ). After the change of variable y = z -A 3 /A 2 , we obtain that is also bounded. It follows that e A4 is bounded and we can conclude easily for q = 2.

A(2) = S m 0 ρ t e A4 ρ 2 t + 2α 2 p + αρ 2 t A 3 ρ 2 t + 2α 2 2 + α 2 ρ 2 t ρ 2 t + 2α 2 . Moreover, if u ≥ t, then ρ 2 t ≥ σ 2 (1 -t) implies that ρ 2 t + 2α 2 ≥ σ 2 (1 -t) + σ 2 u ≥ σ 2 . We have A 4 = - mα 2 2 - p 2 ρ 2 t + α 2 ρ 2 t 2(ρ 2 t + 2α
In a similar way, we can conclude for any q ∈ 2N because we use in particular the property From now on, we can deduce the following results. (1 -t i ) 4 .

Proof. We have S m ti-1 ≤ S m ti-1 + S m ti , and ρ e ti-1 ≥ ρ ti . Furthermore, in virtue of Lemma 6.8, recall that we have

| C xt (t, x)| ≤ c σ 2 t e -ρ 2 t /8
x 1/2 ρ 2 t .

Then, the conclusion follows.

In the same way, we can prove the following results: Lemma 6.19. There exists a constant C such that

E C 4 xtt ( t i-1 , S ti-1 ) ≤ Ce -ρ 2 t i /4
(1 -t i ) 8 .

Proof. The arguments are similar to the previous ones but we also use the inequality: Recall the two following lemmas (see [START_REF] Denis | Modified Leland's strategy for constant transaction costs rate[END_REF]). These results ensures the convergence of the Leland scheme without any hedging error when using the modified Leland strategy. The change of variable x = ρ 2 u appears to be as essential in the following proofs and points out the significative role of the revision dates near the maturity. Lemma 6.21. We have the following equality Proof. We apply Lemma 6.21 with the change of variable x = ρ 2 u . Recall that 0 ≤ 1 -u ≤ c x n -1/2 so that u → 1 as n → ∞ for a given x ≥ 0. We can apply the Lebesgue theorem by dominating the function G 1 (x, S u ) whether x ≤ 1 or not because x ≤ 1 implies that u is sufficiently near from 1 independently of x for n ≥ n 0 . Indeed, outside of the null-set ∪ i {S 1 = K i }, we have 0 < a ≤ | log(S u /K j )| ≤ b for some constants a, b (depending on ω) provided that u is sufficiently near one.

g ′′ (u) g ′ (u) 2 ≤ C (1 -g(u))

1 µ

 1 , µ = 1.5 and n = 10.

Lemma 4 . 4 .

 44 Set y > 0 and v

5. 4 .

 4 Conclusion.Let us summarize the results of the previous theorems:

  n

1 4

 1 +p (V n 1 -h(S 1 ))

  y)ϕ(y)dy, (6.116) with Q 2 (y) := -y 2 -ρy + 1, Q 3 (y) := -y 3 -ρy 2 + 3y + ρ.

y

  k ϕ(y)dy = 0 if k ∈ 2N + 1.

C

  xt (u, S u )du =

ρ 2 s ρ 2 tChC

 2 xt (u, S u ) σ -2 u dx, where u = u(x, n) is defined by x = ρ 2 u and verifies lim n→∞ u(x, n) = 1. Moreover, C xt (u, S u ) σ ′ (S u e √ xy+x/2 )(-y 2 -√ xy+1)ϕ(y)dy satisfies the following inequality | C xt (u, S u ) σ -2 u |du ≤ c G 1 (x, S u ) where G 1 (x, S) Corollary 6.22. Assume that we have two sequences (t k n ) n∈N and (s k n ) n∈N in [0, 1] such that ρ t k n and ρ s k n respectively converge to a ∈ [0, ∞] and b ∈ [0, ∞]. Then, xt (u, S u )du = b a J(x, S 1 )dx < ∞, a.s.

  Corollary 6.14. If m ∈ R and u ≥ t, then there exists a constant c m > 0 such that Proof. Indeed, it suffices to use Lemma 6.8 and apply the previous lemma.Corollary 6.16. If m ∈ R then there exists a constant c m > 0 such that S t ) = E S m t C 3/2 xx (t, S t ) C xx (t, S t) and we apply Cauchy-Schwarz' inequality with p = 4/3 and q = 4 such that p -1 + q -1 = 1. We obtain where the last inequality is deduced from Corollary 6.119. The conclusion follows. Corollary 6.17. If m ∈ R and u ≥ t, then there exists a constant c m > 0 such that Let S ti-1 ∈ [S ti-1 , S ti ] and t i-1 ∈ [t i-1 , t i ] be some random variables. We have the following inequalities: Lemma 6.18. There exists a constant c such that E C 4 xt ( t i-1 , S ti-1 ) ≤

	(6.120)	E S m u C 2 xxx (t, S u ) ≤	c m t ρ 3	e -ρ 2 t /8
	(6.121) (6.122) (6.123)	E S m u C 2 xxt (t, S u ) ≤ E S m u C 4 xxx (t, S u ) ≤ E S m u C 2 xxxx (t, S u ) ≤	c m σ 4 t ρ 5 t c m e -ρ 2 e -ρ 2 t /8 t /8 ρ 7 t c m t ρ 5 e -ρ 2 t /8
	(6.124)	E S m u C 4 xxt (t, S u ) ≤	c m σ 8 t u ρ 11	e -ρ 2 t /8 .
		E S m u C 2 xt (t, S u ) ≤	c m σ 4 t t ρ 3	e -ρ 2 t /8 .
								ce -ρ 2 t i	/4
	(6.118) (6.119)	E S m u C 4 xt (t, S u ) ≤ E S m u C 4 xx (t, S u ) ≤	c m σ 8 t ρ 7 t c m t ρ 3 e -ρ 2 e -ρ 2 t /8 t /4 .
		E S m t C 5/2 xx (t, S t ) ≤	c m t ρ 15/8	e -5ρ 2 t /32 .
	Proof. We write
	E S m t C 5/2 xx (t, E S m t C 5/2 xx (t, S t ) ≤	E S	4m/3 t	C 2 xx (t, S t )	3/4	E C 4 xx (t, S t )	1/4	,
		≤	C m E C 4 xx (t, S t )	3/8	E C 4 xx (t, S t )	1/4	,
		≤	C m	c ρ 3 t	e -ρ 2 t /4	3/8	c t ρ 3	t /4 e -ρ 2	1/4

In a similar way, we have: Corollary 6.15. If m ∈ R and u ≥ t, then there exists a constant C m > 0 such that

  3/2 , ∀u < 1 Lemma 6.20. There exists a constant C such thatE C 4 xxx ( t i-1 , S ti-1 ) ≤ -t i ) 6 f ′ (t i ) (6.126) E C 4 xxxx ( t i-1 , S ti-1 ) ≤

	(6.125)				Ce -ρ 2 t i ti ρ 8	/4
	(6.127)	E C 4 xxt ( t i-1 , S ti-1 ) ≤	Ce -ρ 2 t i n(1 Ce -ρ 2 /4 t i ti ρ 12 .	/4
	6.3. A.2.
	in order to have
		ρ ′′ t ρ t	≤	C (1 -t) 2 .