Another method of viscosity solutions of integro-differential partial equation by concavity

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Another method of viscosity solutions of integro-differential partial equation by concavity

Lamine Sylla

To cite this version:

Lamine Sylla. Another method of viscosity solutions of integro-differential partial equation by con-

cavity. 2018. �hal-01868453�

Another method of viscosity solutions of integro-differential partial equation by concavity

L. SYLLA

September 5, 2018

Abstract

In this paper we consider the problem of viscosity solution of integro-partial differential equa- tion(IPDE in short) via the solution of backward stochastic differential equations(BSDE in short) with jumps where Lévy’s measure is not necessarily finite. We mainly use the concavity of the gener- ator at the level of its second variable to establish the existence and uniqueness of the solution with non local terms.

Keywords: Integro-partial differential equation; Backward stochastic differential equations with jumps; Viscosity solution; Non-local operator; Concavity.

MSC 2010 subject classifications: 35D40, 35R09, 60H30.

†Université Gaston Berger, LERSTAD, CEAMITIC, e-mail: [email protected]

1 Introduction

We consider the following system of integro-partial differential equation, which is a function of(t, x):

∀i∈ {1, . . . , m},









−∂tuⁱ(t, x)−b(t, x)^⊤Dxuⁱ(t, x)−₂¹Tr(σσ^⊤(t, x)D²_xxuⁱ(t, x))−Kiuⁱ(t, x)

−h⁽ⁱ⁾(t, x, uⁱ(t, x),(σ^⊤Dxuⁱ)(t, x),Biuⁱ(t, x)) = 0, (t, x)∈[0, T]×R^k; uⁱ(T, x) =gⁱ(x);

(1.1)

where the operatorsBi andKiare defined as follows:

Biuⁱ(t, x) = Z

E

γⁱ(t, x, e)(uⁱ(t, x+β(t, x, e))−uⁱ(t, x))λ(de); (1.2) Kiuⁱ(t, x) =

Z

E

(uⁱ(t, x+β(t, x, e))−uⁱ(t, x)−β(t, x, e)^⊤Dxuⁱ(t, x))λ(de).

The resolution of (1.1) is in connection with the following system of backward stochastic differential equations with jumps :





dY_s^i;t,x=−f⁽ⁱ⁾(s, X_s^t,x, Y_s^i;t,x, Z_s^i;t,x, U_s^i;t,x)ds+Z_s^i;t,xdBs+ Z

E

U^i;t,x_s (e)eµ(ds,de), s≤T; Y_T^i;t,x=gⁱ(X_T^t,x);

(1.3)

and

the following standard stochastic differential equation of diffusion-jump type:

X_s^t,x=x+ Z s

t

b(r, X_r^t,x) dr+ Z s

t

σ(r, X_r^t,x) dBr+ Z s

t

Z

E

β(r, X_r−^t,x, e)µ(dr, de),e (1.4) fors∈[t, T]andX_s^t,x=xifs≤t.

Several authors have studied (1.1) by various and varied methods. Among others we can quote Barles and al. [1], who use the theorem of comparison by supposition the monotony of the generator; Ouknine and Hamadène [5] use the penalization method.

But recently Hamadène does a relaxation on the hypothesis of monotony on the generator to introduce a new class of functions(see [3] page 216) for the resolution of (1.1).

In this work we propose to solve (1.1) by relaxing the monotonicity of the generator and the class of fonctions introduced in [3] and assuming that λ=∞and the concavity of the generator at the level of its second variable. We recall that this technic was used in [8] for the resolution of BSDE.

Our paper is organized as follows: in the next section we give the notations and the assumptions; in section3we recall a number of existing results; in section4we build estimates and properties for a good resolution of our problem; section5 is reserved to give our main result.

And at the end, classical definition of the concept of viscosity solution is put in appendix.

2 Notations and assumptions

Let (Ω,F,(Ft)t≤T,P) be a stochastic basis such that F0 contains all P−null sets of F, and Ft = Ft+:=T

ǫ>0Ft+ǫ, t≥0, and we suppose that the filtration is generated by the two mutually indepen- dents processes:

(i)B:= (Bt)t≥0 ad-dimensional Brownian motion and,

(ii) a Poisson random measureµonR⁺×EwhereE :=R^ℓ− {0}is equipped with its Borel fieldE(ℓ≥1).

The compensator ν(dt,de) = dtλ(de)is such that{µ([0, t]e ×A) = (µ−λ)([0, t]×A)}t≥0is a martingale for allA∈ E satisfyingλ(A)<∞. We also assume that λis a σ-finite measure on(E,E), integrates the function (1∧ |e|²)andλ(E) =∞.

Let’s now introduce the following spaces:

(iii)P (resp.P)the field on[0, T]×ΩofFt≤T-progressively measurable (resp. predictable) sets.

(iv) Forκ≥1, L²κ(λ)the space of Borel measurable functionsϕ:= (ϕ(e))e∈E fromE into R^κsuch that kϕk²_L2

κ(λ)= Z

E

|ϕ(e)|²_κλ(de)<∞;L²1(λ)will be simply denoted byL²(λ);

(v)S²(R^κ)the space of RCLL (for right continuous with left limits)P-measurable andR^κ-valued pro- cesses such that E[sup_s≤T|Ys|²]<∞;

(vi)H²(R^κ×d)the space of processesZ:= (Zs)s≤T which areP-measurable,R^κ×d-valued and satisfying E

"Z T 0

|Zs|² ds

#

<∞;

(vii)H²(L²_κ(λ))the space of processesU := (Us)s≤T which areP-measurable,L²_κ(λ)-valued and satisfying E

"Z T 0

kUs(ω)k²_L²

κ(λ)ds

#

<∞;

(viii)Πg the set of deterministics functions

̟: (t, x)∈[0, T]×R^κ7→̟(t, x)∈Rof polynomial growth, i.e., for which there exists two non-negative constantsC andpsuch that for any(t, x)∈[0, T]×R^κ,

|̟(t, x)| ≤C(1 +|x|^p).

The subspace ofΠg of continuous functions will be denoted byΠ^c_g;

(ix)Mthe class of functions which satisfy thep-order Mao condition inxi.e. Iff ∈ Mthen there exists a nondecreasing, continuous and concave functionρ(·) :R⁺7→R⁺withρ(0) = 0,ρ(u)>0, foru >0and Z

0⁺

du

ρ(u) = +∞, such thatdP×dt−a.e.,∀x, x^′ ∈R^k and∀p≥2,

|f(t, x, y, z, q)−f(t, x^′, y, z, q)| ≤ρ^1p(|x−x^′|^p);

(x) For any processθ:= (θs)s≤T andt∈(0, T], θt−= limsրtθsand

∆tθ=θt−θt−. Now letbandσbe the following functions:

b: (t, x)∈[0, T]×R^k 7→b(t, x)∈R^k;

σ: (t, x)∈[0, T]×R^k7→σ(t, x)∈R^k×d.

We assume thatb andσbelong toM (2.5)

Letβ : (t, x, e)∈[0, T]×R^k×E7→β(t, x, e)∈R^k be a measurable function such that for some real constant C, and for alle∈E,

(i) |β(t, x, e)| ≤C(1∧ |e|); (2.6)

(ii) β belongs toM;

(iii) the mapping (t, x)∈[0, T]×R^k 7→β(t, x, e)∈R^k is continuous for anye∈E.

The functions(gⁱ)i=1,m and(h⁽ⁱ⁾)i=1,m be two functions defined as follows: fori= 1, . . . , m, gⁱ:R^k −→ R^m

x 7−→ gⁱ(x) and

h⁽ⁱ⁾: [0, T]×R^k+m+d+1 −→ R

(t, x, y, z, q) 7−→ h⁽ⁱ⁾(t, x, y, z, q).

Moreover we assume they satisfy:

(H1): For anyi∈ {1, . . . , m}, the functiongⁱ belongs toM.

(H2): For anyi∈ {1, . . . , m},

(i) the function h⁽ⁱ⁾ is Lipschitz in(y, z, q)uniformly in (t, x), i.e., there exists a real constant C such that for any(t, x)∈[0, T]×R^k,(y, z, q)and(y^′, z^′, q^′)elements ofR^m+d+1,

h⁽ⁱ⁾(t, x, y, z, q)−h⁽ⁱ⁾(t, x, y^′, z^′, q^′)≤C(|y−y^′|+|z−z^′|+|q−q^′|); (2.7) (ii) the(t, x)7→h⁽ⁱ⁾(t, x, y, z, q), for fixed(y, z, q)∈R^m+d+1, belongs uniformly toM,

Next letγⁱ, i= 1, . . . , mbe Borel measurable functions defined from[0, T]×R^k×EintoRand satisfying:

(i) γⁱ(t, x, e)≤C(1∧ |e|);

(ii) γⁱbelongs toM; (2.8)

(iii) the mappingt∈[0, T]7→γⁱ(t, x, e)∈Ris continuous for any(x, e)∈R^k×E.

Finally we introduce the following functions (f⁽ⁱ⁾)i=1,m defined by:

∀(t, x, y, z, ζ)∈[0, T]×R^k+m+d×L²(λ), f⁽ⁱ⁾(t, x, y, z, ζ) :=h⁽ⁱ⁾

t, x, y, z, Z

E

γⁱ(t, x, e)ζ(e)λ(de)

. (2.9) The functions(f⁽ⁱ⁾)i=1,m, enjoy the two following properties:

(a) The function f⁽ⁱ⁾ is Lipschitz in(y, z, ζ)uniformly in (t, x), i.e., there exists a real constant

C such that

f⁽ⁱ⁾(t, x, y, z, ζ)−f⁽ⁱ⁾(t, x, y^′, z^′, ζ^′)≤C(|y−y^′|+|z−z^′|+kζ−ζ^′k_L²(λ)); (2.10) sinceh⁽ⁱ⁾ is uniformly Lipschitz in(y, z, q)andγⁱ verifies (2.8)-(i);

(b) The function (t, x)∈[0, T]×R^k 7→f⁽ⁱ⁾(t, x,0,0,0) belongstoΠ^c_g; and thenE

"Z T 0

f⁽ⁱ⁾(r, X_r^t,x,0,0,0)

2

dr

#

<∞.

3 Some results in backward stochastic differential equation with jumps

3.1 A class of diffusion processes with jumps

Let(t, x)∈[0, T]×R^d and(X_s^t,x)s≤T be the stochastic process solution of (1.4). Under assumptions (2.5)-(2.6) the solution of Equation (1.4) exists and is unique (see [2] for more details). We state some properties of the process{(X_s^t,x), s∈[0, T]}which can found in [2].

Proposition 3.1 For each t ≥0, there exists a version of{(X_s^t,x), s∈ [t, T]} such that s →X_s^t is a C²(R^d)-valued rcll process. Moreover it satisfies the following estimates: ∀p≥2, x, x^′ ∈R^d ands≥t,

E[ sup

t≤r≤s

X_r^t,x−x^p] ≤ Mp(s−t)(1 +|x|^p);

E[ sup

t≤r≤s

X_r^t,x−X_r^t,x′−(x−x^′)^pp] ≤ Mp(s−t)(|x−x^′|^p); (3.11) for some constant Mp.

3.2 Existence and uniqueness for BSDE with jumps

Let(t, x)∈[0, T]×R^d and we consider the following m-dimensional BSDE with jumps:











(i)Y~^t,x:= (Y^i,t,x)i=1,m∈ S²(R^m), Z^t,x:= (Z^i,t,x)i=1,m∈H²(R^m×d), U^t,x:= (U^i,t,x)i=1,m∈H²(L²_m(λ));

(ii)dY_s^i;t,x=−f⁽ⁱ⁾(s, X_s^t,x, Y_s^i;t,x, Z_s^i;t,x, U_s^i;t,x)ds+Z_s^i;t,xdBs+ Z

E

U^i;t,x_s (e)µ(ds,e de), s≤T; (iii)Y_T^i;t,x=gⁱ(X_T^t,x);

(3.12) where for anyi∈ {1, . . . , m}, Y_s^i;t,x is the ith row ofY_s^t,x,Z_s^i;t,x is the ith component ofZ_s^t,xand U_s^i;t,x is the ith component of U_s^t,x.

Proposition 3.2 Assume that assumptions(H1)and(H2)hold. Then for any(t, x)∈[0, T]×R^d, the BSDE (3.12) has an unique solution (Y~^t,x, Z^t,x, U^t,x).

For proof of this proposition we can see [1].

3.3 Viscosity solutions of integro-differential partial equation

Proposition 3.3 (see, [1]) Assume that (H1), (H2), are fulfilled. Then there exists deterministic con- tinuous functions (uⁱ(t, x))i=1,m which belong toΠg such that for any(t, x)∈[0, T]×R^k, the solution of the BSDE (3.12) verifies:

∀i∈ {1, . . . , m}, ∀s∈[t, T], Y_s^i;t,x=uⁱ(s, X_s^t,x). (3.13) Moreover if for any i∈ {1, . . . , m},

(i) γⁱ≥0;

(ii) for any fixed(t, x, ~y, z)∈[0, T]×R^k+m+d, the mapping (q∈R)7−→h⁽ⁱ⁾(t, x, ~y, z, q)∈Ris non-decreasing.

The function(uⁱ)i=1,m is a continuous viscosity solution (in Barles and al. ’s sense, see Definition 5.3 in the Appendix) of (1.1). The solution(uⁱ)i=1,m of (1.1) is unique in the classΠ^c_g.

Remark 1 (see, [1]) Under the assumptions (H1), (H2), there exists a unique viscosity solution of (1.1) in the class of functions satisfying

|x|→+∞lim |u(t, x)|e⁻A[log(|x|)]^{e 2} = 0 (3.14) uniformly for t∈[0, T], for some A >e 0.

4 Estimates and properties

In this section, we will establish a priori estimates concerning solutions of BSDE (3.12), which will play an important role in the proof of our main results.

Lemma 4.1 (see, [3]) Under assumption (H1), (H2), for any p ≥ 2 there exists two non-negative constants C andδ such that,

E



(Z T 0

ds Z

E

U_s^t,x(e)²λ(de) )^p2

=E



(Z T 0

dskU_s^t,xk²_L²

m(λ)

)^p2

≤C

1 +|x|^δ

. (4.15)

Proposition 4.2 For anyi= 1, . . . , m, there exist C≥0,κ >0such that, ∀xandx^′ elements ofR^k

|uⁱ(t, x)−uⁱ(t, x^′)|²≤ρ

M2|x−x^′|²(1 +|x−x^′|²)

[C(1 +|x|^κ)]

Proof. Letxandx^′ be elements ofR^k. Let(Y~^t,x, Z^t,x, U^t,x) (resp. (Y~^t,x′, Z^t,x′, U^t,x′))be the solution of the BSDE with jumps (3.12) associated with (f(s, X_s^t,x, y, η, ζ), g(X_T^t,x))

(resp. f(s, X_s^t,x′, y, η, ζ), g(X_T^t,x′)). Applying Itô formula toY~^t,x−Y~^t,x′2 betweensandT, we have

Y~_s^t,x−Y~_s^t,x′2+ Z T

s

|∆Zr|² dr+ X

s≤r≤T

(∆rY~_r^t,x)² (4.16)

=g(X_T^t,x)−g(X_T^t,x′)

2

+ 2 Z T

s

<

Y~_s^t,x−Y~_s^t,x′

,∆f(r)> dr

−2 Z T

s

Z

E

Y~_r^t,x−Y~_r^t,x′

(∆Ur(e))µ(dr,e de)−2 Z T

s

Y~_r^t,x−Y~_r^t,x′

(∆Zr)dBr; and taking expectation we obtain: ∀s∈[t, T],

E

"Y~_s^t,x−Y~_s^t,x′

2

+ Z T

s

|∆Zr|² dr+ Z T

s

k∆Urk²_L²_(λ)dr

#

(4.17)

≤E"g(X_T^t,x)−g(X_T^t,x′)

2

+ 2 Z T

s

<

Y~_s^t,x−Y~_s^t,x′

,∆f(r)> dr

#

where the processes∆Xr,∆Yr,∆f(r), ∆Zr and∆Urare defined as follows: ∀r∈[t, T],

∆f(r) := ((∆f⁽ⁱ⁾(r))i=1,m = (f⁽ⁱ⁾(r, X_r^i;t,x, ~Y_r^t,x, Z_r^i;t,x, U_r^i;t,x)−f⁽ⁱ⁾(r, X_r^i;t,x′, ~Y_r^i;t,x′, Z_r^i;t,x′, U_r^i;t,x′))i=1,m,

∆Xr=X_r^t,x−X_r^t,x′,∆Y(r) =Y~_r^t,x−Y~_r^t,x′ = (Y_r^j;t,x−Y_r^j;t,x′)j=1,m,

∆Zr =Z_r^t,x−Z_r^t,x′ and∆Ur =U_r^t,x−U_r^t,x′ (<·,·>is the usual scalar product on R^m). Now we will give an estimation of each three terms of the second member of inequality (4.17).

• As for anyi∈ {1, . . . , m}gⁱ belongs toM(forp= 2); therefore

E

g(X_T^t,x)−g(X_T^t,x′)

2

≤ E

ρ

X_T^t,x−X_T^t,x′

2

≤ ρ

E

X_T^t,x−X_T^t,x′

2

(by Jensen’s inequality)

≤ ρ

E

X_T^t,x−X_T^t,x′−(x−x^′)²+ (x−x^′)²

2

and by subsequently using the triangle inequality, the relation (3.11) of proposition 3.1 and the fact that (a+b)^p≤2^p−1(a^p+b^p).

E

g(X_T^t,x)−g(X_T^t,x′)

2

≤ρ

M2|x−x^′|²(1 +|x−x^′|²)

, (4.18)

• To complete our estimation of (4.17) we need to deal with E

"

2 Z T

s

<

Y~_s^t,x−Y~_s^t,x′

,∆f(r)> dr

# .

Taking into account the expression of f⁽ⁱ⁾ given by (2.9) we then split ∆f(r) in the follows way: for r≤T,

∆f(r) = (∆f(r))i=1,m= ∆1(r) + ∆2(r) + ∆3(r) + ∆4(r) = (∆ⁱ₁(r) + ∆ⁱ₂(r) + ∆ⁱ₃(r) + ∆ⁱ₄(r))i=1,m,

where for anyi= 1, . . . , m,

∆ⁱ₁(r) = h⁽ⁱ⁾

r, X_r^t,x, ~Y_r^t,x, Z_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

−h⁽ⁱ⁾

r, X_r^t,x′, ~Y_r^t,x, Z_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

;

∆ⁱ₂(r) = h⁽ⁱ⁾

r, X_r^t,x′, ~Y_r^t,x, Z_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

−h⁽ⁱ⁾

r, X_r^t,x′, ~Y_r^t,x′, Z_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

;

∆ⁱ₃(r) = h⁽ⁱ⁾

r, X_r^t,x′, ~Y_r^t,x′, Z_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

−h⁽ⁱ⁾

r, X_r^t,x′, ~Y_r^t,x′, Z_r^i;t,x′, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

;

∆ⁱ₄(r) = h⁽ⁱ⁾

r, X_r^t,x′, ~Y_r^t,x′, Z_r^i;t,x′, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

−h⁽ⁱ⁾

r, X_r^t,x′, ~Y_r^t,x′, Z_r^i;t,x′, Z

E

γⁱ(r, X_r^t,x′, e)U_r^i;t,x′(e)λ(de)

.

By Cauchy-Schwartz inequality, the inequality2ab≤ǫa²+¹_ǫb², (H2)-(ii), the estimate (3.11) and Jensen’s inequality we have:

E

"

2 Z T

s

<∆Y(r),∆1(r)> dr

#

≤ E

"

1 ǫ

Z T s

|∆Y(r)|²dr+ǫ

Z T s

ρ

|X_r^t,x−X_r^t,x′|² dr

#

≤ E

"

1 ǫ

Z T s

|∆Y(r)|²dr

# +ǫρ

M2|x−x^′|²(1 +|x−x^′|²)

. (4.19) Besides sinceh⁽ⁱ⁾ is Lipschitz w.r.t. (y, z, q)then,

E

"

2 Z T

s

<∆Y(r),∆2(r)> dr

#

≤2CE

"Z T s

|∆Y(r)|²dr

#

, (4.20)

and E

"

2 Z T

s

<∆Y(r),∆3(r)> dr

#

≤E

"

1 ǫ

Z T s

|∆Y(r)|²dr+C²ǫ Z T

s

|∆Z(r)|²dr

#

. (4.21)

It remains to obtain a control of the last term. But for anys∈[t, T]we have, E

"

2 Z T

s

<∆Y(r),∆4(r)> dr

#

(4.22)

≤ 2CE

"Z T s

|∆Y(r)|dr× Z

E

γ(r, X_r^t,x, e)U_r^t,x(e)−γ(r, X_r^t,x′, e)U_r^t,x′(e) λ(de)

# .

Next by splitting the crossing terms as followsγ(r, X_r^t,x, e)U_r^t,x(e)−γ(r, X_r^t,x′, e)U_r^t,x′(e) = ∆Us(e)γ(s, X_s^t,x, e)+

U_s^t,x′

γ(s, X_s^t,x, e)−γ(s, X_s^t,x′, e) and setting∆γs(e) :=

γ(s, X_s^t,x, e)−γ(s, X_s^t,x′, e) , we obtain,

E

"

2 Z T

s

<∆Y(r),∆4(r)> dr

#

≤ 2CE

"Z T s

|^∆Y(r)|×





Z

E

(|U_r^t,x′(e)∆γr(e)|+|∆Ur(e)γ(r,X_r^t,x,e)|)λ(de)



dr

#

≤ 2 ǫE

"Z T s

|∆Y(r)|²dr

#

+C²ǫE

"Z T s

Z

E

(|U_r^t,x′(e)∆γr(e)|λ(de) 2

dr

#

+C²ǫE

"Z T s

Z

E

(|∆Ur(e)γ(r, X_r^t,x, e)|λ(de) 2

dr

#

. (4.23)

By Cauchy-Schwartz inequality, (2.8)-(ii), Jensen’s inequality and (3.11), and the result of Lemma 4.1 it holds:

E

"Z T s

Z

E

(|U_r^t,x′(e)∆γr(e)|λ(de) 2

dr

#

≤ E

"Z T s

dr Z

E

|U_r^t,x′(e)|²λ(de) Z

E

|∆γr(e)|²λ(de)

#

≤ ρ

M2|x−x^′|²(1 +|x−x^′|²)

×E

"Z T s

dr Z

E

|U_r^t,x′(e)|²λ(de)

#

≤ Cρ

M2|x−x^′|²(1 +|x−x^′|²)

(1 +|x|^κ). (4.24)

On the other hand using once more Cauchy-Schwartz inequality and (2.8)-(i) we get E

"Z T s

Z

E

(^|∆Ur(e)γ(r,X_r^t,x,e)|λ(de)

2

dr

#

≤ E

"Z T s

dr Z

E

(^|∆Ur(e)|²λ(de)

Z

E

|γ(r, X_r^t,x, e)|²λ(de)

#

≤ CE

"Z T s

dr Z

E

(|∆Ur(e)|²λ(de)

#

. (4.25)

From (4.19) to (4.25) it follows that:

E

"Y~_s^t,x−Y~_s^t,x′

2

+ Z T

s

|∆Zr|² dr+ Z T

s

k∆Urk²_L²_(λ)dr

#

≤E

"g(X_T^t,x)−g(X_T^t,x′)

2

+ 2 Z T

s

<

Y~_s^t,x−Y~_s^t,x′

,∆f(r)> dr

#

≤ ρ

M2|x−x^′|²(1 +|x−x^′|²) C(1 +|x|^κ) + 1 +ǫ+ǫC³ +

4 ǫ + 2C

E

"Z T s

|∆Y(r)|²dr

#

+C²ǫE

"Z T s

|∆Z(r)|²dr

#

+C³ǫE

"Z T s

dr Z

E

(|∆Ur(e)|²λ(de)

# .

By choosingǫ such that{ǫ+⁴_ǫ +ǫ(2C³+C²) + 2C <1} we deduce the existence of a constantC ≥0 such that for anys∈[t, T],

E

|∆Y(s)|²

≤ρ

M2|x−x^′|²(1 +|x−x^′|²)

[C(1 +|x|^κ)] +E

"Z T s

|∆Y(r)|²dr

#

and by Gronwall lemma this implies that for any s∈[t, T],

E

|∆Y(s)|²

≤ρ

M2|x−x^′|²(1 +|x−x^′|²)

[C(1 +|x|^κ)]. Finally in taking s=t and considering (3.13) we obtain the desired result.

Now we start a point which giving difference of definition viscosity solution between [1] and [3].

It should also be noted that in this part will appear our first contribution after of course the first corre- sponding to the proposition 4.2. It will be mainly about the use of theMclass and the Bihari inequality as in [8].

Proposition 4.3 For anyi= 1, . . . , m,(t, x)∈[0, T]×R^k,

U_s^i;t,x(e) =uⁱ(s, X_s−^t,x+β(s, X_s−^t,x, e))−uⁱ(s, X_s−^t,x), dP⊗ds⊗dλ−a.e. onΩ×[t, T]×E. (4.26)

Proof. Step 1: Truncation of the Lévy measure

For anyk≥1, let us first introduce a new Poisson random measureµ_k (obtained from the truncation of µ) and its associated compensatorν_k as follows:

µk(ds, de) = 1_{|e|≥¹

k}µ(ds, de) andνk(ds, de) = 1_{|e|≥¹

k}ν(ds, de).

Which means that, as usual,µf_k(ds, de) := (µk−νk)(ds, de), is the associated random martingale measure.

The main point to notice is that λk(E) =

Z

E

λk(de) = Z

E

1_{|e|≥¹

k}λ(de)

= Z

{|e|≥¹_k}

λ(de)

= λ({|e| ≥ 1

k})<∞. (4.27)

As in [3], let us introduce the process^kX^t,xsolving the following standard SDE of jump-diffusion type:

kX_s^t,x=x+ Z s

t

b(r,^kX_r^t,x)dr+ Z s

t

σ(r,^kX_r^t,x)dBr

+ Z s

t

Z

E

β(r,^kX_r−^t,x, e)µe_k(dr, de), t≤s≤T; ^kX_r^t,x=xifs≤t.

(4.28) Note that thanks to the assumptions on b, σ, β the process ^kX^t,x exists and is unique. Moreover it satisfies the same estimates as in (3.11) sinceλk is just a truncation at the origin ofλwhich integrates (1∧ |e|²)e∈E.

On the other hand let us consider the following Markovian BSDE with jumps

































 (i)E

"

sup_s≤T^kY_s^t,x2+ Z T

s

^kZ_r^t,x2dr+ Z T

s

k^kU_r^t,xk²_L2(λk)dr

#

<∞

(ii)^kY^t,x:= (^kY^i,t,x)i=1,m∈ S²(R^m), ^kZ^t,x:= (^kZ^i,t,x)i=1,m∈H²(R^m×d),

kU^t,x:= (^kU^i,t,x)i=1,m∈H²(L²_m(λk));

(iii)^kY_s^t,x=g(^kX_T^t,x) + Z T

s

fµk(r,^kX_r^t,x,^kY_r^t,x,^kZ_r^t,x,^kU_r^t,x)dr

− Z T

s

kZ_r^t,xdBr+ Z

E

{^kU_r^t,x(e)}µek(dr, de)

, s≤T; (iv)^kY_T^t,x=g(^kX_T^t,x).

(4.29)

Finally let us introduce the following functions(f⁽ⁱ⁾)i=1,m defined by: ∀(t, x, y, z, ζ)∈[0, T]×R^k×R^m× R^m×d×L²m(λk),

fµk(t, x, y, z, ζ) = (fµ⁽ⁱ⁾k(t, x, y, zi, ζi))i=1,m:=

h⁽ⁱ⁾

t, x, y, z, Z

E

γⁱ(t, x, e)ζi(e)λk(de)

i=1,m

.

First let us emphasize that this latter BSDE is related to the filtration(F_s^k)s≤T generated by the Brownian motion and the independent random measure µ_k. However this point does not raise major issues since for anys≤T,F_s^k⊂ Fsand thanks to the relationship betweenµandµ_k.

Next by the properties of the functionsb,σ,βand by the same opinions of proposition 3.2 and proposition 3.3, there exists an unique quadriple (^kY^t,x,^kZ^t,x,^kU^t,x)solving (4.29) and there also exists a function u^k from [0, T]×R^k intoR^mof Π^c_g such that

∀s∈[t, T], ^kY^t,x:=u^k(s,^kX^t,x), P−a.s. (4.30) Moreover as in proposition 4.2, there exists positive constants C and κwich do not depend on k such that:

∀t, x, x^′, |u^k(t, x)−u^k(t, x^′)| ≤ρ

M2|x−x^′|²(1 +|x−x^′|²)

[C(1 +|x|^κ)]. (4.31) Finally asλk is finite then we have the following relationship between the process^kU^t,x:= (^kU^i;t,x)i=1,m

and the deterministics functionsu^k:= (u^k_i)i=1,m (see [4]): ∀i= 1, . . . , m;

kU_s^i;t,x(e) =u^k_i(s,^kX_s−^t,x+β(s,^kX_s−^t,x, e))−u^k_i(s,^kX_s−^t,x), dP⊗ds⊗dλ_k−a.e.onΩ×[t, T]×E.

This is mainly due to the fact that^kU^t,x belongs toL¹∩L²(ds⊗dP⊗dλ_k)sinceλ_k(E)<∞and then we can split the stochastic integral w.r.t. µe_k in (4.29). Therefore for alli= 1, . . . , m,

kU_s^i;t,x(e)1_{|e|≥¹

k}= (u^k_i(s,^kX_s−^t,x+β(s,^kX_s−^t,x, e))−u^k_i(s,^kX_s−^t,x))1_{|e|≥¹

k}, dP⊗ds⊗dλk−a.e.onΩ×[t, T]×E.

(4.32) Step 2: Convergence of the auxiliary processes

Let’s now prove the following convergence result;

E

"

sup

s≤T

Y_s^t,x−^kY_s^t,x2+ Z T

0

Z_s^t,x−^kZ_s^t,x2 ds+ Z T

0

ds Z

E

λ(de)U_s^t,x(e)−^kU_s^t,x(e)1_{|e|≥¹

k}

2#

k−→+∞−→ 0;

(4.33) where (Y^t,x, Z^t,x, U^t,x)is solution of the BSDE with jumps (3.12).

It should be noted that this convergence (4.33) requires as the technique used in (4.2) the following convergence:

E

sup

s≤T

X_s^t,x−^kX_s^t,x2

k−→+∞−→ 0. (4.34)

Proof.[of 4.34]

X_s^t,x−^kX_s^t,x= Z s

0

(b(r, X_r^t,x)−b(r,^kX_r^t,x))dr+ Z s

0

(σ(r, X_r^t,x)−σ(r,^kX_r^t,x))dBr

+ Z s

0

Z

E

(β(r, X_r−^t,x, e)−β(r,^kX_r−^t,x, e)1_{|e|≥¹

k})µe_k(dr, de).

(4.35) Next let η ∈[0, T]. Since|a+b+c|²≤3(|a|²+|b|²+|c|²)for any real constantsa, band c and by the Cauchy-Schwartz and Burkholder-Davis-Gundy inequalities we have:

E

sup

0≤s≤η

X_s^t,x−^kX_s^t,x2

≤ 3E

"

sup

0≤s≤η

Z s 0

(b(r, X_r^t,x)−b(r,^kX_r^t,x))dr

2

+ sup

0≤s≤η

Z s 0

(σ(r, X_r^t,x)−σ(r,^kX_r^t,x))dB_r

2

+ sup

0≤r≤η

Z r 0

Z

E

(β(r, X_r−^t,x, e)−β(r,^kX_r−^t,x, e)1_{|e|≥¹

k})µek(dr, de)

2#

≤ CE Z η

0

sup

0≤η≤r

n(b(r, X_r^t,x)−b(r,^kX_r^t,x))²+(σ(r, X_r^t,x)−σ(r,^kX_r^t,x))²o dr

+CE Z η

0

Z

E

sup

0≤r≤η

β(r, X_r−^t,x, e)−β(r,^kX_r−^t,x, e)²λ_k(de)dr

+ Z η

0

Z

E

sup

0≤r≤η

β(r, X_r−^t,x, e)²1_{|e|<¹

k}λ(de)dr

(4.36) Sinceb,σandβ belong toM, then we have: ∀r∈[0, T],

sup

0≤τ≤r

n(b(r, X_τ^t,x)−b(r,^kX_τ^t,x))²+(σ(r, X_τ^t,x)−σ(r,^kX_τ^t,x))²o

≤Cρ(X_τ^t,x−^kX_τ^t,x2) (4.37)

and Z

E

sup

0≤r≤η

β(r, X_r−^t,x, e)−β(r,^kX_r−^t,x, e)² λ_k(de)≤Cρ(X_τ^t,x−^kX_τ^t,x2) (4.38)

Plug now those two last inequalities in the previous one to obtain: ∀η ∈[0, T], E

sup

0≤s≤η

X_s^t,x−^kX_s^t,x2

≤ CE

"Z η 0

ρ(X_τ^t,x−^kX_τ^t,x2)dr+ Z

{|e|<¹_k}

(1∧ |e|²)λ(de)

#

≤ C Z η

0

ρ(EhX_τ^t,x−^kX_τ^t,x2i

)dr+CE

"Z

{|e|<_k¹}

(1∧ |e|²)λ(de)

#

(by Jensen),

By Bihari’s inequality (see [8] page 171 and [9]) and the fact of Z

{|e|<¹_k}

(1∧ |e|²)λ(de)k−→+∞^−→ 0; we obtain our result the (4.34).

We now focus on (4.33). Note that we can apply Ito’s formula, even if the BSDEs are related to filtrations and Poisson random measures which are not the same, since:

(i)F_s^k ⊂ Fs,∀s≤T; (ii) for any s ≤ T,

Z s 0

Z k E

U^i;t,x(e)µek(dr, de) = Z s

0

Z k E

U^i;t,x(e)1_{|e|≥¹

k}µe(dr, de) and then the first (F_s^k)s≤T−martingale is also an(Fs)s≤T−martingale. ∀s∈[0, T],

Y~_s^t,x−^kY_s^t,x

2

+ Z T

0

Z_s^t,x−^kZ_s^t,x2ds+ X

s≤r≤T

(^k∆rY~_r^t,x)²

=g(X_T^t,x)−g(^kX_T^t,x)²+ 2 Z T

s

Y~_r^t,x−^kY_r^t,x

×^k∆f(r)dr

−2 Z T

s

Z

E

Y~_r^t,x−^kY_r^t,x

k∆Ur(e) e

µ(dr,de)−2 Z T

s

Y~_r^t,x−^kY~_r^t,x

k∆Zr

dB_r;

and taking expectation we obtain: ∀s∈[t, T],

E

"Y~_s^t,x−^kY_s^t,x

2

+ Z T

0

Z_s^t,x−^kZ_s^t,x2+ Z

E

U_s^t,x−^kU_s^t,x1_{|e|≥¹

k}

2

λ(de)

ds

#

≤E

"

g(X_T^t,x)−g(^kX_T^t,x)²+ 2 Z T

s

Y~_r^t,x−^kY_r^t,x

×^k∆f(r)dr

#

;

(4.39) where the processes^k∆Xr,^k∆Yr,^k∆f(r), ^k∆Zrand^k∆Ur are defined as follows: ∀r∈[0, T],

k∆f(r) := ((^k∆f⁽ⁱ⁾(r))i=1,m = (f⁽ⁱ⁾(r, X_r^t,x, ~Y_r^t,x, Z_r^i;t,x, U_r^i;t,x)−f_k⁽ⁱ⁾(r,^kX_r^t,x,^kY_r^t,x,^kZ_r^t,x,^kU_r^t,x))i=1,m,

k∆Xr = X_r^t,x−^k X_r^t,x, ^k∆Y(r) = Y~_r^t,x−^k Y_r^t,x = (Y_r^j;t,x −^k Y_r^j;t,x)j=1,m, ^k∆Zr = Z_r^t,x−^k Z_r^t,x and

k∆Ur=U_r^t,x−^kU_s^t,x1_{|e|≥¹

k}. Next let us set forr≤T,

k∆f(r) = (f(r, X_r^t,x, ~Y_r^t,x, Z_r^t,x, U_r^t,x)−f_k(r,^kX_r^t,x,^kY_r^t,x,^kZ_r^t,x,^kU_r^t,x)) =A(r) +B(r) +C(r) +D(r);

where for any i= 1, . . . , m, A(r) =

h⁽ⁱ⁾

r, X_r^t,x, ~Y_r^t,x, Z_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

−h⁽ⁱ⁾

r,^kX_r^t,x, ~Y_r^t,x, Z_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

i=1,m

; B(r) =

h⁽ⁱ⁾

r,^kX_r^t,x, ~Y_r^t,x, Z_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

−h⁽ⁱ⁾

r,^kX_r^t,x,^kY_r^t,x, Z_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

i=1,m

; C(r) =

h⁽ⁱ⁾

r,^kX_r^t,x,^kY_r^t,x, Z_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

−h⁽ⁱ⁾

r,^kX_r^t,x,^kY_r^t,x,^kZ_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

i=1,m

; D(r) =

h⁽ⁱ⁾

r,^kX_r^t,x,^kY_r^t,x,^kZ_r^i;t,x, Z

E

γⁱ(r, X_r^t,x, e)U_r^i;t,x(e)λ(de)

−h⁽ⁱ⁾

r,^kX_r^t,x,^kY_r^t,x,^kZ_r^i;t,x, Z

E

γⁱ(r,^kX_r^t,x, e)^kU_r^i;t,x(e)λk(de)

i=1,m

.

Sinceg belongs toMand by (4.34) we have,

Ehg(X_T^t,x)−g(^kX_T^t,x)²i

−→0

k→+∞ (4.40)

Now we will interest toE

"Z T s

Y~_r^t,x−^kY_r^t,x

×^k∆f(r)dr

#

for found (4.33).

By (2.7), we have: ∀r∈[0, T]

|A(r)|² ≤ ρ(X_r^t,x−^kX_r^t,x2);