BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

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stochastic games [5, 8, 9]. BSDE also appear as a powerful tool to give probabilistic formulas for solution of partial differential equations [15, 16, 18, 19].
Communications on Stochastic Analysis Vol. 2, No. 2 (2008) 277-288

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BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS ´ ASSOCIATED WITH LEVY PROCESSES AND PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS MOHAMED EL OTMANI Abstract. In this paper, we deal with a class of backward stochastic differential equations driven by Teugels martingales associated with a L´ evy process (BSDELs). The comparison theorem is obtained. It is also shown that the solution of BSDE provides a viscosity solution of the associated system with partial integro-differential equations.

1. Introduction Backward stochastic differential equations (BSDEs), in the nonlinear case, have been introduced by Pardoux and Peng [17] in 1990. They gave the first existence and uniqueness result under suitable assumptions on the coefficient and the terminal value of the BSDE. Since then, BSDE have gradually become an important mathematical tool in mathematical finance [6, 7], optimal stochastic control and stochastic games [5, 8, 9]. BSDE also appear as a powerful tool to give probabilistic formulas for solution of partial differential equations [15, 16, 18, 19]. On another context, Situ [24] studied BSDEs driven by a Brownian motion and a Poisson point process, Ouknine [14] studied BSDEs driven by a Poisson random measure, Nualart and Schoutens established in [13] the existence and uniqueness of solutions for BSDEs driven by a L´evy process and later Bahlali et al. [1] treated the case where the BSDE is driven by a Brownian motion and the martingales of Teugels associated with an independent L´evy process. The comparison theorem for BSDEs turns out to be one of the classic results of this theory. But is not true in general case (see Remark 2.7 in [2] for a counterexample). We refer the reader to [2, 12, 22, 25] for some particular cases of the comparison theorem for BSDEs with jumps. The main results of this paper are the proof of a comparison theorem for a classes of BSDEs driven by a L´evy process and a representation that identifies any solution of these classes of BSDEs as a viscosity solution of the associated partial integro-differential equations (PIDEs). The paper is organized as follows. In Section 2, we introduce some notations and we discuss a pricing problem by replication in a market controlled by a L´evy process: we verify that it can be written in terms of linear BSDELs. In Section 3, 2000 Mathematics Subject Classification. Primary 60H30; Secondary 60H10. Key words and phrases. Backward stochastic differential equations, L´ evy process, viscosity solution, partial integro-differential equations. 277

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we prove the comparison theorem of BSDELs under some appropriate conditions on the generator f . By means of our comparison theorem, one can show in Section 4 that a certain function defined through the solution of Markovian BSDELs is a viscosity solution of the associated system of P.I-D.Es. 2. Preliminaries Let (Ω, F, P) be a completed probability space on which a real-valued L´evy process (Lt )t∈[0,T ] with c`adl` ag paths is defined. Let F = (Ft )t≥0 be the rightcontinuous filtration generated by L: (Ft = σ{Ls ; s ≤ t}) and assume that F0 contains all P-null sets of F. The process L is characterized by its so-called local characteristics in the L´evy-Khintchine formula. So that Z ¡ iux ¢ σ2 2 iuLt −tΨ(u) Ee =e with Ψ(u) = −ibu + u − e − 1 − iuxI(|x|≤1) ν(dx). 2 R Thus L is characterized by its L´evy triplet (b, σ, ν) where b ∈ R, σ 2 ≥ 0 and ν is a measure defined in R \ {0} and satisfies R (i) R (1 ∧ x2 )ν(dx) < +∞, R (ii) ∃ ε > 0 and λ > 0 as (−ε,ε)c eλ|x| ν(dx) < +∞. This implies that the random variables Lt have moments of all orders, i.e. Z +∞ |x|i ν(dx) < ∞, ∀i ≥ 2. −∞

For background on L´evy processes, we refer the reader to [3, 23]. We denote by Lt− = lims%t Ls and ∆Lt = Lt − Lt− . We define the power jumps of the L´evy process L by X (1) (i) (∆Ls )i , i ≥ 2. Lt = Lt and Lt = 0 0 such that for any y1 , y2 ∈ R, z1 , z2 ∈ `2 and t ∈ [0, T ], |f (t, y1 , z1 ) − f (t, y2 , z2 )| ≤ κ (|y1 − y2 | + kz1 − z2 k`2 ) 2

P-a.s.

A solution of the BSDEL is a pair of processes (Y, Z) ∈ S × P (`2 ) such that, for all t ∈ [0, T ], Z T ∞ Z T X (2.1) Yt = ξ + f (s, Ys− , Zs )ds − Zs(k) dHs(k) . t

k=1

2

t

Nualart and Schoutens proved the following theorem. Theorem 2.1 ([13], Theorem 1). Given standard data (ξ, f ), there exists a unique solution of the BSDEL (2.1). 2.1. BSDEL in finance. Consider a market where the risk-neutral dynamics of the asset can be described by the Dol´eans-Dade exponential dXt = Xt− dLt ,

X0 = x.

We suppose additionally that x > 0 and the L´evy measure is supported on a subset of [−1, +∞). This ensures that Xt > 0, ∀t ≤ T a.s. The value of the risk-free bond at time t ≤ T is given by Xt0 = ert where the constant r is the riskless rate of interest. We suppose that there exists a risk-neutral ˜ t = e−rt Xt is a martingale and (L ˜ t )t≤T = (Lt − rt)t≤T will measure Q such that X ˜ is a martingale. Let (H ˜ (i) )i≥1 be the be a L´evy process; moreover the process L ˜ orthonormalized power jump processes for L under the measure Q. We Enlarge ˜ (i) defined, for all t ≤ T , by the market with price processes X ˜ t(i) = ert H ˜ t(i) , X

i ≥ 2.

Note that ˜ t(i) /Fs ] = EQ [H ˜ t(i) /Fs ] = H ˜ s(i) ∀ 0 ≤ s ≤ t ≤ T. EQ [e−rt X ˜ and the power jump assets (X ˜ (i) )i≥2 remains arbitrage It follows that the stock X free.

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˜ T ) is defined as a conThe value of a European option with terminal payoff φ(X ditional expectation of its actualized terminal payoff under risk neutral probability Q: h i ˜ T )/Ft . Vt = EQ e−r(T −t) φ(X ˜ T )|2 ] < ∞. For all (t, x) ∈ [0, T ] × [0, +∞), we define the Suppose that EQ [|φ(X function v as follows h i ˜ T )/X ˜t = x . v(t, x) = EQ e−r(T −t) φ(X ˜ t ). We suppose that Clearly, Vt = v(t, X σ 2 > 0 or

Z lim ε−β

∃β ∈ (0, 2),

ε↓0

ε

|x|2 v(dx) > 0.

−ε

Under this regularity condition, the function v is continuous on [0, T ] × [0, +∞), C 1,2 on (0, T ) × [0, +∞) and v solves the partial integro-differential equation (see e.g. [4], Proposition 2) ( ∂v (t, x) + Lv(t, x) − rv(t, x) = 0, ∀(t, x) ∈ [0, T ) × [0, +∞), ∂t v(T, x) = φ(x), ∀x ∈ [0, +∞), where Lv(t, x)

= m1 x

1 ∂2v ∂v (t, x) + σ 2 x2 2 (t, x) ∂x Z µ 2 ∂x + R

¶ ∂v v(t, x(1 + y)) − v(t, x) − (t, x)xy ν(dy). ∂x

Next, let us define the following precesses  ˜  Y˜t = v(t, Z hXt ), i (k) ˜ t (1 + y)) − v(t, X ˜ t ) pk (y)˜ = v(t, X ν (dy),  Z˜t

k ≥ 1.

R

³ ´ It is not hard to see that Y˜ , (Z˜ (k) )k≥1 ∈ S 2 × P 2 (`2 ). So, we have ³ ´ Proposition 2.2. The process Y˜ , (Z˜ (k) )k≥1 is the unique solution of the following BSDEL Z T ∞ Z T X ˜T ) − ˜ s(k) , 0 ≤ t ≤ T. Y˜t = φ(X rY˜s ds − Z˜s(k) dH t

k=1

t

Proof. This is a direct consequence Itˆo’s formula and the uniqueness result of the solution for BSDEL (2.1). ¤ 3. The Comparison Theorem The aim of this section is to establish the comparison theorem when the coefficient f made up of: P∞ (k) (H.2) (1) f (t, ω, y, z) = f 1 (t, ω, y) + k=1 γ (k) (s, hRω)z . i T (2) f 1 is progressively measurable and E 0 |f 1 (t, 0)|2 dt < ∞.

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f 1 is uniformly κ-Lipschitz with respect to y. P∞ (k) (k) > −1 for all (t, ω). k=1 γt ∆Ht

(3) (4)

Theorem 3.1 (Comparison Theorem). Let (Y, Z) and (Y 0 , Z 0 ) be solution of the BSDEL (2.1) with data (ξ, f ) and (ξ 0 , f 0 ) respectively which satisfies (H.0)−(H.2). 1 We suppose that ξ ≤ ξ 0 and f 1 (t, y) ≤ f 0 (t, y) for all (t, y) dP × dt–a.s. Then, 0 Yt ≤ Yt , ∀t ≤ T P–a.s. Proof. Using the Meyer-Itˆo formula with the convex function x 7→ (x+ )2 with Y − Y 0 implies that (Yt − Yt0 )+

2

Z

2

= (Y0 − Y00 )+ − 2 −2 +2

∞ Z X

t

k=1 0 ∞ Z t X 0

k=1

t

0

³ ´ 1 (Yr − Yr0 )+ f 1 (r, Yr ) − f 0 (r, Yr0 ) dr (k)

(Yr − Yr0 )+ γr(k) (Zr(k) − Z 0 r )dr (k)

(Yr − Yr0 )+ (Zr(k) − Z 0 r )dHr(k) + At ,

(3.1)

where A is a continuous non decreasing process (see e.g. [20], Theorem 66 P. 210). Let (Γt,s , s ∈ [t, T ]) be solution of the linear stochastic differential equation Γt,s = 1 +

∞ Z X k=1

t

s

Γt,r− γr(k) dHr(k) ,

which we can write as (see e.g. [20], p. 84) Ã Γt,s

= exp

∞ Z X

k=1

Y t 0.

k=1

k=1

On the other hand,from the integration by part formula combining with (3.1), we have for all t ≤ s ≤ T 2

2

(Yt − Yt0 )+ = Γt,s (Ys − Ys0 )+ Z Z s ³ ´ 1 Γt,r (Yr − Yr0 )+ f 1 (r, Yr ) − f 0 (r, Yr0 ) dr + +2 +

t ∞ Z s X

Γt,r dAr

t

n ³ ´o (k) dHr(k) Γt,r (Yr − Yr0 )+ (Yr − Yr0 )+ γr(k) + 2 Zr(k) − Z 0 r

k=1 t ∞ Z s X

+2

i,j=1

s

t

³ ´ (j) Γt,r γr(i) (Yr − Yr0 )+ (Zr(j) − Z 0 r ) d[H (i) , H (j) ]r − hH (i) , H (j) ir .

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In particular for s = T . It follows that h i 2 E (Yt − Yt0 )+ Z T ³ ´ 1 0 +2 ≤ E[Γt,T (ξ − ξ ) ] + 2E Γt,r (Yr − Yr0 )+ f 1 (r, Yr ) − f 0 (r, Yr ) dr Z

t

T

+2κE Z ≤ 2κ t

t T

Γt,r (Yr − Yr0 )+ |Yr − Yr0 |dr

h i 2 E Γt,r (Yr − Yr0 )+ dr.

Using Gronwall’s lemma we conclude that Yt ≤ Yt0 P–a.s for all 0 ≤ t ≤ T . This completes the proof of the theorem. ¤ 4. Viscosity Solution of Partial Integro-differential Equation In this section, we prove that, if the BSDEL (2.1) has an adapted solution, then it will provide a solution to a system of integro-partial differential equations in the sense of viscosity. 4.1. Forward-Backward SDE with L´ evy process. Suppose that: (H.3) The autonomous function F : R 7→ R is κ-Lipschitz; that is there exists a constant κ > 0 such that |F (x) − F (x0 )| ≤ κ|x − x0 |,

x, x0 ∈ R.

Proposition 4.1. ([21], Lemma 5.1) For every initial condition (t, x) ∈ [0, T ]×R, the stochastic differential equation Z s∨t t,x Xst,x = x + )dLr , (4.1) F (Xr− t

has a unique solution such that E sup |Xst,x |p ≤ C(p, κ)(T − t)(1 + |x|p ),

for all p ≥ 1.

(4.2)

t≤s≤T

Proposition 4.2. Assume that X t,x is the solution of the SDE (4.1). Then, there exists a constant C > 0 such that, for all t, t0 ∈ [0, T ] and x, x0 ∈ R, t,x 2 2 0 0 • E|Xst,x 0 − Xs | ≤ C(1 + |x| )|s − s |, ∀ s, s ∈ [t, T ]. t,x 2 2 • E sup |Xr − x| ≤ C(1 + |x| )|t − s|. t≤r≤s © ª 0 0 • E sup |Xst,x − Xst ,x |2 ≤ C |t − t0 | + |x − x0 |2 . t∨t0 ≤s≤T

Proof. Due to Lemma 4.1 in [21], we can write for s < s0 Z s0 t,x 2 t,x 2 t,x E|Xs − Xs0 | ≤ C(b, σ, m1 ) E|F (Xr− )| dr s

≤ C|s − s0 |(1 + E sup |Xrt,x |2 ). t≤r≤T

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The first point follows by virtue (4.2). For the second, by using Burkholder-DavisGundy inequality, we get Z s t,x 2 E sup |Xrt,x − x|2 ≤ 4E )| d[L, L]r |F (Xr− t≤r≤s

t

≤ C(1 + |x|2 )|t − s|. On the same way, For t < t0 we have E sup t0 ≤s≤T

Z +E ( ≤

C

t

T

|Xst,x



0 0 Xst ,x |2

(

≤C

Z t

) t,x |F (Xr− )



t,x 2 |F (Xr− )| d[L, L]r

t0 ,x0 2 F (Xr− )| d[L, L]r

Z 0 2

t0

|x − x0 |2 + E

0

|x − x | + |t − t | + E t

T

) |Xrt,x



0 0 Xrt ,x |2 dr

.

The result then follows from Gronwall’s lemma.

¤

Now, Let f : [0, T ] × R2 × `2 → R and g : R → R two real functions such that (H.4) (1) The coefficient f is continuous and κ-Lipschitz respect to x. (2) The map f (., X t,x , ., .) verifies the hypothesis (H.2) with γ(s, ω) = γ, ∀(s, ω) such that 0 < ε < kγk`2 < ∞. (3) The function g is κ-Lipschitz. For each (t, x), let {(Yst,x , Zst,x ); t ≤ s ≤ T } be the unique solution of the BSDEL described by Z T ∞ Z T X t,x t,x , Zrt,x )dr − Zr(k)t,x dHr(k) . (4.3) , Yr− f (r, Xr− Yst,x = g(XTt,x ) + s∨t

k=1

s∨t

Existence and uniqueness follow from Theorem 2.1. Moreover, we have Z T ¡ ¢ E sup |Yst,x |2 + E kZst,x k`2 ds ≤ C 1 + |x|2 . 0≤s≤T

(4.4)

0

Let us define the deterministic function u by u(t, x) = Ytt,x ,

(t, x) ∈ [0, T ] × R.

(4.5)

So, we have Proposition 4.3. The following inequality holds, ¡ ¢ |u(t, x) − u(t0 , x0 )| ≤ C |x − x0 |2 + |t − t0 | . Proof. First, we remark that 0

|u(t, x) − u(t0 , x0 )|2 = |Ytt,x − Ytt,x |2 ½ ¾ 0 0 0 0 0 0 2 ≤ C E sup |Yst,x − Yst ,x |2 + E|Ytt,x − Ytt ,x |2 + |Ytt0 ,x − Ytt,x . 0 | t∨t0 ≤s≤T

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If we apply Itˆo’s formula, we compute that Z t0 2 | + E kZst,x k`2 ds E|Ytt,x − Ytt,x 0 t

Z

t0

= 2E

¡

t

¢ Yst,x − Ytt,x f (s, Xst,x , Yst,x , Zst,x )ds 0 Z

t0

≤ C(κ, ε)E

¯ t,x ¯2 ¯Ys − Y t,x ¯ ds 0 t

t

µ ¶ Z t0 kγk`2 0 t,x 2 t,x 2 +C|t − t | 1 + E sup [|Xs | + |Ys | ] + E kZst,x k`2 ds. ε 0≤s≤T t Gronwall’s lemma yields 2 0 2 E|Ytt,x − Ytt,x 0 | ≤ C|t − t |(1 + |x| ).

On the other hand, using (H.4), we can write for t ∨ t0 ≤ s ≤ T Z T 0 0 0 0 E|Yst,x − Yst ,x |2 + E kZrt,x − Zrt ,x k2`2 dr s

0

0

≤ E|g(XTt,x ) − g(XTt ,x )|2 Z T ¯ ¯ 0 0 ¯ 0 0 0 0 ¯ +2E |Yrt,x − Yrt ,x | ¯f 1 (r, Xrt,x , Yrt,x ) − f 1 (r, Xrt ,x , Yrt ,x )¯ dr s

Z

T

+2E s

¯ ¯ 0 0 ¯ 0 0 ¯ |Yrt,x − Yrt ,x | ¯hγ, Zrt,x − Zrt ,x i`2 ¯ dr Z 0

0

≤ 2κ E sup [|Xrt,x − Xrt ,x |2 ] + C(κ, ε)E 2

s≤r≤T

kγk`2 E + ε

Z

T

s

0

s

T

¯ ¯ 0 0 ¯2 ¯ t,x ¯Yr − Yrt ,x ¯ ds

0

kZrt,x − Zrt ,x k2`2 dr.

The result follows by Burkholder-Davis-Gundy inequality.

¤

4.2. Viscosity solution. To begin, let us consider the following system for parabolic integral-partial differential equation  ν 1 ∞   ∂t u(t, x) + L u(t, x) + f (t, x, u(t, x), (uk (t, x))k=1 ) = 0, ∀(t, x) ∈ [0, T ) × R, (4.6)   u(T, x) = g(x), ∀x ∈ R, where 1 m1 F (x)∂x φ(t, x) + σ 2 F (x)2 ∂xx φ(t, x) 2 ¶ Z µ ∂φ + φ(t, x + F (x)y) − φ(t, x) − (t, x)F (x)y ν(dy) ∂x R

Lν φ(t, x) =

and

Z φ1k (t, x) =

(φ(t, x + F (x)y) − φ(t, x)) pk (y)ν(dy) for R

k ≥ 1.

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Definition 4.4. (i) u ∈ C([0, T ] × R, R) is a viscosity subsolution of (4.6), if u(T, x) ≤ g(x); ∀x ∈ R, and for ϕ ∈ C 1,2 ([0, T ] × R) if u − ϕ has a local maximum at (t, x) then −∂t ϕ(t, x) − Lν ϕ(t, x) − f (t, x, u(t, x), (ϕ1k (t, x))∞ k=1 ) ≤ 0. (ii) u ∈ C([0, T ]×R, R) is a viscosity supersolution of (4.6), if u(T, x) ≥ g(x); ∀x ∈ R, and for ϕ ∈ C 1,2 ([0, T ] × R) if u − ϕ has a local minimum at (t, x) then −∂t ϕ(t, x) − Lν ϕ(t, x) − f (t, x, u(t, x), (ϕ1k (t, x))∞ k=1 )) ≥ 0. (iii) u ∈ C([0, T ] × R, R) is a viscosity solution of (4.6) if it is both a viscosity subsolution and supersolution. We give now the main result of this part: Theorem 4.5. The function u given by the formula (4.5) is a viscosity solution of the system (4.6). Proof. First, by uniqueness of the solution of BSDEL (4.3), we can write for any s,X t,x

s ∈ [t, T ] that Yst,x = Ys s = u(s, Xst,x ). Now, we suppose that (t, x) is a local maximum of u − ϕ where ϕ ∈ C 1,2 ([0, T ] × R, R) and u(t, x) = ϕ(t, x). We can assume additionally that ϕ and its derivatives have at most polynomial growth in x uniformly on t. Let h > 0 and consider (Y¯ h , Z¯ h ) the unique solution of the BSDEL given on [t, t + h] by, Z t,x Y¯sh = ϕ(t + h, Xt+h )+

t+h

s

t,x ¯ h ¯ h f (r, Xr− , Yr− , Zr )dr



∞ Z X k=1

t+h

s

Z¯r(k)h dHr(k) .

t,x t,x However, u(t + h, Xt+h ) ≤ ϕ(t + h, Xt+h ). Then Theorem 3.1 implies that

Y¯sh ≥ Yst,x ,

∀ s ∈ [t, t + h].

(4.7)

On the other hand, from Itˆo’s formula (see [20], p. 78), we have t,x ϕ(t + h, Xt+h )

=

ϕ(s, Xst,x ) Z

Z

+ s

t+h

Z ∂t ϕ(r, Xrt,x )dr

+ s

t+h

t,x t,x F (Xr− )∂x ϕ(r, Xr− )dLr

t+h

1 2 + σ F (Xrt,x )2 ∂xx ϕ(r, Xrt,x )dr 2 s X © ª t,x t,x + ϕ(r, Xrt,x ) − ϕ(r, Xrt,x . − ) − ∂x ϕ(r, Xr − )F (Xr− )∆Lr s 0 such that Z ¤ 1 t+h £ εh = (∂t ϕ + Lν ϕ) (r, Xrt,x ) + f (r, Λt,x r ) dr ≤ −δ, h t

∀ 0 < h < h0 ,

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¡ ¢ t,x t,x 1 t,x ∞ where Λt,x r = r, Xr , ϕ(r, Xr ), (ϕk (r, Xr ))k=1 . So, using (4.7), we have Yˆth = Y¯th − ϕ(t, x) = Y¯th − u(t, x) = Y¯th − Ytt,x ≥ 0. Therefore, we deduce by using (4.9) that Z t+h ³ ´ 1 1 δ ≤ E| Yˆth − εh | ≤ E |Yˆrh | + kZˆrh k dr h h t à !1/2 ¶1/2 µ Z t+h 1 +C ≤ C E sup |Yˆsh |2 E kZˆr k2 dr h t≤s≤t+h t ≤ Ch1/2 ,

∀ 0 < h ≤ h0 .

That is impossible. Consequently −∂t ϕ(t, x) − Lν ϕ(t, x) − f (t, x, u(t, x), (ϕ1k (t, x))∞ k=1 ) ≤ 0. By the same argument, we prove that u is a viscosity supersolution. From where u is a viscosity solution of the system (4.6). ¤ References 1. Bahlali, K.; Eddahbi, M. and Essaky, E.: BSDE associated with L´ evy processes and application to PDIE, Journal of Applied Mathematics and Stochastic Analysis. 16, part 1 (2003) 1–17. 2. Barles, G.; Buckdan, R. and Pardoux, E.: BSDEs and integral-partial differential equations, Stochastics. 60 (1997) 57–83. 3. Bertoin, J.: L´ evy processes, Cambridge University Press, 1996. 4. Cont, R. and Voltchkova, E.: Integro-differential equations for option prices in exponential Levy models, Finance and Stochastics. 9 (2005) 299–325. 5. El Karoui, N. and Hamad` ene, S.: BSDEs and risk sensitive control, zero-sum and nonzerosum game problems of stochastic functional differential equations, Stochastics Processes and their Applications. 107 (2003) 145–169. 6. El Karoui, N.; Peng, S. and Quenez, M. C.: Backward stochastic differential equations in finance, Mathematical Finance. 7 (1997) 1–71. 7. El Karoui, N. and Quenez, M. C.: Non-linear pricing theory and backward stochastic differential equations, In W.J. Runggaldier (ed.), Financial Mathematics, Lecture Notes in Math. (1656) 191–246. Berlin: Springer-Verlag. 8. Hamad` ene, S. and Lepeltier, J. P.: Zero-sum Stochastic Differential games and BSDEs, Systems and Control letters. 24 (1995) 259–263. 9. Hamad` ene, S. and Lepeltier, J. P.: Backward equations, stochastic control and zero-sum stochastic differential games, Stochastics and stochastics reports. 54, part 3-4 (1995) 221– 231 10. Ikeda, N. and Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, North-Holland, 1981. 11. L´ eon, J. A.; Sol´ e, J. L.; Utzet, F. and Vives, J.: On L´ evy processes, Malliavin calculus and market models with jumps, Finance and Stochastic. 6 (2002) 197–225. 12. Lin, Q.: Nonlinear Doob-Meyer Decomposition with jumps, Acta Mathematica Sinica: English Series. 19(1) (2003) 69–78. 13. Nualart, D. and Schoutens, W.: BSDEs and Feynman Kac-Formula for L´ evy Processes with Applications in Finance, Bernoulli. 7 (2001) 761–776. 14. Ouknine, Y.: Reflected backward stochastic differential equations with jumps, Stochastic Stochastic Rep. 65 (1998) 111–125. 15. Pardoux, E.: Backward Stochastic differential equations and viscosity solutions of systems of semi-linear parabolic and elliptic PDEs of second order, Stochastic analysis and related

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Mohamed EL OTMANI: Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, BP 2390, Marrakesh, Morocco E-mail address: [email protected] URL: http://elotmani.site.voila.fr/index.html