P Barrillon

S Blin

C Cheikali

D Cuisy

M Gaspard

D Fournier

M Heller

W Iwanski

B Lavigne

C De La Taille

P Puzo

J-L Socha

PMF: the front end electronic of the ALFA detector

The front end electronic (PMF) of the future ATLAS luminometer is described here. It is composed by a MAPMT and a compact stack of three PCBs which deliver the high voltage, route and readout the output signals. The third board contains a FPGA and MAROC, a 64 channels ASIC which can correct the non uniformity of the MAPMT channels gain thanks to a variable gain preamplifier. Its main role is to shape and discriminate the input signals at 1/3 photo-electron and produce 64 trigger outputs. Laboratory tests performed on prototype and pre-series PMFs have showed performances in good agreement with the requirements.

I. INTRODUCTION

The PMF (section B) is the front end electronics designed for the ATLAS luminometer (section A). It consists of a 64 channels photomultiplier (MAPMT) and a very compact stack of three different PCBs, mounted directly on the back and in the shadow of the MAPMT. The ASIC MAROC (section C) is used for the signal readout and treatment. ALFA stands for Absolute Luminosity For ATLAS [START_REF] Socha | ATLAS Forward Detectors for Measurement of Elastic Scattering and Luminosity Determination[END_REF]. This detector is located at 240 m from the interaction point (Fig. 1) and is meant to detect high energy diffused protons from the LHC beam. It is made of 20 staggered U-V scintillating fiber layers (Fig. 2) inserted in Roman Pots (eight in total). Each of these plans is made of 64 fibers which produce light when a particle comes through them. The light is then collected by multi-anode photomultiplier tubes H7546 from Hamamatsu [2], that will run at 800 to 950 V which corresponds to a gain of 3.10 5 to 2.10 6 . These PMTs have an important non-uniformity that can not be corrected by applying a different high voltage to each channel. Therefore one has to amplify the output signal with different factors for each channel. Figure 3 shows a schematic of a Roman Pot and the front end electronic. The whole readout system of ATLAS luminometer is presented in reference [START_REF] Blanchot | System Design of the ATLAS Absolute Luminosity Monitor[END_REF]. The PMFs will be arranged in a 5 by 5 matrix for each Roman Pot. Each line of up to 5 PMFs will be linked to the mother board through a kapton cable. In total 23 PMFs per Roman Pot will be installed, 20 for the standard scintillating fibers layers and 3 for another detector (overlap detector) described in [START_REF] Socha | ATLAS Forward Detectors for Measurement of Elastic Scattering and Luminosity Determination[END_REF].

A. ALFA detector

The active board is technologically challenging since it consists of 10 different layers, has MAROC located on the sixth layer through a hole in the upper layers and needs different types of vias (see Fig. 5). The bonding wires (<1cm long) are split between the first two layers.

C. MAROC MAROC (Multi Anode ReadOut Chip

) is a 64 inputs ASIC which allows correcting for the gain spread of MAPMT channels thanks to a 6 bits variable gain preamplifier. For each channel the signal is shaped (fast shaper, 15 ns) and discriminated to produce a trigger output. The discriminator thresholds are set by an internal 12 bit DAC, made of a 4 bit thermometer DAC for coarse tuning and a 8 bit mirror for fine tuning [START_REF] Barrillon | channel Front-End readout chip -MAROC datasheet[END_REF].

A multiplexed charge output is also produced both in analog and digital thanks to a Wilkinson ADC. The block diagram represented on figure 6 summarizes the different features of this chip [START_REF] Barrillon | channel Front-End readout chip -MAROC datasheet[END_REF]. The main requirements are the following: 100 % trigger efficiency for a signal greater than 1/3 of a photoelectron, a charge measurement up to 30 photoelectrons with linearity of 2 % or better and a low cross talk. The performances of the second version of MAROC were checked successfully during the year 2007 at LAL-Orsay [START_REF] Barrillon | MAROC: Multi-Anode ReadOut Chip[END_REF]. A nice dispersion of the trigger output (± 5 fC) was, in particular, observed.

II. LABORATORY TESTS

A. Description Lots of PMTs and PCBs were tested separately respectively at DESY and LAL-Orsay.

As far as what concerns the PCBs, the critical one to test is the active board due to its complexity and the many steps in its production. It is tested coupled with a passive board. A capacitor is used to inject signal in a single channel. A dedicated test board (left part of figure 7) has been designed for this purpose. Its role is to communicate with up to 5 PMFs via a kapton cable (central part of figure 7) sending configuration and collecting output signals. Standalone C and Labview programs were developed in order to enable testing.

A first batch of 8 active boards, considered as the prototype, was produced during autumn 2007. Thorough laboratory tests were performed at LAL after a period dedicated to the development of both test board (Xilinx) and PMF (Lattice) FPGA firmwares.

One of this active board was used to build a full PMF (3 PCBs and a PMT) in order to carry out a test with a LED illuminating a single scintillating fiber or the whole PMT (right part of the figure 7). A second lot of 32 pieces was produced end of spring 2008 in order to equip a complete Roman Pot for beam tests during following summer.

B. Prototypes 1) Standard electrical tests

All the main characteristics were measured for five active boards out of 8: pedestals, DAC linearity, S-curves (trigger efficiency as a function of the injected charge or the threshold), cross talk and charge.

All of them were found in good agreement with the results obtained during the characterisation of MAROC2 with particularly a nice homogeneity of the pedestals and scurves.

2) Tests with a LED

Trigger measurements were performed illuminating one or all channels of the MAPMT with a LED for different settings. The parameters tuned were the threshold, the diode pulse amplitude (Vdiode) and the high voltage (HV). Left part of figure 8 represents the trigger efficiency, for fixed Vdiode and HV, as a function of the threshold applied to the PMF for 8 different channels. As expected the so called s-curves are observed with a trigger efficiency going from a constant (plateau) value to 0.

Increasing the amplitude of the LED increases the level of this plateau as it can be seen on the right side of the figure 8.

A test of the gain equalization was tried by illuminating all channels as uniformly as possible. Figure 9 shows the trigger efficiency as a function of the channel number before and after gain equalization. Dispersion was improved by a factor 3.5 from 13.3 % to 3.8 %.

C. Pre-series

Here are presented results of the measurements performed on 28 active boards out of the 32.

1) Pedestals

Fast shaper pedestals were measured and a good homogeneity was found as it can be seen on figure 10 which represents the distribution of the pedestal for all the channels of the 28 active boards (i.e. 1792 channels in total). The dispersion is small and equal to 0.4 %. Fig. 10. Fast shaper pedestal distribution for pre-series active boards.

2) DAC linearity

Similarly the DAC linearity of the 28 active boards has been measured. It consisted in measuring the amplitude (Vdac) obtained for different DAC register values. Top part of figure 11 gives the evolution of Vdac as a function of this register for all boards. The nice homogeneity observed is confirmed by the linear fit (between 0 and 3000) parameters represented as a function of the PMF number (bottom part of figure 10).

3) Trigger outputs

The trigger output behaviour was studied by scanning the injected charge for a fixed threshold value and a unity gain applied to all channels.

Figure 12 represents the s-curves obtained for 1785 channels out of 1792. The 7 remaining ones are a bit shifted but nothing that can not be corrected by tuning the gain.

III. CONCLUSIONS

The laboratory tests carried out on prototype and preseries PMFs have showed that they are matching the requirements. The technological challenge consisting in producing a 10 layers PCB with several different vias and an ASIC bonded directly on it seemed to be achieved.

The pre-series batch of active boards showed a really high yield with no board rejected. Only 0.4 % of the channels (7 out of 1792) had an s-curve shifted with respect to the others.

LED tests performed at CERN and DESY proved that the full PMF was also working well.

23 pre-series PMFs were used during beam tests of a complete Roman Pot which took place during August 2008. Results should give the final green light for full production of all three kinds of board.

Fig. 1 .

 1 Fig. 1. Schematic view of the Roman Pots location with respect to the LHC beam.

Fig. 2 .

 2 Fig. 2. Schematic view of location of the scintillating fibers with respect to the beam. The U/V layers are arranged orthogonally to each other.

Fig. 3 .

 3 Fig. 3. Exploded schematic view of a Roman Pot and its front end electronic. B. PMF The Photomultiplier Front end (PMF) electronic is represented on the Fig. 4. It is made of a MAPMT and three boards (3 cm × 3 cm) in its shadow: • The HV board which brings the high voltage to the MAPMT; • The passive (or intermediate) board which routes the signals to connectors located on the edge; • The active board which has the read out chip MAROC directly wire-bonded on the PCB on one side and a FPGA (Lattice) on the other side.

Fig. 4 .

 4 Fig. 4. PMF schematic and picture.

Fig. 5 .

 5 Fig. 5. Schematic view of the active board layer structure.

Fig. 6 .

 6 Fig. 6. Block diagram of MAROC second version.

Fig. 7 .

 7 Fig. 7. Left: PMF test board. Center: kapton cable with 5 couples activepassive boards. Right: LED test set-up.

Fig. 8 .

 8 Fig. 8. Left: S-curves of 8 channels for a fixed Vdiode value. Right: Scurves of a single channel for four different Vdiode values.

Fig. 9 .

 9 Fig. 9. Trigger efficiency as a function of channel number before and after gain equalization.

Fig. 11 .

 11 Fig. 11. Top: DAC linearity for the 28 active boards. Bottom left: slope of the DAC linearity fit as a function of PMF number. Bottom right: intercept (Vdac at DAC=0) of the DAC linearity fit as a function of PMF number.

Fig. 12

 12 Fig. 12. S-curves for the 28 active boards distinguished by different colors. The good homogeneity observed can be characterized by the distribution of the injected charge needed to get 50 % trigger efficiency represented on figure 13.

Fig. 13 .

 13 Fig. 13. Distribution of the injected charge needed to get 50 % trigger efficiency among all 1792 channels.


Anne Eyraud-Loisel

BY AN INFLUENT INFORMED INVESTOR

Keywords: Enlargement of filtration, FBSDE, insider trading, influent investor, asymmetric information, martingale representation. 2000 Mathematics Subject Classification: Primary 60H10 Secondary 60G44, 91G20

In this paper a model with an influent and informed investor is presented from a hedging point of view. The financial agent is supposed to possess an additional information, and is also supposed to influence the market prices. The problem is modeled by a forward-backward stochastic differential equation (FBSDE), to be solved under an initial enlargement of the Brownian filtration. An existence and uniqueness Theorem is proved under standard assummptions. The financial interpretation is derived, together with an example of such influenced informed model.

Introduction

In this article a hedging problem is studied from an influent informed investor's point of view. The agent is supposed to possess an additional information on the market, and is also supposed to influence market prices. This is a natural extension of the work of Eyraud-Loisel [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF], where the informed agent was only supposed to be a small investor, with no influence on asset prices. We now study an influent informed agent, who wants to hedge against an option.

The presence of an asymmetrical information will be modeled by an initial enlargement of the Brownian filtration, as developed by the German school to model insider proof in the Brownian filtration case by [START_REF] Pardoux | Forward-backward stochastic differential equations and quasilinear parabolic PDEs[END_REF] [START_REF] Pardoux | Forward-backward stochastic differential equations and quasilinear parabolic PDEs[END_REF], whose details have been put in Appendix A. We prove, under similar hypotheses on the driver, coupled with hypothesis (H 3 ) on the additional information, that the FBSDE has a unique solution in the enlarged space. This result is obtained in 3 cases of influence on the drift and volatility of prices. The first case is a case of "weak influence", where the Lipschitz coefficient of the drift and volatility of prices with respect to the wealth process and the investment strategy are not too large; the second one is satisfied when the payoff is independent of the price process, and the last case concerns models where the investment strategy of the large trader do not influence the volatility of prices.

Under such conditions, the influent informed agent has a unique hedging strategy, and we give in Section 4 a financial interpretation of this result in terms of completeness of the informed market, and incompleteness of the market for a non informed trader's point of view.

Finally in Section 5, we present an example of influence satisfying all our hypotheses.

Model

Market model with influence

Let be a -dimensional standard Brownian motion, and (Ω, (ℱ ) 0≤ ≤ , ℙ) filtered probability space, where Ω = ([0, ]; R ). (ℱ ) ∈[0, ] denotes the natural Brownian filtration , completed with the -null sets. On the market, there are risky assets and a riskless asset. The asset prices dynamics is supposed to be influenced by the wealth and portfolio strategy of an agent, called the large investor, or influent investor.

denotes the wealth at time or this agent and her portfolio.

Prices of risky assets are driven by the following diffusion :

= ( , , , ) + ( ( , , , ),

), 0 = > 0, ∀ ∈ [1, ], (1)
where and are supposed to satisfy :

( , , , ) = ′ ( , , , ) , ( , , , ) = ′ ( , , , ) .

The riskless asset evolves according to the following equation :

0 = 0 ( , , ) , 0 0 = 1. (3)
Functions ′ , ′ and defined on Ω × [0, ] × ℝ × ℝ × ℝ are supposed to be ℱ-adapted for fixed , , , respectively ℝ , ℝ × , ℝ-valued.

The financial agent wants to hedge against a given contingent claim. She tries to determine her positive initial wealth, and her portfolio between 0 and maturity in order to possess as terminal wealth the payoff of the contingent claim.

In the standard self-financing framework, the wealth of the agent may be written as the solution of the following stochastic differential equation :

= ( ( - ∑ =0 ) ( , , )
) + ∑ =1 ′ ( , , , ) + ∑ =1 < ′ ( , , , ), > .
As prices satisfy > 0, ⊗ ℙ almost surely, thanks to the exponential form of SDE

(1) and (3), and from the hypothesis on the form of coefficients and in Equation

(2), this is equivalent to

= - ∫ ( , , , ) - ∫ < , >, (4)
where

( , , , ) = ( , , )+ < ′ * -1 , ′ ( , , , ) -( , , ).1I >, and = ′ * ( , , , )
The forward equation of prices [START_REF] Amendinger | Initial enlargement of filtrations and additionnal information of financial markets[END_REF] and the backward equation of wealth ( 4) are now fully coupled, because of the influence hypothesis :

⎧  ⎨  ⎩ = 0 + ∫ 0 ( , , , ) + ∫ 0 < ( , , , ), > = - ∫ ( , , , ) - ∫ < , > . (5)
Hypotheses are still to be specified in order to ensure such that this system admits a unique solution. This is the aim of Section 3.

Informed agent

The influent agent described in the previous section is also supposed to be an insider trader. She has an additional information at time = 0. This information is supposed to be ℱ ′ -measurable, where < ′ : it will be public at time ′ . The global information available at time to the informed agent is not ℱ any more, but it has been augmented with information . To model it, we introduce the enlarged filtration

:= ∩ > (ℱ ∨ ( )). (6)
It is an initial enlargement of the Brownian filtration (as developped by Jacod [START_REF] Jacod | Grossissement Initial, Hypothèse H' et Théorème de Girsanov[END_REF]).

This kind of information is known as strong initial information. We will work under the following usual hypothesis, introduced by Jeulin [START_REF] Jeulin | Grossissement d'une filtration et applications[END_REF][START_REF] Jeulin | Comportement des semi-martingales dans un grossissement de filtration[END_REF], and extensively used by Grorud and Pontier [START_REF] Grorud | Insider trading in a continuous time market model[END_REF] and Eyraud-Loisel [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF] :

Hypothesis 1. (H 3 ) There exists a probability measure ℚ equivalent to ℙ such that under ℚ, ℱ and ( ) independent for all < ′ . This hypothesis (H 3 ) is known to be equivalent to "the conditional distribution of given ℱ is equivalent to the distribution of , for all < ′ ". Under (H 3 ), a remarkable property is that is still a ( , ℚ)-Brownian motion. Moreover, ℚ may be chosen such that, for all ≤ < ′ , ℚ| ℱ = ℙ| ℱ , and on , ℚ = 1 ( , ) ℙ. For a reference on the existence of such a probability, and on the properties under (H 3 ), the reader may refer to Jeulin [START_REF] Jeulin | Comportement des semi-martingales dans un grossissement de filtration[END_REF], Amendinger [START_REF] Amendinger | Initial enlargement of filtrations and additionnal information of financial markets[END_REF], Grorud and Pontier [START_REF] Grorud | Insider trading in a continuous time market model[END_REF].

Solution of the enlarged FBSDE

Enlarged FBSDE and space of solutions

We have supposed that the informed investor may have an influence on asset prices.

To solve the hedging problem described in Section 2, we have to find a solution of the coupled forward-backward SDE [START_REF] Bouchard | Discrete-time approximation of bsdes and probabilistic schemes for fully nonlinear pdes[END_REF]. The mathematical problem lays in the space of solutions : this equation has to be solved in the enlarged space (Ω, , ℚ). As under hypothesis (H 3 ), is still a Brownian motion under ℚ, the FBSDE is the same in the enlarged space :

⎧  ⎨  ⎩ = 0 + ∫ 0 ( , , , ) + ∫ 0 < ( , , , ), > = ( ) - ∫ ( , , , ) - ∫ < , > . (7)
with a ( , ℚ)-Brownian motion, 0 ∈ 0 and = ( ) ∈ 2 (Ω, , ℚ).

As there exists existence and uniqueness results (see Eyraud-Loisel [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF]) for BSDE under initially enlarged filtration, the main difficulty is linked to the coupling between the forward equation of prices and the backward equation of wealth, which can not be solved separately. Pardoux and Tang [START_REF] Pardoux | Forward-backward stochastic differential equations and quasilinear parabolic PDEs[END_REF] show the existence of a unique solution in the case of a Brownian filtration (i.e. trivial initial -field), which is not the case here.

We will have to adapt the proof to the present case.

Remark :

Let us remark here the main difference with the model in [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF], where all parameters of the price diffusion were adapted to the Brownian filtration, and so ℱ = ℱ , the filtration generated by prices was the Brownian filtration. In the present model, wealth and portfolio may be -adapted and depend on . So the influence hypotheses induces that ℱ ∕ = ℱ . We have now ℱ ⊂ : a part of the additional information of the influent insider is "revealed" by the observation of prices. This is mathematically induced by the coupling of both SDEs.

If ℍ denotes a general Euclidian space, 2 (0, ; ℍ) denotes the set of all -progressively measurable ℍ-valued process { ( ), 0 ≤ ≤ } such that

∥ (.) ∥ 0 := ( ℚ ∫ 0 | ( )| 2 ) 1/2 < +∞.
For any process in 2 (0, ; ℝ ), we denote also by

∥ (.) ∥ := ( ℚ ∫ 0 -| ( )| 2 ) 1/2 < +∞.
a family of equivalent norms, indexed by ℝ.

2

ℚ denotes the set of all -progressively measurable process such that

∥ ∥ 2 := ℚ [ sup 0≤ ≤ | | 2 ] < +∞.
and 2 ℚ, the subset of continuous processes of 2 ℚ . A strong solution of system ( 7) is a -adapted solution ( , , ) 0≤ ≤ such that :

ℚ ∫ 0 ∥ ∥ 2 < +∞. (8)
It financially means that the agent looks for an admissible portfolio hedging the contingent claim.

Existence and Uniqueness Theorem

, , and are supposed to satisfy the following hypotheses (( 1) to ( 8)), which are derived from those for which E. Pardoux and S. Tang obtained existence of a unique solution in the case where there is no additional information (trivial initial -field):

(A1) is invertible × IP-a.s., ′ and ′-1 ( ′ -) are bounded.

(A2) functions , , , are continuous w.r.t. , , in

ℝ × ℝ × ℝ , for all ( , ) ∈ Ω × [0, ]. (A3) ∃ 1 , 2 ∈ ℝ tel que ∀ , , 1 , 2 , , 1 , 2 , , ℙ-p.s. : < ( , 1 , , ) -( , 2 , , ), 1 -2 > ≤ 1 | 1 -2 | 2 , < ( , , 1 , ) -( , , 2 , ), 1 -2 > ≤ 2 | 1 -2 | 2 .
(A4) is globally Lipschitz w.r.t. and , and at most linearly increasing w.r.t. , and is globally Lipschtiz w.r.t. and , and at most linearly increasing w.r.t.

: ∃ , such that ℙ-a.s. ∀ , , ,

| ( , , 1 , 1 ) -( , , 2 , 2 )| ≤ 1 | 1 -2 | + 2 ∥ 1 -2 ∥, | ( , , , )| ≤ | ( , 0, , )| + (1 + | |), | ( , 1 , , 1 ) -( , 2 , , 2 )| ≤ 3 | 1 -2 | + 4 ∥ 1 -2 ∥, | ( , , , )| ≤ | ( , , 0, )| + (1 + | |).
(A5) is globally Lipschitz w.r.t. , and : ∃ such that ℙ-a.s. ∀ , , ,

∥ ( , 1 , 1 , 1 ) -( , 2 , 2 , 2 ) ∥ 2 ≤ 2 5 | 1 -2 | 2 + 2 6 | 1 -2 | 2 + 2 7 ∥ 1 -2 ∥ 2 . (A6) is globally Lipschitz w.r.t. : ∃ 8 such that ℙ-a.s. ∀ | ( 1 ) -( 2 )| 2 ≤ 2 8 | 1 -2 | 2 .
(A7) ∀ , , , (., , , ), (., , , ) and (., , , ) are ℱ-adapted processes and

( ) is ℱ -measurable. Moreover : ℙ ∫ 0 | ( , 0, 0, 0)| 2 + ℙ ∫ 0 | ( , 0, 0, 0)| 2 + ℙ ∫ 0 ∥ ( , 0, 0, 0) ∥ 2 + ℙ | (0)| 2 < +∞. (9)
(A8) is also supposed to be globally Lipschitz w.r.t. : ∃ 9 such that ℙ-a.s. ∀ , , ,

| ( , 1 , , ) -( , 2 , , )| 2 ≤ 2 9 | 1 -2 | 2 .
Remarks :

• Hypothesis ( 1) guarantees IP ( exp 1 2 ∫ 0 ′2
) < +∞, which implies that the price process is a true martingale, as a uniformly integrable local martingale (Doleans exponential of ′ ˜ under a risk-neutral probability), and also insures that prices are positive ℙ-a.s. (see D. Lépingle et J. Mémin [START_REF] Lépingle | Sur l'intégrabilité uniforme des martingales exponentielles[END_REF]).

• Moreover, from Hypothesis ( 1), ℰ(-′-1 ( ′ -1l). ) is IP-integrable. This guarantees the existence of a risk-neutral probability (and so no arbitrage opportunity) when there is no additional information, see F. Delbaen et W. Schachermayer [START_REF] Delbaen | Non-arbitrage and the fundamental theorem of asset pricing: summary of main results[END_REF].

• We can notice that these two remarks are also true under probability ℚ of Hypothesis (H 3 ).

These hypotheses guarantee existence in (Ω, ℱ, ℙ) of a unique solution, and so of a unique portfolio (only in certain cases of coupling, see E. Pardoux et S. Tang [START_REF] Pardoux | Forward-backward stochastic differential equations and quasilinear parabolic PDEs[END_REF], and

Conditions ( 1) to ( 3) defined hereafter) in the case where there is no additional information (Brownian filtration). With the additional information satisfying hypothesis (H 3 ), on space (Ω, , ℚ), is there still a unique solution to the system ?

Hypotheses ( 1) to ( 8) are still satisfied, except [START_REF] Cuoco | Optimal consumption choices for a "large" investor[END_REF], under the enlarged space (Ω, , ℚ), with the new filtration and new probability ℚ, as ℚ and ℙ are equivalent.

We will assume the following additional integrability condition, in order to have [START_REF] Cuoco | Optimal consumption choices for a "large" investor[END_REF] also satisfied under ℚ:

( 7 ′ ) ℚ ∫ 0 | ( , 0, 0, 0)| 2 + ℚ ∫ 0 | ( , 0, 0, 0)| 2 + ℚ ∫ 0 ∥ ( , 0, 0, 0) ∥ 2 + ℚ | (0)| 2 < +∞. (10)
Remark:

Let us notice that this condition is always satisfied in our financial framework, as all parameters are null in 0.

Notice also that results are proved in the present section in the general dimension case for processes ( , , ) ∈[0, ] taking values in ℝ , ℝ and ℝ × . In the financial setting = , = 1 et = .

Three different influence cases are treated here :

(I1) The influence is weak : the forward and backward equations are weakly coupled:

∃ 0 > 0 depending on 3 , 4 , 5 , 8 , 1 , 2 and such that 1 , 2 , 6 , 7 ∈ [0, 0 ).

(I2) , ℱ -measurable, is independent from the price process, and 1 and 2 from hypothesis (A3) satisfy also:

∃ > 0, = 1, 2, 3, 4, 4 < -1 4 , > 0 such that 1 + 2 < - 1 2 [ 3 3
(

2 2 + 2 7 1 -4 4 + 1 1 + 2 6 ) + 1 -1 1 + 2 -1 2 + 3 -1 3 + 4 -1 4 + 2 5 + ] . (11)
(I3) is independent from : the portfolio does not influence the volatility of prices and 1 and 2 from hypothesis (A3) satisfy also:

∃ > 0, = 1, 3, 4, 4 < -1 4 , > 0, > 0 such that 1 + 2 < - 1 2 [ (1 + ) [ 1 1 + 2 6 + 2 2
(1 -4 4 )

] (

2 8 + 3 3
)

+ 1 -1 1 + 3 -1 3 + 4 -1 4 + 2 5 + ] . (12)
In these 3 cases, the existence and uniqueness Theorem given by E. Pardoux and S.

Tang [START_REF] Pardoux | Forward-backward stochastic differential equations and quasilinear parabolic PDEs[END_REF] is still satisfied in the enlarged space.

Theorem 1. We suppose Hypotheses ( 1) to ( 6), ( 7 ′ ) and ( 8) satisfied, and also one of the cases (I1) to (I3) satisfied. Then the following forward-backward stochastic differential equation:

⎧  ⎨  ⎩ = 0 + ∫ 0 ( , , , ) + ∫ 0 < ( , , , ), > = - ∫ ( , , , ) - ∫ < , > . (13)
admits in the space (Ω, , ) a unique solution ( , , ) -adapted such that

ℚ ∫ 0 ( sup 0≤ ≤ | | 2 + sup 0≤ ≤ | | 2 + ∥ ∥ 2 ) < +∞. (14)
Proof :

We derive the four majoration lemma and use the same arguments as in Pardoux and Tang [START_REF] Pardoux | Forward-backward stochastic differential equations and quasilinear parabolic PDEs[END_REF]. Theorem 2.1 in Eyraud-Loisel [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF] for BSDE with enlarged filtration allows us to construct strict contractions and conclude to the existence and uniqueness of the solution in the enlarged space. Even if the standard proofs in the Brownian case may be adapted to the case of an enlarged filtration, even with a change of filtration and a non trivial initial -algebra, there are some difficulties within the adaptation of these proofs. The key point that leads hidden in Theorem 2.1 of [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF] is the existence of a martingale representation Theorem under hypothesis (H 3 ) in the enlarged space. This point is hidden in the problem of solving the FBSDE, but it is nevertheless crucial. See in Appendix A the main steps of this proof. □

Financial interpretation 4.1. Bound on the wealth process

We also derived another result, unknown in the literature, up to the knowledge of the author, which may financially be interpreted as a bound on the wealth process.

Proposition 1. Under Hypotheses ( 1) and ( 8), and ( 7 ′ ), ∀ ∈ 2 (Ω, , ℚ), if

3 =0 and if ( , 0, 0, 0) is bounded, then solution of the FBSDE (13) satisfies:

| | 2 ≤ -( -) ℚ ( | | 2 ) + 1 ℚ ( ∫ -( -) | ( , 0, 0, 0)| 2 ) , (15)
where = -(

2 2 + + 4 -1 4
), > 0 and 0 < 4 ≤ -1 4 .

Proof: This is obtained by using Itô's formula, taking conditional expectation w.r.t. and using standard inequalities, as well as Hypotheses (A1) to (A8).

Apply Itô's lemma to -| | 2 between and , with ∈ ℝ :

-| | 2 + ∫ 0 -| | 2 = -| | 2 + ∫ -| | 2 +2 ∫ ( , , , ) - -2 ∫ - . (16)
Taking conditional expectation of this expression w.r.t. under probability measure ℚ eliminates the last term, as the increments of the ( , ℚ)-Brownian motion are independent. Hypotheses ( 3) and ( 4) give a majoration to the other terms, and as 3 = 0, we obtain, ∀ , 3 , 4 > 0 :

| | 2 -≤ - ℚ ( | | 2 ) + ( + 2 2 + + 4 4
)

ℚ ( ∫ -| | 2 ) + ( 4 4 -1) ℚ ( ∫ -| | 2 ) + 1 ℚ ( ∫ -| ( , 0, 0, 0)| 2
) .

Choosing = -(2 2 + + 4 - 1
4 ), and 0 < 4 ≤ -1 4 , as solution of FBSDE ( 13)

satisfies ( ∫ 0 | | 2 )
< +∞, we obtain the expected inequality :

| | 2 -≤ - ℚ ( | | 2 ) + 1 ℚ ( ∫ -| ( , 0, 0, 0)| 2
) .

□

Corollary 1. If moreover is bounded, then process is bounded.

Proof: This appears naturally from equation [START_REF] Gobet | A regression-based monte-carlo method to solve backward stochastic differential equations[END_REF] as the right term is bounded. □ Toy example : We can find a simple example to illustrate this property, in the case of prices driven by a geometric Brownian motion (Black-Scholes model), without influence:

( , , , ) = + ′-1 ( ′ -) .
does not depend on variable , so 3 = 0, and ( , 0, 0, 0) = 0.

In any cases of to hedge, satifies :

| | 2 ≤ -( -) ℚ ( | | 2 ℱ ) . (17)
And for a pay-off bounded (for instance a European put option ( -) + , bounded by ), we have :

| | 2 ≤ -( -) 2 ≤ - 2 . ( 18
)
Wealth is bounded by 2 -.

Market incompleteness derived by the influent informed agent

In these three cases of influence, ( 1), ( 2) et ( 3), the forward-backward stochastic differential equation has a unique adapted solution in the enlarged space. This means that the influent agent has a unique hedging strategy adapted to his information. We have supposed invertible, so from , it is possible to derive the unique portfolio hedging the pay-off : = ′ -1 . If the solution of the insider trader is adapted to the Brownian filtration, then it is the same as if there was no additional information.

Any contingent claim in 2 (Ω, , ℚ) satisfying hypothesis ( 6) is attainable. In fact, the market is complete for the informed investor, relatively to the enlarged filtration.

The hedging problem in the market is reduced to a resolution of a FBSDE, whose coupling depends on the influence of the insider in the market.

We obtain here the same results for the insider's point of view as in [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF], for the existence and uniqueness of the solution of a BSDE under an enlarged filtration.

What is different in the model with an influent agent, compared to the model without influence developed in [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF], is the behavior of the market from the point of view of a normally informed agent (when the additional information is unknown). The market from a non insider's point of view is incomplete. This incompleteness is due to a lack of information, as in a model developed by H. Föllmer and M. Schweizer (1991) [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF]. Our model is an example of complete market which becomes incomplete from a small non informed investor point of view. The study of such a market uses tools of quadratic hedging in incomplete market, or under incomplete information, and will be developed separately in a further work, see Eyraud-Loisel [START_REF] Eyraud-Loisel | Quadratic hedging in an incomplete market derived by an influent informed investor[END_REF].

Example

In this last section, we present a model of influence that satisfies all hypotheses of this study. Suppose that the price process is driven by the following dynamics (stochastic volatility model) :

= ′ ( , , ) + ′ ( ) , (19)
where

′ ( ) = 0 ℐ [0, [ ( ) + 1 ℐ [ , ] ( ) , 0 , 1 ∕ = 0. ( 20
)
The volatility of this model is piecewise constant, taking two possible values 0 and 1 fixed by the model, is a random variable satisfying hypothesis (H 3 ), taking his values in [0, + ].

The information of the insider trader is = ℱ + -measurable . This is an example of strong initial information.

Case without influence

If we choose a drift parameter ′ ( , , ) = 0 constant, we obtain a model without influence. The coupling between the BSDE and the FSDE vanishes, and it leads to a resolution of BSDE with initial enlargement of filtration. All hypotheses

( 1) to ( 8) and case 1 (no dependence, so a fortiori weak dependence) are satisfied, and even without influence, the market is complete from the insider point of view, and incomplete from the non insider point of view. This gives a toy example where an additional information may "complete" the market.

Case with influence

The drift parameter is chosen as the following :

′ ( , , ) = 0 + 1 (1 + )(1 + 2 ) , 0 , 1 ∈ ℝ fixed. (21)
The interest rate is supposed to be constant.

Drift ′ is bounded, and may vary between two thresholds 0 and 0 + . Two cases may appear, depending on the sign of 1 . If 1 < 0, the influence is a positive influence: the bigger is the investment porfolio, the higher is the drift of the prices. This is moderated by the level of prices: the higher are the prices, the lower is the influence. If 1 > 0, it is the converse principle: when the level of the portfolio increases, the drift of the prices decreases, and the influence is stronger when the level of prices is high. Remark that the case 1 = 0 is the case treated in the previous section, without influence.

Depending on the sign of 1 , representing the amplitude of the influence, this influence will have either a leverage effect or a return effect on the drift of the price process around the value 0 . The influence is from the insider's portfolio on the price process, which remains bounded according to the hypotheses.

We can also notice that

′-1 ( ′ -) = ′-1 ( 0 -+ 1 (1 + )(1 + 2 ) )
is bounded, as well as ′ . So there exists a risk-neutral probability measure Q under which = ′ ˜ is a positive uniformly integrable martingale (see Lépingle and

Mémin [START_REF] Lépingle | Sur l'intégrabilité uniforme des martingales exponentielles[END_REF] Theorems II-2 and III-7).

Remark: We don't have here constraints on the signs of 0 or 1 , whereas it is often the case in previous influence models developed in the literature, such as in the model introduced by Cuoco and Cvitanic (1998) [START_REF] Cuoco | Optimal consumption choices for a "large" investor[END_REF], and treated deeply in [START_REF] Grorud | Fiancial market model with influential informed investors[END_REF] [START_REF] Grorud | Fiancial market model with influential informed investors[END_REF] (their influence form is slightly different from the one treated in this work).

This may be explained by the fact that we consider a hedging problem, whereas they considered an optimization problem, and therefore we do not need the convexity of the parameters here.

For the present model, considering the hedging of a European call option of maturity and exercise price , the parameters are the following :

( , , , ) = + ( 0 -+ 1 (1 + )(1 + 2 )
) ,

( ) = ( -) + ( 22
)
which leads to the following FBSDE modeling the hedging problem of the informed agent :

⎧  ⎨  ⎩ = 0 + ∫ 0 ( 0 + 1 (1+ )(1+ 2 ) ) + ∫ 0 ( ) = ( -) + - ∫ ( + ( 0 + ℎ( )) - ) - ∫ ( ) . ( 23
)
We have to check if all hypotheses of Theorem 1 are satisfied.

(A1) ′ is bounded and invertible, since 0 and 1 are constant and not null.

Moreover, ′-1 ( ′ -) is bounded, since ′ and ′-1 are bounded.

(A2) , , and are continuous w.r.t. variables , , .

(A3) This hypothesis is satisfied by taking 1 = sup( 0 , 0 + 1 ) :

( ( , 1 , , ) -( , 2 , , ), 1 -2 ) = ( 0 1 + 1 (1 + 1 )(1 + ′-2 2 ) -0 2 - 1 (1 + 2 )(1 + ′-2 2 ) ) ( 1 -2 ) = 0 ( 1 -2 ) 2 + 1 (1 + ′-2 2 ) × ( 1 -2 ) 2 (1 + 1 )(1 + 2 ) ≤ sup( 0 , 0 + 1 )( 1 -2 ) 2 , ( 24
)
and 2 = : ( ( , , 1 , ) -( , , 2 , ), 1 -2 ) = ( 1 -2 ) 2 .
(A4) 1 = 0 since does not depend on .

( , , , ) is 1 , with first derivative w.r.t. given by :

∂ ( , , , ) ∂ = 1 1 + (-2 ′-1 ) (1 + ′-2 2 ) 2 ≤ | 1 | 1 1 + ′-2 2 × (-2 ′-1 ) (1 + ′-2 2 ) ≤ | 1 |.
So is uniformly Lipschitz w.r.t. , with 2 = | 1 |.

Moreover, since ( , 0, , ) = 0, we can write

| ( , , , )| ≤ (| 0 | + | 1 |) | |. (25)
So

≥ | 0 | + | 1 |.
is also 1 , with first derivative w.r.t. and given by :

∂ ( , , , ) ∂ = 1 1 + 2 1 (1 + ) 2 ≤ | 1 | 2 , ∂ ( , , , ) ∂ = 1 ′-1 (1 + ) 1 -2 (1 + 2 ) 2 ≤ | 1 | inf( 0 , 1 )
.

So is uniformly Lipschitz w.r.t. and , with

3 = | 1| 2 and 4 = | 1 | inf( 0 , 1 ) . Finally is linear w.r.t. , so ≥ . = max( , | 0 | + | 1 |) is convenient. (A5) is uniformly Lipschitz w.r.t.
(piecewise linear), and does not depend on and , so this hypothesis is satisfied with 5 = ( 0 , 1 ), 6 = 0, 7 = 0.

equations, which is the difficulty to express explicitely the solutions. But all these new schemes may open a way to simulate and use more extensively such study, because these new works and their future extensions give interesting tracks for efficient schemes for FBSDEs.

∈ ℝ, , 3, 4 > 0, such that 0

< 4 < -1 4 , define ¯ 2 := --2 2 -3 -1 3 -4 -1 4
-..

We have :

∥ (.) ∥ 2 ≤ 1 --¯ 2 ¯ 2 [ 2 8 (1 + ) - ℚ | | 2 + 3 3 ∥ (.) ∥ 2 + ( 1 + 1 ) - ℚ | (0)| 2 + 1 ∥ (., 0, 0, 0) ∥ 2 ] . (28)
Furthermore if ¯ 2 ≥ 0 :

∥ (.) ∥ 2 ≤ 1 1 -4 4 [ 2 8 (1 + ) - ℚ | | 2 + 3 3 ∥ (.) ∥ 2 + ( 1 + 1 ) - ℚ | (0)| 2 + 1 ∥ (., 0, 0, 0) ∥ 2 ] . (29)
Lemma 3. Suppose Hypotheses ( 3) -( 5), ( 7) and ( 7 ′ ) hold. Let (.) satisfy the forward equation of ( 13) associated to ( (.), (.)) = ( (.), (.))

∈ 2 (0, ; ℝ )× 2 (0, ; ℝ × ), = 1, 2. Define, for all ∈ ℝ, 1, 2 > 0, ¯ 1 := -2 1 -1 -1 1 -2 -1 2 -2 5 . Then ∥ Δ ∥ 2 ≤ 1 --¯ 1 ¯ 1 [ ( 1 1 + 2 6 ) ∥ Δ ∥ 2 +( 2 2 + 2 7 ) ∥ Δ ∥ 2 ] , (30)
and

- ℚ |Δ | 2 ≤ [ 1 ∨ -¯ 1 ] ( 1 1 + 2 6 ) ∥ Δ ∥ 2 +( 2 2 + 2 7 ) ∥ Δ ∥ 2 . ( 31
)
Moreover if 1 ≥ 0, more simply it leads to :

- ℚ |Δ | 2 ≤ ( 1 1 + 2 6 ) ∥ Δ ∥ 2 +( 2 2 + 2 7 ) ∥ Δ ∥ 2 . ( 32
)
Lemma 4. Suppose Hypotheses ( 3), ( 4),( 6), ( 7) and ( 7 ′ ) hold. Let ( (.), (.)) satisfy the backward equation of ( 13) associated to (.) = (.)

∈ 2 (0, ; ℝ ), = 1, 2. For all ∈ ℝ, 3, 4 > 0 such that 0 < 4 < -1 4 , define ¯ 2 := --2 2 -3 -1 3 -4 -1 4 . Then ∥ Δ ∥ 2 ≤ 1 --¯ 2 ¯ 2 [ 2 8
-

ℚ |Δ | 2 + 3 3 ∥ Δ ∥ 2 ] , (33)
and

∥ Δ ¯ ∥ 2 ≤ 2 8 -¯ 2 - ℚ |Δ | 2 + 3 3(1 ∨ -¯ 2 ) ∥ Δ ∥ 2 (1 -4 4)(1 ∧ -¯ 2 ) . ( 34
)
Moreover if 2 ≥ 0, more simply it leads to :

∥ Δ ¯ ∥ 2 ≤ 1 1 -4 4 [ 2 8 - ℚ |Δ | 2 + 3 3 ∥ Δ ∥ 2 ] . (35)
Proof. These lemmas are derived using several standard inequalities, such as Burkholder-Davis-Gundy inequality, and the proof uses extensively the Lipschitz and linear growth assumptions of hypotheses ( 1) to ( 7), ( 7 ′ ) and [START_REF] Cvitanić | Hedging options for a large investor and forward-backward SDE's[END_REF]. □

We will suppose from now on that all hypotheses ( 1) to ( 7), as well as ( 7 ′ ) and ( 8) are satisfied. We prove the existence and uniqueness of the solution of the forward equation at fixed solution of the backward equation, and then construct a strict contraction in the space of solutions, which will prove the existence and uniqueness of the entire solution of Equation [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF].

Let ∈ 2 ℚ, , ∈ 2 (Ω, , ℚ), ( , ) ∈ 2 (0, ; ℝ ) × 2 (0, ; ℝ × ). Define ∀ ∈ [0, ] ( ) := + ∫ 0 ( , , , ) + ∫ 0 ( , , , ) . (36)
( ) is well defined and continuous, as ∈ 2 ℚ, , and thanks to properties of and .

Lemma 5. ( ) belongs to 2 ℚ, as soon as ∈ 2 ℚ, , which means

∥ ( ) ∥ 2 = ℚ [ sup 0≤ ≤ | ( ) | 2 ] < +∞.
Proof of Lemma 5 : Let 1, 2 ∈ 2 , . Using Doob and Hölder inequalities, we have

ℚ [sup 0≤ ≤ | ( 1) -( 2) | 2 ] ≤ 2 ℚ [∫ 0 | ( , 1( )) -( , 2( ))| 2 ] +8 ℚ [∫ 0 ∥ ( , 1( )) -( , 2( )) ∥ 2 ] .
As and are uniformly Lipschitz, there exists > 0 such that :

ℚ [ sup 0≤ ≤ | ( 1) -( 2) | 2 ] ≤ 2 2 ( + 4) ℚ [∫ 0 | 1( ) -2( )| 2 ] ≤ 2 2 ( + 4) ℚ [∫ 0 sup 0≤ ≤ | 1( ) -2( )| 2 ] . (37)
Moreover, using again Doob and Hölder's inequalities,

ℚ [ sup 0≤ ≤ | (0) | 2 ] ≤ 3
( ℚ [ 2 ] + [ sup 0≤ ≤ | ∫ 0 ( , 0) | 2 ] + ℚ [ sup 0≤ ≤ | ∫ 0 ( , 0) | 2
]) .

Applying ( 4) and ( 5) between ( , ) and (0, 0) as well as ( 7 ′ ) we obtain the following majoration :

ℚ [ sup 0≤ ≤ | (0) | 2 ] ≤ 3 ( ℚ ( 2 ) + 2 2 + 4 2 )
.

Hence

[ sup 0≤ ≤ | ( ) | 2 ] ≤ 2 [ sup 0≤ ≤ | ( ) -(0) | 2 + | (0) | 2 ] ≤ 4 2 ( + 4) [ sup 0≤ ≤ | ( )| 2 ] + 3 [ 2 ] + 2 2 + 4 2 . (38)
And then ( ) ∈ 2 ℚ, as soon as ∈ 2 ℚ, . □ Let us prove now that the forward equation of ( 13) has a unique solution in 2 (0, ; ℝ ). Lemma 6. Let ( (.), (.)) fixed in 2 (0, , ℝ × ℝ × ) and let ∈ ℒ 2 (Ω, 0 = ( ), ℚ).

Then equation

( ) = + ∫ 0 ( , , , ) + ∫ 0 ( , , , ) (39)
has a unique -adapted solution (.) in 2 (0, ; ℝ ).

Proof. The proof is similar to the standard one. Two things change : (0) is not deterministic any more, and the used filtration is not the Brownian one. We have to solve a standard SDE in (Ω, , ℚ), as under ( 3), is a (ℱ, ℙ) and a ( , ℚ)-Brownian.

Fix and two -adapted processes in 2 (0, , ℝ × ℝ × ).

Proof of existence

By induction we construct a sequence od processus in 2 ℚ, :

Then using majoration (37) and Cauchy formula we deduce :

ℚ [ sup 0≤ ≤ | +1 - | 2 ] ≤ 2 2 ( + 4) ∫ 0 ℚ [ sup 0≤ ≤ | - -1 | 2 ] ≤ 1 ( -1)! ∫ 0 ( -) -1 ℚ [ sup 0≤ ≤ | 1 | 2 ] ≤ ! ℚ [ sup 0≤ ≤ | 1 | 2 ] .
Denote by the majorant in Equation ( 38) associated to = 0 = 0,

ℚ [ sup 0≤ ≤ | +1 - | 2 ] ≤ ! .
Then it follows

∑ ≥0 ∥ sup 0≤ ≤ | +1 - | ∥ 1 ≤ ∑ ≥0 ∥ sup 0≤ ≤ | +1 - | ∥ 2 ≤ √ ∑ ≥0 ( ) /2 √ ! < +∞.
The sequence converges ℚ-a.s., so converges uniformly ℚ-a.s. on [0, ] to ˜ , which is continuous. Moreover ˜ ∈ 2 ℚ, since the convergence is in 2 ℚ . Finally ˜ satisfies SDE (39) est solution de l'EDS par passage à la limite dans +1 = ( ).

Proof of Uniqueness

Let 1 and 2 be two solutions of SDE (39) in 2 ℚ, . We have

[ sup 0≤ ≤ | 1 -2 | 2 ] ≤ 2 2 ( + 4) ∫ 0 [ sup 0≤ ≤ | 1 -2 | 2 ] ,
so applying Gronwall's Lemma, there are indistinguishable:

[ sup 0≤ ≤ | 1 -2 | 2 ] = 0.
This proves the uniqueness of the solution of the forward equation of (13) at fixed ( , )

∈ 2 (0, , ℝ × ℝ × ). □
The forward equation induces an application 1 : ( , ) ∈ 2 (0, , ℝ × ℝ × ) → 1( (.), (.)) = (.), unique solution of the forward equation at fixed ( (.), (.)). This is justified by Lemma 6.

In the same way, we define

2 : ∈ 2 (0, , ℝ ) → 2( (.)) = ( (.), (.)) ∈ 2 (0, , ℝ × ℝ × ),
the unique solution of the backward equation at fixed (.). This is possible thanks to the existence and uniqueness Theorem 2.1 given in Eyraud-Loisel [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF].

Finally this allows us to define the two following operators

Γ1 := 2 ∘ 1 et Γ2 := 1 ∘ 2.
From the previous existence and uniqueness results we deduce in particular that Γ1 maps 2 (0, ; ℝ × ) into itself, and Γ2 maps 2 (0, ; ℝ ) into itself.

Let ( , ) ∈ 2 (0, ; ℝ × ℝ × ), = 1, 2. Define := 1( , ) and ( ¯ , ¯ ) = Γ1(( , )), = 1, 2.
The existence and uniqueness of the solution of the enlarged FBSDE will be obtained using the majoration Lemmas in order to prove that Γ1 or Γ2 is a strict contraction, so have a unique fixed point, which is, by construction, the unique solution of the enlarged FBSDE.

We will have to distinguish three cases. In each case, we will need to put an upper bound on 1 + 2 depending on the .

(I1)/ Weak coupling between forward equation and backward equation

We prove here that Γ is a strict contraction for norm ∥ . ∥ , for ∈ ℝ.

From majorations (30), (31), (33) and (34), we can write :

∥ Δ ¯ ∥ 2 + ∥ Δ ¯ ∥ 2 ≤ [ 1 --¯ 2 ¯ 2 + 1 1 -4 4 ] ( 2 8 (1 ∨ -¯ 1 ) + 3 3 1 --¯ 1 ¯ 1 ) × ( ( 1 1 + 2 6 ) ∥ Δ ∥ 2 +( 2 2 + 2 7 ) ∥ Δ ∥ 2 ) . (40)
Then, as soon as

1 1 + 2 6 < [ 1 --¯ 2 ¯ 2 + 1 1 -4 4 ] -1 ( 2 8 (1 ∨ -¯ 1 ) + 3 3 1 --¯ 1 ¯ 1 ) -1 , 2 2 + 2 7 < [ 1 --¯ 2 ¯ 2 + 1 1 -4 4 ] -1 ( 2 8 (1 ∨ -¯ 1 ) + 3 3 1 --¯ 1 ¯ 1 ) -1 ,
Γ1 is a strict contraction. So the existence of 0 which depend on 3, 4, 8, 1, 2, such that 1, 2, 6, 7 ∈ [0, 0) is enough to insure that Γ1 is a strict contraction, which ends the proof of Theorem 1 by proving that there exists a unique fixed point which is the unique solution of enlarged FBSDE [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF].

(I2)/ The contingent claim is independent of prices.

We suppose here that is a ℱ -measurable random variable independent of prices, which means 8 = 0. Suppose moreover that ( 3) is such that 1 and 2 satisfy ∃ > 0, = 1, 2, 3, 4, 4 < -1 4 , > 0 such that Then by definition of ¯ 2 in Lemma 2 and of ¯ 1 in Lemma 1,

1 + 2 < - 1 2
¯ 2 = > 0, ¯ 1 = -(2 1 + 2 2 + 1 -1 1 + 2 -1 2 + 3 -1 3 + 4 -1 4 + 2 5 + ).
So from (41),

¯ 1 > 3 3( 2 2 + 2 7 1 -4 4 + 1 1 + 2 6 ) > 0.
From Equations (30) and (33) in Lemmas 3 and 4, we get :

∥ Δ ¯ ∥ 2 ≤ 1 --¯ 1 ¯ 1 3 3 ( 2 2 + 2 7 1 -4 4 + 1 1 + 2 6 ) ∥ Δ ∥ 2 . ( 42
)
As soon as (41) is satisfied, the coefficient of ∥ Δ ∥ 2 is nonnegative and strictly bounded by 1. Hence Γ2 is a strict contraction, and admits a unique fixed point, which is the only solution of FBSDE [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF]. This ends the proof of Theorem 1 in case ( 2).

( 3)/ The volatility does not depend on the portfolio.

We suppose here that 7 = 0 and also that 1 and 2 in ( 3) satisfy ∃ > 0, = 1, 3, 4, 4 < -1

4 , > 0, > 0 such that

1+ 2 < - 1 2
[ (1 + )

( 1 1 + 2 6 + 2 2
(1 -4 4)

) (

2 8 + 3 3
) + > (1 + )

[ 1 1 + 2 6 + 2 2
(1 -

] (

2 8 + 3 3
) > 0.

As this is true for any 2 > 0, we deduce that ¯ 2 > 0. Moreover, as 7 = 0 and ¯ 1, ¯ 2 > 0, we deduce from majorations (30), (32), ( 33) and (35) from Lemmas 3 and 4 :

∥ Δ ¯ ∥ 2 ≤ 1 ¯ 2 ( 2 8 + 3 3 1 --¯ 1 ¯ 1 ) [ ( 1 1 + 2 6 ) ∥ Δ ∥ 2 +( 2 2) ∥ Δ ∥ 2 ] , ∥ Δ ¯ ∥ 2 ≤ 1 1 -4 4 ( 2 8 + 3 3 1 --¯ 1 ¯ 1 ) [ ( 1 1 + 2 6 ) ∥ Δ ∥ 2 +( 2 2) ∥ Δ ∥ 2 ] .
Let = (1 -4 4)( ¯ 2) -1 . We obtain :

∥ Δ ¯ ∥ 2 + ∥ Δ ¯ ∥ 2 ≤ 1 + ¯ 2 ( 2 8 + 3 3 1 --¯ 1 ¯ 1 ) ×( 1 1 + 2 6 ) [ ∥ Δ ∥ 2 + 2 2 (1 -4 4)( 1 1 + 2 6 ) ¯ 2 ∥ Δ ∥ 2 ] . Choosing -1 2 = 2 ¯ 2 (1-4 4 )( 1 1 + 2 6
) , and combining it with previous inequalities leads to

∥ Δ ¯ ∥ 2 + ∥ Δ ¯ ∥ 2 ≤ 1 + ¯ 2 ( 2 8 + 3 3
)

( 1 1 + 2 6 ) [ ∥ Δ ∥ 2 + ∥ ¯ ∥ 2 ] .
and

¯ 2 > (1 + ) ( 2 8 + 3 3
)

( 1 1 + 2 6
) , such that Γ1 is a strict contraction, as soon as Equation ( 43) is satisfied. It admits a unique fixed point, which is the unique solution of FBSDE [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF]. This ends the proof of Theorem 1.

From

= 0, +1 = ( ), ∀ ≥ 0.

(A6) is uniformly Lipschitz w.r.t. , with 8 = 1.

(A7) (., , , ), (., , , ) and (., , , ) are -adapted processes and ( ) is deterministic for fixed .

Moreover, the integrability equation in 0 is satisfied : all terms are null.

(A8) is uniformly Lipschitz in , with coefficient

We are in the case of weak influence ( 1) of existence and uniqueness Theorem 1. Indeed, whatever is the value of 1 , 2 may be chosen small enough so that Equation (41) (see proof of the Theorem 1 in Appendix A) is satisfied. In the present example, this equation becomes, as 1 = 6 = 7 = 0 :

which is always true as 2 may be chosen arbitrarily small.

Let us notice that if 0 and 1 are negative enough to satisfy Equation (43), we could also use the case ( 3), because there is no influence from the insider on the volatility of prices.

So all hypotheses of Theorem 1 are satisfied, and EDSPR (13) admits a unique solution.

This allows us to conclude to the existence of a unique solution of the insider hedging problem.

Remark : Despite several studies developed on the subject, from Chevance (1997) [START_REF] Chevance | Résolution numériques des équations différentielles stochastiques rétrogrades[END_REF], to Delarue (2002) [START_REF] Delarue | On the existence and uniqueness of solutions to fbsdes in a non-degenerate case[END_REF], Gobet, Lemor and Warin (2005) [START_REF] Gobet | A regression-based monte-carlo method to solve backward stochastic differential equations[END_REF], or more recently Bouchard and Elie (2008) [START_REF] Bouchard | Discrete time approximation of decoupled forward-backward sde with jumps[END_REF], or Bouchard, Elie and Touzi (2009) [START_REF] Bouchard | Discrete-time approximation of bsdes and probabilistic schemes for fully nonlinear pdes[END_REF] with discretization schemes of BSDE and FBSDE, there is still a difficulty that leaves in the use of such

Appendix A. Proof of Theorem 1

The first step of the proof of Theorem 1 consist in deriving the first four majoration lemmas on the norms of solutions. These fundamental estimates lemmas are the same as those in Pardoux and Tang [START_REF] Pardoux | Forward-backward stochastic differential equations and quasilinear parabolic PDEs[END_REF]. Even if the considered space is not the same (the initial sigma-field is not trivial in our case), these majoration lemmas still hold in the enlarged space. The main reason is that the only tools used to prove these lemmas are Itô Formula, Lipschitz property and linear growth of the different functions and coefficients in the hypotheses ( 1) to [START_REF] Cuoco | Optimal consumption choices for a "large" investor[END_REF], which are also supposed to hold under ℙ and, as seen page 9, remain true under ℚ. We recall the statement of these four lemmas in the way we use it (slightly different from the lemmas given in [START_REF] Pardoux | Forward-backward stochastic differential equations and quasilinear parabolic PDEs[END_REF] called Lemmas 2.1 to 2.4).

Lemma 1. Suppose ( 3) -( 5), ( 7) and ( 7 ′ ) hold. Let ( (.), (.)) ∈ 2 (0, ; ℝ ) × 2 (0, ; ℝ × ), and ( (.)) ∈ 2 (0, ; ℝ ) solutions of forward equation of [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF]. Then, for all ∈ ℝ, , 1, 2 > 0,

with

Lemma 2. Suppose Hypotheses ( 3), ( 4),( 6), ( 7) and ( 7 ′ ) hold. Let ( (.)) ∈ 2 (0, ; ℝ ), and ( (.), (.)) ∈ 2 (0, ; ℝ ) × 2 (0, ; ℝ × ) satisfy backward equation of [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF]. For all


Anne Eyraud-Loisel

Quadratic hedging in an incomplete market derived by an influent informed investor

Keywords: Enlargement of filtration ⋅ FBSDE ⋅ quadratic hedging ⋅ risk minimization ⋅ insider trading ⋅ influent investor ⋅ asymmetric information ⋅ AMS Classification (2000): 60H10, 60G44, 60H07, 60J75, 91G20, 91B70, 93E11. JEL Classification: C60, G11, G14 ∃, such that ℙ-a.s. ∀, | (, 1, 1 ) -(, 2

In this paper a model with an influent and informed investor is presented. The studied problem is the point of view of a non informed agent hedging an option in this influenced and informed market. Her lack of information makes the market incomplete to the non informed agent. The obtained results, by means of Malliavin calculus and Clark-Ocone Formula, as well as Filtering Theory are the expressions and a comparison between the strategy of the non informed trader, and the strategy of the informed agent. An expression of the residual risk a non informed trader keeps by detaining an option in this influenced and informed market is derived using a quadratic approach of hedging in incomplete market. Finally, the analysis leads to a measure of the lack of information that makes the incompleteness of the market. The financial interpretation is explained throughout the theoretical analysis, together with an example of such influenced informed model.

Introduction

This article deals with a hedging problem in an incomplete market, where the incompleteness of the market is due to the presence of an influent informed investor. This influent informed agent is supposed to possess an additional information, and is also supposed to influence the market. This is a natural extension of the work of Eyraud-Loisel [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF], where the informed agent is supposed to be a small investor, who does not influence asset prices. [START_REF] Eyraud-Loisel | Option hedging by an influent informed investor[END_REF] [START_REF] Eyraud-Loisel | Option hedging by an influent informed investor[END_REF] studies the case of an influent informed agent, who wants to hedge an option.

The chosen model is the same as in [START_REF] Eyraud-Loisel | Option hedging by an influent informed investor[END_REF], where the market prices process is influenced by the portfolio of an informed agent. It is more realistic to consider large traders in the market, who may influence the evolution of asset prices, either by their large investment depth, or by their notoriety, when a charter phenomenon appears. Then asset prices may be influenced by certain big agents in the market, and it is quite natural to suppose that such large agents may have more easily access to additional information on the market. This is the reason why it is interesting to develop a model with an influent informed investor. The additional information is supposed to be a strong initial information, modeled by an initial enlargement of the Brownian filtration, as developed for insider trader modeling in Grorud and Pontier [START_REF] Grorud | Insider trading in a continuous time market model[END_REF], or Amendinger [START_REF] Amendinger | Initial enlargement of filtrations and additionnal information of financial markets[END_REF]. This influenced market has been already studied in [START_REF] Eyraud-Loisel | Option hedging by an influent informed investor[END_REF] from the influent informed agent's point of view, by solving the FBSDE appearing when modeling the hedging problem of the large trader, under the enlarged filtration induced by the additional information. The present article is devoted to the study of the non-informed agent's strategy in this influenced market. Despite the completeness of the influenced informed market, the non-informed market is incomplete : the common agent does not have access to the entire information driving the prices process.

In Section 2, we present the model and quickly recall the results of [START_REF] Eyraud-Loisel | Option hedging by an influent informed investor[END_REF] about the hedging strategy of the influent informed agent, and the completeness of the influenced informed market.

In Section 3 we study the incompleteness of the market from a non informed trader's point of view. As a consequence of the influence of the large informed trader, the filtration generated by prices reflect a part of the additional information, and is not the Brownian filtration anymore. In order to compare both strategies, of the informed influent agent, and of the small common agent, we use a quadratic hedging approach in this incomplete market. As there is no predictable representation property in this market, we use the local risk minimization approach, introduced by Föllmer and Sonderman [START_REF] Föllmer | Hedging of nonredundant contingent claims[END_REF] in the case where prices are martingales, and developed by Schweizer [START_REF] Schweizer | Option hedging for semimartingales[END_REF] when prices are only semi-martingales, and Föllmer and Schweizer [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF] in terms of minimal martingale measure. This approach shows the existence of a unique risk-minimizing strategy, which may be expressed using the Galtchouk-Kunita-Watanabe decomposition (as developed in Ansel and Stricker [START_REF] Ansel | Décomposition de Kunita-Watanabe[END_REF]). Another possible approach would be the mean-variance hedging method, developed by Gouriéroux, Laurent and Pham [START_REF] Gourieroux | Mean-variance hedging and numéraire[END_REF], and Rheinlander and Schweizer [START_REF] Rheinländer | On 2 -projections on a space of stochastic integrals[END_REF], who obtained an expression of the optimal strategy in terms of variance-optimal martingale measure. A very clear review of the subject and of these two quadratic hedging methods in a continuous market is provided in Pham (2000) [START_REF] Pham | On quadratic hedging in continuous time[END_REF].

We have chosen the local risk-minimization approach developed in Föllmer and [START_REF] Schweizer | Option hedging for semimartingales[END_REF] [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF], as our model fits into their global idea : considering an incomplete market where the incompleteness is due to a lack of information. In our model, when adding the additional information, the market is complete. We compare the hedging strategy of a non informed agent to the strategy of the informed influent agent, in an incomplete market, under Hypothesis (H 3 ), which differs from classical hedging hypotheses studied in Föllmer and Schweizer [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF]. We obtain an expression of the optimal strategy in terms of projection of the strategy of the informed influent agent, with respect of the filtration generated by prices (Filtering Theory is used). Moreover, we give a version of the Clark-Ocone formula in our framework of enlarged filtration, as well as an expression of the informed agent strategy in terms of Malliavin derivatives. We derive also an expression of the residual risk linked to the lack of information of the non informed agent. In the last section, we give an example of information and influence that satisfies all hypotheses of this model, and give the expressions of the strategies and residual risk of the lack of information of the non informed trader derived in the previous section and applied in this case.

Model

We recall here the model introduced in Eyraud-Loisel [START_REF] Eyraud-Loisel | Option hedging by an influent informed investor[END_REF], where an informed and influent agent wants to hedge against a given contingent claim. The market is supposed to be driven by a multidimensional Brownian motion. The wealth process and investment strategy of the influent agent are expressed as the solution of a backward stochastic differential equation. This investor may influence asset prices either by his wealth , or by his portfolio strategy , which may influence the drift of the volatility of prices. The BSDE driving the wealth process and the investment strategy, modeling the hedging problem, is then fully coupled with the forward equation of prices. The equation to be solved for the hedging problem is the following forwardbackward stochastic differential equation (FBSDE):

{ = 0 + ∫ 0 ( , , , ) + ∫ 0 < ( , , , ), > = - ∫ ( , , , ) - ∫ < , >, (2.1)
where ( , , , ) = ( , , , ), ( , , , ) = ′ ( , , , ), and = ′ * ( , , , ) . As the influent agent has an additional information on the market, this FBSDE has to be solved under an initial enlargement of filtration of the Brownian (A8) is also supposed to be globally Lipschitz w.r.t. :

∃ 9 such that ℙ-a.s. ∀ , , , | ( , 1 , , ) -( , 2 , , )| 2 ≤ 2 9 | 1 -2 | 2 .
We suppose also that one of the 3 following cases ( 1) to ( 3) is satisfied :

(I1)
The influence is weak : the forward and backward equations are weakly coupled: ∃ 0 > 0 depending on 3 , 4 , 5 , 8 , 1 , 2 and such that 1 , 2 , 6 , 7 ∈ [0, 0 ). (I2) , ℱ -measurable, is independent from the price process, and 1 and 2 from hypothesis (A3) satisfy also:

∃ > 0, = 1, 2, 3, 4, 4 < -1 4 , > 0 such that 1 + 2 < - 1 2 [ 3 3 ( 2 2 + 2 7 1 -4 4 + 1 1 + 2 6 ) + 1 1 + 2 2 + 3 3 + 4 4 + 2 5 +
] .

(I3) is independent from : the portfolio does not influence the volatility of prices, and 1 and 2 from hypothesis (A3) satisfy also:

∃ > 0, = 1, 3, 4, 4 < -1 4 , > 0, > 0 such that 1 + 2 < - 1 2 [ (1 + ) [ 1 1 + 2 6 + 2 2
(1 -4 4 )

] (

2 8 + 3 3
)

1 1 + 3 3 + 4 4 + 2 5 +
] .

Under these hypotheses, Eyraud-Loisel [START_REF] Eyraud-Loisel | EDSR et EDSPR avec grossissement de filtration, problèmes d'asymétrie d'information et de couverture sur les marchés financiers[END_REF][START_REF] Eyraud-Loisel | Option hedging by an influent informed investor[END_REF] derived an existence and uniqueness theorem of such FBSDE under an initial enlargement of filtration satisfying Hypothesis (H 3 ) :

Theorem 2.2 (Theorem 1 of [START_REF] Eyraud-Loisel | Option hedging by an influent informed investor[END_REF]) Under Hypotheses (H 3 ), ( 1) to ( 8), and under one of the cases of influence ( 1) to ( 3), the following forward-backward stochastic differential equation:

{ = 0 + ∫ 0 ( , , , ) + ∫ 0 < ( , , , ), > = - ∫ ( , , , ) - ∫ < , > (2.2)
admits in the space ( , , ) a unique -adapted solution ( , , ), such that

ℚ ∫ 0 ( sup 0≤ ≤ | | 2 + sup 0≤ ≤ | | 2 + ∥ ∥ 2 ) < +∞. (2.3)
The FBSDE has a unique adapted solution in the enlarged space. This means that the influent agent has a unique hedging strategy adapted to his information. As volatility is supposed to be invertible, it is possible to derive from the unique portfolio hedging the pay-off . If the solution of the insider trader is adapted to the Brownian filtration, then it is the same as if there was no additional information. Any contingent claim in 2 ( , , ℚ) is attainable. The market is complete for the informed investor, w.r.t. to the enlarged filtration. The hedging problem in the market is reduced to a resolution of a FBSDE, whose coupling depends on the influence of the insider in the market. The results from the insider's point of view are the same as in [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF], with no influence.

What is different in the model with an influent agent, is the behavior of the market from the point of view of a normally informed agent (for whom the additional information is unknown). The market is incomplete for the non informed trader. This incompleteness is due to his lack of information, as in [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF] [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF]. The present model is an example of complete market which becomes incomplete from a small non informed investor point of view. The study of such a market uses tools of quadratic hedging in incomplete market, or under incomplete information, and it is the aim of this paper.

3 Quadratic hedging in this incomplete market

Introduction of a non informed agent

In this model, the market prices are influenced by the insider's investment process. We suppose from now on that hypotheses of Theorem 2.2 are satisfied. We consider a non informed agent, who wants to hedge against an option in this market. This non informed agent is supposed to be a small investor, which means that she can not influence asset prices. This can be viewed as a problem of hedging in partial information, or lack of information : the non informed agent can only observe prices. The historic of prices of the risky assets is the only available information. The non informed agent knows the filtration generated by prices :

F := ℱ := ( , 0 ≤ ≤ )
As the coefficients in the diffusion of prices are adapted to the enlarged filtration, due to the influence of the informed agent, even the Brownian filtration is not available to the small trader. Other processes that appear in the BSDE of the hedging problem of the non informed trader, in particular Brownian motion, are not directly observable on the market. Her portfolio strategy and wealth processes have to be F-adapted. As the investment strategies of the informed and non informed traders are adapted to different filtrations, their sets of admissible strategies differ. Even if the insider market is complete, it may not be complete from the non informed trader's point of view. If the non informed trader has a hedging strategy, it may be different from the strategy of the influent informed trader, whose exact replicating strategy is -adapted. Completeness of a market (defined as "any square integrable contingent claim is attainable") depends on the considered filtration and probability measure : everything depends on the existence f a martingale representation Theorem. In our context, under Hypothesis ( 3 ), there exists a martingale representation Theorem in the enlarged space , under probability ℚ, and in the Brownian filtration ℱ under probability ℙ, but not under filtration F (see Jacod and Shiryaev [START_REF] Jacod | Limit theorems for stochastic processes[END_REF] Eyraud-Loisel [START_REF] Eyraud-Loisel | Backward stochastic differential equations with enlarged filtration. option hedging of an insider trader in a financial market with jumps[END_REF]). The problem becomes a hedging problem in an incomplete market. We will use a quadratic local risk-minimization approach for this study.

We first study in Subsection 3.2 Proposition 3.5 the Kunita-Watanabe decomposition of the contingent claim in the prices filtration. In Subsection 3.3 a Clark-Ocone formula is obtained in the enlarged space, from which we derive an expression of the integrand of the representation Theorem in this space. Finally we give an expression of the residual quadratic risk linked to the lack of information of the non informed trader, successively under two families of risk-neutral probabilities : the risk-neutral probabilities from the informed trader's point of view in Paragraph 3.4.1, and the risk-neutral probabilities from the non informed trader's point of view in Paragraph 3.4.2.

Hedging strategy and Kunita-Watanabe decomposition

The different considered -fields are linked by the following relations :

F ⊂ , and ℱ ⊂ = ∩ > ℱ ∨ ( ) = ∩ > F ∨ ( ).
To keep notations as light as possible, we consider only one risky asset here, but all results may be generalized to the -dimensional case. The dynamics of the risky asset satisfies the following forward equation :

= 0 + ∫ 0 ′ ( , , , ) + ∫ 0 ′ ( , , , ) (3.1)
where , and are respectively the price of the risky asset, and the wealth and hedging strategy of the informed agent, deduced from the unique solution of FBSDE (2.1) in the enlarged space (Theorem 2.2). The non informed trader wants to hedge against a European contingent claim = ℎ( ) ∈ ℒ 2 ( , F, ℚ). This contingent claim is not necessarily the same as the contingent claim replicated by the informed trader. This point of view may be used to compare investments strategies of two small agents, one who has access to the additional information and the other who can only observe prices and has just a part of the total information. If the same contingent claim is considered, it may lead to a comparison of the strategies of both agents, the first one being influent and informed and the other one not influent and not informed.

Under ℚ, probability measure defined in the previous section (Definition 2.1) Hypothesis (H 3 ) is satisfied, is a -Brownian motion, and is a ( , )-semimartingale. As is F-adapted, it is also a ( F, ℚ)-semimartingale. The market is not complete: as is attainable in ( , ℚ), the replicating strategy is adapted to filtration , and so is not admissible for the non informed trader in the smaller space. We will develop quadratic hedging arguments in order to find the hedging strategy of the non informed trader that minimize the intrinsic residual risk of the contingent claim, risk linked to the lack of information on this influenced informed market.

Denote by the set of all equivalent martingale measures under filtration , i.e. the set of all probability measures equivalent to ℚ (and so equivalent to ℙ) under which is a -martingale. Financially speaking it is the set of all risk-neutral probability measures from the informed trader's point of view. Denote also by the set of all equivalent martingale measures under filtration F, equivalent to ℙ, i.e. the set of all probability measures equivalent to ℙ (and so equivalent to ℚ) under which is a F-martingale. It is the set of all risk-neutral probability measures from the non informed trader's point of view. We have clearly ⊂ .

Indeed, intuitively, from her lack of information, the non informed agent will have a larger range of risk-neutral probability measures. Some of the riskneutral probability measures will appear to be risk-neutral from the non informed agent's point of view, whereas with the additional information, they will appear not to be risk-neutral. Conversely, any risk-neutral probability measure for the informed trader is always also risk-neutral for the non informed trader: if is a -martingale, as it is F-adapted, it is a F-martingale. Moreover, is not empty: from Assumption ( 1), ′ is invertible and ℰ(-′-1 ( ′ -) ) is integrable, which are classical arguments implying no arbitrage opportunities (see Delbaen and Schachermayer [START_REF] Delbaen | A general version of the fundamental theorem of asset pricing[END_REF]).

Recall the definition of orthogonality for two martingales (see Protter [START_REF] Protter | Stochastic integration and differential equations, A new approach[END_REF], and Pham [START_REF] Pham | On quadratic hedging in continuous time[END_REF]): Definition 3.1 Two martingales , are (strongly) orthogonal if their product is a local martingale with initial value 0 0 = 0. If and ∈ ℳ 2 , this is equivalent to < , >= 0.

We now give the definition of the decomposition of Kunita-Watanabe (see Ansel and Stricker [START_REF] Ansel | Décomposition de Kunita-Watanabe[END_REF] or Pham [START_REF] Pham | On quadratic hedging in continuous time[END_REF]) on a general probability space ( , ℱ, IP) :

Definition 3.
= 0 + ∫ 0 + , IP -. . 0 ≤ ≤ (3.2)
where ∈ 2 ( ) ( ), ∈ ℳ 2 0,( ) (ℱ, IP) (square integrable (ℱ, IP)-local martingale with initial value 0) and orthogonal to in (ℱ, IP). Let Q ∈ be risk-neutral from the informed agent,

Q equivalent to ℚ, under which is a ( , Q)-martingale. As is F-adapted, it is a ( F, Q)- martingale.
∈ ℒ 2 ( , F, Q) is attainable with respect to the largest filtration. The martingale representation Theorem in ( , , ℚ) gives the existence of

∈ 2 ( × [0, ], Q ⊗ ⟨ ⟩) such that = Q( | 0 ) + ∫ 0 = Q( | ( )) + ∫ 0 .
(3.3) Remark 3.4 This decomposition does not depend on the choice of the risk neutral probability Q ∈ from the informed agent's point of view. Indeed, according to Grorud and Pontier [START_REF] Grorud | Probabilités neutres au risque et asymétrie d'information[END_REF], if Q1 and Q2 are two risk neutral probabilities in , then Q1 = ( ) Q2 , where ( ) is a nonnegative random variable, with mean 1 under Q2 . One can then write :

Q1 ( | ( )) = Q2 ( ( ) | ( )) Q2 ( ( )| ( )) = ( ) Q2 ( | ( )) ( ) = Q2 ( | ( )) .
Hence, Q( | ( )) does not depend on the choice of Q risk neutral probability in . Then, does not either depend on the considered risk-neutral probability measure in .

represents the contingent claim hedged by the informed agent, and = the unique -adapted hedging strategy of the informed agent (see Eyraud-Loisel [START_REF] Eyraud-Loisel | Option hedging by an influent informed investor[END_REF]). In this continuous framework, all martingales are locally square integrable. So the unique Kunita-Watanabe decomposition of w.r.t. the ( F, Q)-martingale , for any Q ∈ , is the following:

= Q( ) + ∫ 0 Q + Q . ( 3.4)
Moreover,

:= Q( | F ) = Q( ) + ∫ 0 Q + Q , (3.5)
where Q is F -measurable and belongs to 2 ( × [0, ], Q ⊗ ⟨ ⟩) and where the rest ( Q ) is a ( F, Q)-local martingale orthogonal to the stable space generated by , such that Q( Q ) = 0.

We can derive the expression of the integrand (see [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF]):

Q = < , > < > .
This expression is not easily exploitable in our case. By using filtering Theory, we derive another expression in terms of projection on the space of all Fadapted processes under any risk neutral probability measure Q ∈ .

Theorem 3.5 Under any equivalent martingale measure Q ∈ , the Kunita-Watanabe decomposition of w.r.t. ( F, Q) and process is the following:

= Q[ ] + ∫ 0 Q ( | F ) + Q a.s. ,
or in other terms First of all, (

Q = Q ( | F ) . Proof If Q( | F )
Q( | F ) ) 0≤ ≤
is F-adapted. Define :

= Q( | F ) -Q( ) - ∫ 0 Q( | F ) .
It is sufficient to prove that is a ( F, Q)-martingale with mean 0, orthogonal to .

is indeed a ( F, Q)-martingale:

is a ( F, Q)-martingale, so ∫ 0 Q( | F ) is a ( F, Q)-martingale as integral of a F-adapted square integrable process against a ( F, Q)-martingale, and Q( | F ) is a ( F, Q)- martingale. Moreover, Q( ) = Q( ) -Q( ) -Q (∫ 0 Q( | F ) ) = 0
the last term being the expectation of a ( F, Q)-martingale null in 0.

It remains to prove that is orthogonal to all martingales generated by , stable space generated by , denoted by ℳ( ). Using decomposition (3.3) of in ( , Q), we obtain:

= Q ( Q( | ) + ∫ 0 F ) -Q( ) - ∫ 0 Q( | F ) = Q [ ∫ 0 | F ] - ∫ 0 Q [ | F ] + Q ( Q( | ( ))| F ) -Q( ) , (3.6)
where the last term is F -measurable. The following lemma is a classical result of Filtering Theory (see for example the Saint Flour summer school in Probability Theory course by Pardoux in 1989 on non linear filtering theory [START_REF] Pardoux | Filtrage non linéaire et équations aux dérivées partielles stochastiques associées[END_REF]). Lemma 3.6 (Pardoux (1989) [START_REF] Pardoux | Filtrage non linéaire et équations aux dérivées partielles stochastiques associées[END_REF]) Let be a (ℱ, Q)-martingale, ∈ 2 ( ) on the filtered probability space ( , , Q), such that ℱ satisfied ℱ ⊂ ∀ ∈ [0, ]. Then we have:

Q ( ∫ 0 ℱ ) = ∫ 0 Q ( | ℱ ) .
As a consequence of the lemma, we can write

Q (∫ 0 | F ) = ∫ 0 Q( | F )
Using the law of iterated expectations, together with the previous lemma, we can simplify the expression of in Equation (3.6):

= Q [ ∫ 0 | F ] - ∫ 0 Q( | F ) + = Q [ Q [ ∫ 0 F ] F ] - ∫ 0 Q( | F ) + = Q [ ∫ 0 Q ( F ) F ] - ∫ 0 Q( | F ) + = ∫ 0 Q ( F ) - ∫ 0 Q ( F ) + = . (3.7)
The third equality is obtained because

∫ . 0 Q ( F ) is a ( F , Q)-martingale.
Let ( ) be a F-adapted bounded process. Equation (3.7) leads to:

Q ( ∫ 0 ) = Q ( ∫ 0 ) .
Recall that may be written as:

= Q ⎛ ⎜ ⎜ ⎝ Q( | ( )) ( ) F ⎞ ⎟ ⎟ ⎠ -Q( ).
As Q( | ( )) is measurable with respect to the -algebra generated by , it may be written as ( ), where is a measurable function. Hence :

Q [ ∫ 0 ] = Q [ [ Q ( ( )| F ) -Q( ) ] ∫ 0 ] = Q ( Q ( ( ) ∫ 0 F )) -Q( ) Q (∫ 0 ) = Q ( ( ) ∫ 0 ) .
Literature on initial enlargement of filtration (see Grorud and Pontier [START_REF] Grorud | Probabilités neutres au risque et asymétrie d'information[END_REF]) has already established that if there are two risk neutral probability measures ℚ 1 and ℚ 2 , then there exists measurable such that ℚ 2 = ( )ℚ 1 , where ( ) is a 0 -measurable nonnegative random variable, with expectation 1 under ℚ 1 , and conversely any such probability measure is risk neutral. For this reason, the previous computations do not depend on the choice of the risk neutral probability measure Q ∈ . Then let ℚ * be the probability measure defined as followed:

ℚ * = ( ) Q( ( )) Q ∈ .
As ℚ * is a risk neutral probability measure, from the previous argument, ∫ 0 is a F-martingale under ℚ * . We get

0 = ℚ * (∫ 0 ) = 1 Q( ( )) Q ( ( ) ∫ 0 ) .
We deduce

Q ( ( ) ∫ 0 ) = 0
and we have finally

Q [ ∫ 0 ] = 0, ∀ F -measurable.
Under any equivalent martingale measure Q ∈ , we have the expected orthogonality. This proves that the Kunita-Watanabe decomposition of has integrand Q ( | F ) under any risk-neutral probability measure Q in , which ends the proof of Theorem 3.5. □ Remark 3.7 As this integrand determines the unique Kunita-Watanabe decomposition, and as it is the same under any risk-neutral probability measure of , then the orthogonal rest is the same and the quadratic residual risk is the same : it may be interpreted as the intrinsic risk of the contingent claim .

Remark 3.8 This also proves that Q( | F ) does not depend on the choice of Q ∈ . This can also be proved by the following way : Let ∈ F , 2 = ( ) 1 , from the previous argument, it follows that

1 [ 2 [ / F ] 1 [ ( )/ F ]] = 1 [ ( )].
The main consequence of the previous Theorem is then: Proposition 3.9 Prices are the same under two different risk-neutral probability measures.

Remark 3.10 We obtain here a result which is coherent with the result of Grorud and Pontier [START_REF] Grorud | Probabilités neutres au risque et asymétrie d'information[END_REF] in a different framework, according to whom all riskneutral probability measures in an incomplete market (where incompleteness is due to additional information) give the same market price.

Remark 3.11 Our study is also coherent with Föllmer and Schweizer's work in [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF], but several points differ: their hypothesis of invariance of the decomposition of Doob-Meyer of the prices does not remain valid in our case. They suppose that the decomposition of Doob-Meyer of the prices under filtration F and probability ℚ, = 0 + + where is F-predictable, and where is a ( F, ℚ)-martingale, stays the decomposition of Doob-Meyer of the prices in the larger filtration, in other words, remain a ( , ℚ)-martingale, although adapted to F (See Hypotheses (4.1) to (4.3) in [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF]). These hypotheses, although weaker, are similar to the so-called Hypothesis (H) in Filtering Theory (used also in progressive enlargement of filtration, in models as those of Jeanblanc [START_REF] Jeanblanc | Default risk and hazard process[END_REF][START_REF] Blanchet-Scalliet | Hazard rate for credit risk and hedging defaultable contingent claims[END_REF]). Instead of such hypothesis, our study takes place under Hypothesis (H 3 ). Furthermore, we distinguish here three different filtrations: the filtration of the Brownian motion, the filtration of prices, and the filtration enlarged with the information. On the other hand, as pointed out before, our study joins perfectly within the framework of incompleteness that they studied: the incompleteness due to a lack of information.

Finally, our results are coherent with those in previous literature, whereas the model is a bit different. The present study illustrates in a different framework the same kind of incompleteness already studied in this literature: incompleteness due to a lack of information.

Clark-Ocone Formula

Since

Q may be expressed from , it is important to study how to obtain . We express it as a function of the Malliavin derivative of , via the use of a Clark-Ocone Formula. We can not use directly the standard formula in our case, under enlarged filtration , as 0 = ( ) is not trivial. Nevertheless, thanks to a transformation and the independence of ( ) and ℱ under ℚ, we will use the existing Clark-Ocone Formula under filtration ℱ, and obtain an expression of .

Theorem 3.12 If ∈ 2 ( , , ) and ∀ , (., ) ∈ 1,2 , we have

= ( ℚ [ ( (., ))|ℱ ]| = ) ( ) -1 . (3.8)
Proof In the enlarged space , under (H 3 ), ( ) and ℱ are independent. Moreover, ℚ| ℱ = ℙ and ℚ| ( ) = ℙ (law of ). is measurable w.r.t. , so may be written as = ( , ) (thanks to the independence and because is generated by ℱ and ( )). Then, at fixed = , ( , ) is ℱ -measurable. So its representation is given by the standard Clark-Ocone Formula:

( , ) = ℙ ( ( , )) + ∫ 0 ℙ ( ( ( , ))|ℱ ) ℙ . . = ℚ ( | ) | = + ∫ 0 ℚ ( ( ( , ))|ℱ ) .
Indeed, derive w.r.t. with = has a sense under ℚ as and are independent, and derive → ( , ) under ℚ or ℙ is the same, as is a ℙand also a ℚ-Brownian motion. Again using independence between ( ) and ℱ , we can write ( , ) = ( , )| = and an identification leads to:

= ℚ [ ( ( , ))|ℱ ]| = . (3.9)
Equation (3.8) directly comes from Equation (3.9) and the fact that is invertible, which ends the proof of Theorem 3.12. □

Residual Risk

Expression of the residual risk under a risk neutral probability in

We derivean expression of the intrinsic residual risk of the contingent claim under a risk neutral probability in .

Proposition 3.13 Let ℚ * ∈ . The residual risk ℚ * has the following expression:

ℚ * = ∫ 0 [ ℚ * ( | F ) -ℚ * ] + ℚ * [ ℚ * ( | ( ))| F ] -ℚ * ( ). (3.10)
A measure of the risk of detaining the contingent claim in this incomplete market is given by the variance of ℚ * at terminal time:

ℚ * ( ℚ * ) = ℚ * ( ∫ 0 [ ℚ * ( | F ) -ℚ * ] × ( ℚ * [ ℚ * ( | ( ))| F ] -ℚ * ( ) ) ) + ℚ * ( ( ℚ * [ ℚ * ( | ( ))| F ] -ℚ * ( ) ) 2 ) . (3.11)
Proof Let ℚ * ∈ a risk neutral probability measure for the non informed agent. The expression of the residual risk ℚ * at time is given by Equations (3.4) and (3.5), which are true under any ℚ * ∈ :

ℚ * = ℚ * [ ∫ 0 | F ] - ∫ 0 ℚ * + ℚ * [ ℚ * ( | ( ))| F ] -ℚ * ( ).
The second part of this expression corresponds to the difference of prices of the contingent claim under ℚ * , whether the agent has or does not have the additional information, which is the difference of prices on the informed (complete) market and the non informed (incomplete) market. In the first case, the price will be ℚ * ( | ( )) projected on the F -measurable random variables (which means attainable prices), and in the other case, the price is ℚ * ( ).

The first part may be rewritten thanks to Lemma 3.6:

ℚ * [ ∫ 0 | F ] = ℚ * [ ℚ * ( ∫ 0 | F ) | F ] = ℚ * [ ∫ 0 ℚ * ( | F ) | F ] = ∫ 0 ℚ * ( | F ) ,
as is a ( F, ℚ * )-martingale.

The residual risk may be rewritten as:

ℚ * = ∫ 0 [ ℚ * ( | F ) -ℚ * ] + ℚ * [ ℚ * ( | ( ))| F ] -ℚ * ( ).
The sum of these two terms forms the residual risk due to the lack of information of the non informed agent. This risk is the minimal risk under ℚ * which the non informed agent can hope to take by holding the option (see Föllmer and Schweizer [START_REF] Föllmer | Hedging of contingent claims under incomplete information[END_REF] and Pham [START_REF] Pham | On quadratic hedging in continuous time[END_REF]). It is the constituent of the option which is orthogonal to the market prices (see Kunita-Watanabe decomposition). A measure of this risk, represented by ℚ * , martingale with null expectation, is its variance ℚ * ( ℚ * ), which represents the quadratic residual risk at terminal time .

At terminal time, we can write:

ℚ * = ∫ 0 [ ℚ * ( | F ) -ℚ * ] + ℚ * [ ℚ * ( | ( ))| F ] -ℚ * ( ). So ℚ * ( ℚ * ) = ℚ * ⎛ ⎝ ( ∫ 0 [ ℚ * ( | F ) -ℚ * ] ) 2 ⎞ ⎠ +2 ℚ * ( ∫ 0 [ ℚ * ( | F ) -ℚ * ] × ( ℚ * [ ℚ * ( | ( ))| F ] -ℚ * ( ) ) ) + ℚ * ( ( ℚ * [ ℚ * ( | ( ))| F ] -ℚ * ( ) ) 2 ) .
As ℚ * is orthogonal to , this expression may be slightly simplified, and the variance may be rewritten as:

ℚ * ( ℚ * ) = ℚ * ( ∫ 0 [ ℚ * ( | F ) -ℚ * ] × ( ℚ * [ ℚ * ( | ( ))| F ] -ℚ * ( ) ) ) + ℚ * ( ( ℚ * [ ℚ * ( | ( ))| F ] -ℚ * ( ) ) 2 ) .
This is the general expression of the quadratic residual risk under a probability measure in . □

In the case of a measure in but not in , we can not simplify better the previous expression. Nevertheless, with orthogonality arguments we obtain a simpler expression. Proposition 3.14 A measure of the residual risk under a risk neutral probability measure ℚ * ∈ ∖ is given by:

ℚ * ( ℚ * ) = ℚ * ( ( -ℚ * ( )) 2 ) -ℚ * ⎛ ⎝ ( ∫ 0 ℚ * ) 2 ⎞ ⎠ . (3.12)
Proof Let ℚ * ∈ be a probability measure that does not belong to . Then the Kunita-Watanabe decomposition may be written as follows:

-ℚ * ( ) = ∫ 0 ℚ * + ℚ * ,
being a ( F, ℚ * )-martingale, but not a ( , ℚ * )-martingale (otherwise ℚ * would belong to ). There is no martingale representation Theorem. As ℚ * is orthogonal to the price process, we get Equation (3.12) and the expected expression of the quadratic residual risk. □

Expression of the residual risk under a risk neutral probability in

In the case of a measure in ⊂ , Equations (3.10) and (3.12) of Proposition ?? may be simplified. We obtain the following result: Proposition 3.15 A measure of the residual risk under a risk neutral probability measure ℚ * in is: ) measures the revelation of the information in the prices, because the not informed agent does not possess the whole information, but she "sees" all the same a part of the information showing through in market prices. A minimal risk exists, which is taken over all possible equivalent martingale measures in and it is the following:

( ℚ * ) = ℚ * ( ℚ * [ ℚ * ( | ( ))| F ] 2 ) -ℚ * ( )
inf * ∈ { ℚ * ( ℚ * [ ℚ * ( | ( ))| F ] 2 ) -ℚ * ( ) 2 } .
It can be approached by a minimizing sequence.

Remark 3.17 We can notice that measuring the residual risk under a riskneutral probability is arbitrary, because choosing this equivalent martingale measure is already an arbitrary choice, and also because at first, the model took into account a historic probability, which we do not take into account in this measure of risk. Moreover the estimated risk is the risk of this model under this measure, and not the intrinsic risk.

Existence of a minimal martingale measure

Let us write the Doob-Meyer decomposition of the ( F, ℚ)-semi-martingale of prices :

= 0 + + ,
where is a ( F, ℚ)-martingale of null expectation (as a uniformly integrable local martingale, from Assumption ( 1), see Lépingle and Mémin [START_REF] Lépingle | Sur l'intégrabilité uniforme des martingales exponentielles[END_REF] Theorems II-2 and III-7), and is a F-predictable finite-variations process, whose increments are square integrable under ℚ and satisfy 0 = 0. By Girsanov Theorem, and as Assumption ( 1) ensures the existence of an equivalent martingale measure, may be written as follows:

= ∫ 0 < > , 0 ≤ ≤ , 4
Example Let us give an example satisfying all hypotheses of the previous model. This example was first introduced in the last section of [START_REF] Eyraud-Loisel | Option hedging by an influent informed investor[END_REF]. Suppose that the price process is driven by the following dynamics (stochastic volatility model) :

= ′ ( , , ) + ′ ( ) , (4.1)
where

′ ( ) = 0 ℐ [0, [ ( ) + 1 ℐ [ , ] ( ) , 0 , 1 ∕ = 0. (4.2)
The volatility of this model is piecewise constant, taking two possible values 0 and 1 fixed by the model, is a random variable satisfying Hypothesis (H 3 ), taking its values in [0, + ].

The information of the insider trader is the following strong initial information: = , ℱ + -measurable . The drift parameter is chosen as the following:

′ ( , , ) = 0 + 1 (1 + )(1 + 2 ) , 0 , 1 ∈ ℝ fixed.
Interest rate is supposed to be constant. Drift ′ is bounded, and may vary between two thresholds 0 and 0 + . Two cases may appear, depending on the sign of 1 . If 1 < 0, the influence is a positive influence: the bigger is the investment portfolio, the higher is the drift of the prices. This is moderated by the level of prices: the higher are the prices, the lower is the influence. If 1 > 0, it is the converse principle: when the level of the portfolio increases, the drift of the prices decreases, and the influence is stronger when the level of prices is high. Remark that the case 1 = 0 is the case treated in the previous section, without influence. Depending on the sign of 1 , representing the amplitude of the influence, this influence will have either a leverage effect or a return effect on the drift of the price process around the value 0 . The influence is from the insider's portfolio on the price process, which remains bounded according to the hypotheses. We can also notice that

′-1 ( ′ -) = ′-1 ( 0 -+ 1 (1 + )(1 + 2 ) )
is bounded, as well as ′ . So there exists a risk-neutral probability measure Q under which = ′ ˜ is a positive uniformly integrable martingale.

Remark 4.1 We don't have here constraints on the signs of 0 or 1 , whereas it is often the case in previous influence models developed in the literature, such as in the model introduced by Cuoco and Cvitanic (1998) [START_REF] Cuoco | Optimal consumption choices for a "large" investor[END_REF], and treated deeply in [START_REF] Grorud | Fiancial market model with influential informed investors[END_REF] [21] (their influence form is slightly different from the one treated in this work). This may be explained by the fact that we consider a hedging problem, whereas they considered an optimization problem, and therefore we do not need the convexity of the parameters here.

For the present model, considering the hedging of a European call option with maturity and strike , the parameters are the following :

( , , , ) = +

( 0 -+ 1 (1 + )(1 + 2 )
) , ( ) = ( -) + .

This leads to the following FBSDE modeling the hedging problem of the informed agent :

{ = 0 + ∫ 0 ( 0 + 1 (1+ )(1+ 2 ) ) + ∫ 0 ( ) = ( -) + - ∫ ( + ( 0 + ℎ( )) - ) - ∫ ( ) .
According to [START_REF] Eyraud-Loisel | Option hedging by an influent informed investor[END_REF], hypotheses of Theorem 2.2 are satisfied, and the influent informed agent has a unique hedging strategy in this market. As explained before, the informed market is complete, whereas the non informed market is incomplete. From Subsection 3.3, we derive an expression of the strategy of the informed agent:

= ℚ [ ( ( , ))|ℱ ]| = .
In the case of a European call option ( , ) = ( -) + , where is the price at terminal time when the volatility jump appeared at time ,

= ℚ [1l > |ℱ ] = .
We obtain an expression of the optimal hedging strategy of a non informed agent in the market, thanks to results of Section 3 (Equation (3.8)):

ℚ * = ℚ * [ ( ) -1 ℚ [1l > |ℱ ] = F ] .
We derive also the expression of the quadratic residual risk from Equation (3.13):

( ℚ * ) = ℚ * ( ℚ * [ ℚ * (( -) + | ) F ] 2 ) -ℚ * (( -) + ) 2 .
Remark 4.2 We have in this example

( ) 2 = 1 2 < > .
So ( ) 2 is F-measurable, hence observable.

Conclusion

As a consequence of the last remark, imagine a weak informed agent, who only knows the law of (such information is commonly called weak information). Then, as soon as the agent observes the jump in the volatility process, she has the same information as the strong informed agent. This would mean intuitively that in this example, the weak information would be enough to complete the market, because it would be sufficient to hedge against this contingent claim. A possible continuation of this model would be the study of BSDE or FBSDE with other types of additional information (weak, progressive), and thus under other types of enlargement of filtration. Other techniques were proposed to study the presence of asymmetry of information, as Malliavin calculus (Imkeller [22], Nualart et al. [START_REF] Leon | An anticipating calculus approach to the utility maximization of an insider[END_REF]), either models of weak information (Baudoin [START_REF] Baudoin | Modeling anticipations on financial markets[END_REF]) or models of punters (Corcuera, Imkeller, Kohatsu-Higa and Nualart [START_REF] Corcuera | Additional utility of insiders with imperfect dynamical information[END_REF]).

Several studies have been developed on discretization schemes for BSDE and FBSDE, from [START_REF] Chevance | Résolution numériques des équations différentielles stochastiques rétrogrades[END_REF] [START_REF] Chevance | Résolution numériques des équations différentielles stochastiques rétrogrades[END_REF], to [START_REF] Delarue | On the existence and uniqueness of solutions to fbsdes in a non-degenerate case[END_REF] [START_REF] Delarue | On the existence and uniqueness of solutions to fbsdes in a non-degenerate case[END_REF], Gobet, Lemor and Warin (2005) [START_REF] Gobet | A regression-based monte-carlo method to solve backward stochastic differential equations[END_REF], or more recently Bouchard and Elie (2008) [START_REF] Bouchard | Discrete time approximation of decoupled forward-backward sde with jumps[END_REF], or Bouchard, Elie and Touzi (2009) [START_REF] Bouchard | Discrete-time approximation of bsdes and probabilistic schemes for fully nonlinear pdes[END_REF]. Even if these new schemes may open a way to simulate and use more extensively such study, and even if these new works and their future extensions give interesting tracks for efficient schemes for FBSDEs, there is still a difficulty that leaves in the use of such equations. The main difficulty is to express explicitly the solutions, especially when the filtration is not easy to express as an enlarged filtration of the Brownian filtration. Since then, the risk minimization and the quadratic hedging approaches allows to have all the same an expression of the hedging strategy, and of the minimal risk a non informed trader is taking by detaining the contingent claim.

Remark 3 . 3

 33 Decomposition (3.2) is unique in the sense that if ( 0 , , ) et ( ˜ 0 , ˜ , ˜ ) satisfying the Kunita-Watanabe hypotheses, then 0 = ˜ 0 , -a.s. ∀0 ≤ ≤ . Remark also that if and are in ℳ 2 (ℱ, IP), then ∈ 2 ( ) and ∈ ℳ 2 0 (ℱ, IP).

  satisfies hypothesis of Kunita-Watanabe decomposition given in Definition 3.2, by uniqueness of this decomposition, it is Q .

  Proof In the case of a risk neutral probability ℚ * in , the first term of ℚ * in Equation (3.10) nullifies, in application of Proposition 3.5. The expression becomes simpler, and we deduce Equation (3.13), which gives a measure of the residual risk under a risk neutral probability ℚ * ∈ . □

		2 ,	(3.13)
	Remark 3.16	( ℚ *

where is a F-predictable process. Define

and denote by ˆ the process solution of the following SDE:

Then:

ˆ is a positive uniformly integrable ( F, ℚ)-local martingale, so it is a ( F, ℚ)martingale. This process defines a probability measure

As is a continuous process and ˆ ∈ ℳ 2 (ℚ), Proposition 4.3 of Pham [START_REF] Pham | On quadratic hedging in continuous time[END_REF] may apply, and we establish the following result in our case: Proposition 3.18 Probability measure Q is an equivalent martingale measure, called minimal martingale measure. It satisfies the following property: any square integrable ( F, ℚ)-martingale orthogonal to under ℚ remains a ( F, Q)-martingale:

Also following Pham [START_REF] Pham | On quadratic hedging in continuous time[END_REF] (Paragraph 4.3 Theorem 4.2), we can mention the following result, which gives an expression of the strategy minimizing the local risk, in the case where is uniformly bounded: Proposition 3.19 If is uniformly bounded, there exists a strategy minimizing the local risk, given by:

where is the integrand of the Föllmer-Schweizer decomposition of under ℚ, which is:

Proof It is the Kunita-Watanabe decomposition of the ( F, Q)-martingale * against the ( F, Q)-martingale . Indeed, is a ( F, ℚ)-martingale orthogonal to , and so remains a martingale under Q. □ Remark 3.20 By uniqueness of the Kunita-Watanabe decomposition under Q, we have 0 = Q( ), which is the right term of the Föllmer-Schweizer decomposition.