Discrete-time Approximation of Multidimensional BSDEs with oblique reflections

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Discrete-time Approximation of Multidimensional BSDEs with oblique reflections

Romuald Elie, Idris Kharroubi, Jean-François Chassagneux

To cite this version:

Romuald Elie, Idris Kharroubi, Jean-François Chassagneux. Discrete-time Approximation of Multidi- mensional BSDEs with oblique reflections. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2012, 22 (3), pp 971 - 1007. �hal-00475628�

Discrete-time Approximation of Multidimensional BSDEs with oblique reflections

Jean-Francois CHASSAGNEUX

Département de Mathématiques, Université d’Evry Val d’Essone,

and CREST

[email protected]

Romuald ELIE

CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine,

and CREST [email protected]

Idris KHARROUBI

LPMA, CNRS, UMR 7599 Université Paris 7,

and CREST [email protected]

This version: April 2010

Abstract

In this paper, we study the discrete-time approximation of multi-dimensional reflected BSDEs of the type of those presented by Hu and Tang [16] and generalized by Hamadène and Zhang [15]. In comparison to the penalizing approach followed by Hamadène and Jeanblanc [14] or Elie and Kharroubi [12], we study a more natural scheme based on oblique projections. We provide a control on the error of the algorithm by introducing and studying the notion of multidimensional discretely reflected BSDE. In the particular case where the driver does not depend on the variableZ, the error on the grid points is of order ¹₂−ǫ,ǫ >0.

Key words: BSDE with oblique reflections, discrete time approximation, Switching problems.

MSC Classification (2000): 93E20, 65C99, 60H30.

1 Introduction

The main motivation of this paper is the discrete-time approximation of the following system of reflected Backward Stochastic Differential Equations (BSDEs)







Y˙_tⁱ=gⁱ(X_T) +R_T

t fⁱ(X_s,Y˙_sⁱ,Z˙_sⁱ)ds−R_T

t Z˙_sⁱdW_s+ ˙K_Tⁱ −K˙_tⁱ, 0≤t≤T , Y˙_tⁱ≥max_j∈I{Y˙_t^j−c^ij(X_t)}, 0≤t≤T ,

R_T

0 [ ˙Y_tⁱ−max_j∈I{Y˙_t^j−c^ij(X_t)}]dK˙_tⁱ = 0, i∈ I ,

(1.1)

whereI := {1, . . . , d},f, g and (c^ij)_i,j∈I are Lipschitz functions andX is the solution of a forward Stochastic Differential Equation (SDE).

These equations are linked to the solutions of optimal switching problems, arising for example in real option pricing. In the particular case wheref does not depend of( ˙Y ,Z),˙ a first study of these equations was made by Hamadène and Jeanblanc [14]. They derive existence and uniqueness of solution to this problem in dimension 2. The extension of this result to optimal switching problems in higher dimension is studied by Djehiche, Hamadène and Popier [9], Carmona and Ludkovski [5], Porchet, Touzi and Warin [24]

and by Pham, Ly Vath and Zhou [23] for an infinite time horizon consideration. In this last paper, the resolution of optimal switching problems relies mostly on the link with systems of variational inequalities.

Considering deterministic costs, Hu and Tang [16] derive existence and uniqueness of solution to this type of BSDE and relate it to optimal switching problems between one dimensional BSDEs. Extensions developed in [15] and [8] cover in particular the exis- tence of a unique solution to the BSDE (1.1). Recently two of the authors related in [11] the solution of (1.1) to corresponding constrained BSDEs with jumps. As presented in [12], this type of BSDE can be numerically approximated combining a penalization procedure with the use of the moonwalk scheme for BSDEs with jumps. Unfortunately, no convergence rate is available for this algorithm. We present here a more natural dis- cretization scheme based on a geometric approach. For any t≤T, all the components of theY˙_tprocess are interconnected, so that the vectorY˙_tlies in a random closed convex set Q(X_t) characterized by the cost functions (c^ij)_i,j∈I. The vector process Y˙ is thus obliquely reflected on the boundaries of the domainQ(X) and we plan to approximate these continuous reflections numerically.

As in [19, 1, 6], we first introduce a discretely reflected version of (1.1), where the reflection occurs only on a deterministic grid ℜ={r₀ := 0, . . . , r_κ :=T}:

Y_T =Ye_T :=g(X_T)∈ QT, and, forj≤κ−1and t∈[r_j, r_j+1), ( Ye_t = Y_rj+1+Rrj+1

t f(X_u,Ye_u, Z_u)du−Rrj+1

t Z_udW_u ,

Yt = Yet1{t /∈ℜ}+P(Xt,Yet)1{t∈ℜ}, (1.2) where P(X_t, .) is the oblique projection operator on Q(X_t), for t≤ T. Extending the approach of Hu and Tang [16], we observe that the solution to (1.2) interprets as the value process of a one-dimensional optimal BSDE switching problem with switching times belonging to ℜ. This allows us to prove a key stability result for this equation.

We control the distance between(Y, Z)and( ˙Y ,Z˙)in terms of the mesh of the reflection grid. Due to the obliqueness of the reflections, the direct argumentation of [1, 6] does not apply. Using the reinterpretation in terms of switching BSDEs, we first prove that Y approaches Y˙ on the grid points with a convergence rate of order 1−ε, ε >0 uni- formly in ℜ, whenever the cost function is Lipschitz andf is bounded in z. Imposing more regularity on the cost function, we control the convergence rate of (Y_t, Z_t)_0≤t≤T to ( ˙Y_t,Z˙_t)_0≤t≤T.

We then consider an Euler type approximation scheme associated to the BSDE (1.2) defined on π={t₀, . . . , t_n} by Y_T^π := g(X_T^π)and, fori∈ {n−1, . . . ,0},









Z¯_t^π_i := (t_i+1−t_i)⁻¹E h

Y_t^π_i+1(W_ti+1−W_ti)^′ | Fti

i , Ye_t^π_i := E

hY_t^π_i+1 | Fti

i+ (t_i+1−t_i)f(X_t^π_i,Ye_t^π_i,Z¯_t^π_i),

Y_t^π_i := Ye_t^π_i1{ti∈ℜ}/ +P(X_t^π_i,Ye_t^π_i)1{ti∈ℜ},

(1.3)

where X^π is the Euler scheme associated to X. It is now well known, see e.g. [2, 25], that the convergence rate of the scheme (1.3) to the solution of (1.2) is controled by the regularity of (Y, Z) through the quantities

E

"

X

i<n

Z _ti+1

ti

|Y_t−Y_ti|²dt

#

and E

"

X

i<n

Z _ti+1

ti

|Z_t−Z¯_ti|²dt

# ,

withZ¯_ti = _t ¹

i+1−tiEhR_ti+1

ti Z_tdt| Fti

i

, fori≤n.

Using classical Malliavin differentiation tools, we prove a representation forZ, extending the results of [1, 6] to the system of discretely reflected BSDEs (1.2). We deduce the expected regularity results on (Y, Z) and, using the technics of [7], Chapter 3, we obtain in a very general setting the convergence of (1.3) to (1.2). However, due to the obliqueness of the reflections, the projection operator P(X, .) is only L-lipschitz with L > 1, leading to a convergence rate of order L^κ|π|^1/4, where we recall that κ is the

number of points in the reflection gridℜ. The termL^κ can be very large even for small κ and leads to a poor logarithmic convergence rate when passing to the limit κ → ∞ for the approximation of (1.1). In the particular case where f does not depend on z, we are able to get rid of theL^κ term.

Our innovative approach relies on on the use of comparison results to get a control of the involved quantities:

• we interpret the solution of (1.2) as a value process of an optimization problem, which allows to get a control of the distance between the continuously and dis- cretely reflected BSDEs,

• we introduce a convenient auxiliary process dominating both solutions (1.2) and (1.3), to get a control of the distance between these quantities.

Combining the previous estimates, we deduce the convergence of the discrete time scheme (1.3) to the solution of (1.1) with a convergence rate of order |π|^12−ε on the grid points, whenever ℜ =π and f is independent of Z. Whenever the cost functions are constant, all the previous estimates hold true with ε= 0. We want to emphasize that all these results are obtained without any assumption on the non-degeneracy of the volatility matrix σ.

The rest of the paper is organized as follows. In Section 2, we introduce the notion of discretely obliquely reflected BSDEs, connect it with optimal switching problems and give the fundamental stability result. Section 3 focuses on the regularity of the solution to this new type of BSDE. This analysis leads to precious estimates allowing to deduce the convergence of the associated discrete time scheme, see Section 4. Afterwards, Section 5 focuses on the extension to the continuously reflected case and provides a convergence rate of the discretely reflected BSDE to the continuously one, whenever the driverf is bounded in the variable Z. The global error of the scheme is provided at the end of this section. Somea priori estimates are reported in the Appendix.

Notations. Throughout this paper we are given a finite time horizon T and a prob- ability space (Ω,F,P) endowed with a d-dimensional standard Brownian motion W = (Wt)t≥0. The filtration F= (F^t)_t≤T generated by the Brownian motion is supposed to satisfy the usual conditions. Here, P denotes theσ−algebra on[0, T]×Ωgenerated by F−progressively measurable processes. Any elementx∈R^ℓwithℓ∈Nwill be identified to a column vector with i-th component xⁱ and Euclidian norm|x|. For x, y∈R^ℓ,x.y denotes the scalar product ofxandy, andx^′ denotes the transpose ofx. We denote by

the component by component partial ordering relation on vectors. M^m,d denotes the set of real matrices withmlines and dcolumns. We denotes byC_b^k the set of functions from R^d to R with continuous and bounded derivatives up to order k. For a function f ∈C¹, ∇^xf denotes the Jacobian matrix of f with respect to x. Finally, for ease of notation, we will sometimes writeE_t[·]instead ofE[·|Ft],t∈[0, T]. In the following, we shall use these notations without specifying the dimension nor the dependence inω∈Ω when it is clearly given by the context. Finally, for any p≥1, we introduce:

• the setS^p of real-valuedcàdlàg P-measurable processes Y = (Y_t)_0≤t≤T satisfying kYkSp :=E

sup_0≤t≤T|Y_t|^p1_p

< ∞.

• the set H^p of R^d-valued P-measurable processesZ = (Z_t)_0≤t≤T such that kZk^Hp :=

E h

(RT

0 |Zt|²dt)^p2i¹_p

< ∞.

•the closed subsetA^p ofS^p consisting of nondecreasing processesK satisfyingK₀ = 0.

In the sequel we denote byC_L a constant whose value may change from line to line but which depends only on L. We use the notation C_L^p whenever it depends on some other parameter p >0.

2 Discretely obliquely reflected BSDE

In the beginning of this section, we define and study discretely obliquely reflected BSDEs in a general setting. In particular, we show how their solutions relate to the solutions of one-dimensional optimal switching problems, where the switching times are restricted to lie in a discrete time set. This allows to prove a stability result for obliquely RBSDEs, which will be use several times in the paper.

2.1 Definition

A discretely obliquely reflected BSDE is a reflected BSDE where the reflection is only allowed on a discrete time set.

We thus consider a gridℜ:={r₀ = 0, . . . , r_κ=T}of the time interval [0, T]satisfying

|ℜ|:= max

1≤k≤κ|r_k−r_k−1| ≤ L

κ. (2.1)

We also consider a matrix valued processC = (C^ij)_1≤i,j≤m such thatC^ij belongs to S² for i, j∈ {1, . . . , d}and satisfies the following structure condition









C_tⁱⁱ= 0, for 1≤i≤d and0≤t≤T ; inf_0≤t≤T C_t^ij > _L¹ , for 1≤i, j≤d , withi6=j; inf_0≤t≤T C_t^ij+C_t^jl−C_t^il >0, for 1≤i, j, l≤d, withi6=j, j 6=l.

(2.2)

We introduce a random closed convex set family associated toC:

Qt:=

y∈R^d|yⁱ≥max

j (y^j−C_t^ij), 1≤i≤d

, 0≤t≤T ,

and the oblique projection operator onto Qt, denoted Pt and defined by Pt:y∈R^d 7→

maxj∈I

n

y^j−C_t^ijo

1≤i≤d

.

which is P⊗ B(R^d)-measurable.

Remark 2.1. (i) It follows from the structure condition (2.2) thatP is increasing with respect to the partial ordering relation , whereyy^′ means yⁱ≥(y^′)ⁱ for alli∈ I. (ii) An easy calculation leads to

|Pt(y₁)− Pt(y₂)| ≤√

d|y₁−y₂|. (2.3)

ThusP is L- Lipschitz continuous with L >1, recalling thatd≥2.

Finally, we are also given a random variable ξ ∈ [L²(FT)]^d valued in QT, representing the terminal value of the BSDE and a random functionF : Ω×[0, T]×R^d× M^d,q →R^d which is P⊗ B(R^d)⊗ B(M^d,q)−measurable and satisfies the Lipschitz property:

|F(t, y, z)−F(t, y^′, z^′)| ≤L(|y−y^′|+|z−z^′|), for all (t, y, y^′, z, z^′)∈[0, T]×(R^d)²×(M^d,q)²,P−a.s.

We shall also assume that

(HF) The component iofF(t, y, z) depends only on the componentiof the vector y and on the rowiof the matrix z, i.e. Fⁱ(t, y, z) =Fⁱ(t, yⁱ, zⁱ).

Given this set of data(ℜ, C, F, ξ), a discretely obliquely reflected BSDE, denotedD(ℜ, C, F, ξ), is a triplet(Y , Y, Ze )∈(S²× S²× H²)^I satisfying Y_T =Ye_T :=ξ∈ Q(T), and defined in a backward manner, for j≤κ−1 and t∈[r_j, r_j+1), by

( Ye_t = Y_rj+1+R_rj+1

t F(u,Ye_u, Z_u)du−R_rj+1

t Z_udW_u ,

Y_t = Ye_t1{t /∈ℜ}+Pt(Ye_t)1{t∈ℜ}. (2.4) This rewrites equivalently fort∈[0, T]as

( Ye_t = ξ+R_T

t F(u,Ye_u, Z_u)du−R_T

t Z_udW_u+ (Ke_T −Ke_t), Ke_t := P

r∈ℜ\{0}∆Ke_r1{r≤t} with ∆Ke_t:=Y_t−Ye_t=−(Ye_t−Ye_t−), (2.5)

Observe that Ke ∈(A²)^I, since C^ij is positive and valued in S², for any i, j∈ I. We shall also use the following integrability condition for somep≥2

(C_p) |ξ|^p+ sup

t∈[0,T]|C_t|^p+ Z _T

0 |F(s,0,0)|^pds ≤ β ,

where β is a positive random variable satisfying E[β] ≤ C_L. Importantly, β does not depend onℜ.

The proof of the following a priori estimates is postponed in Section 6.1 of the Appendix.

Proposition 2.1. Assume that(C_p)holds for some given p≥2, there exists a unique solution(Y , Y, Z)e to (2.4) and it satisfies

kYek^Sp +kZk^Hp +kKeTkLp ≤C_L^p .

2.2 Corresponding optimal switching problem

In this subsection, we interpret the solution of the discretely obliquely RBSDE (2.5) as the value process of a corresponding optimal switching problem, where the possible switching times are restricted to belong to the grid ℜ. Our approach relies on similar arguments as the one followed by Hu and Tang [16] in a framework with continuous reflections.

A switching strategyais a nondecreasing sequence of stopping times(θ_j)_j∈N, combined with a sequence of random variables(α_j)_j∈Nvalued inI, such thatα_jisFθj−measurable, for any j ∈N. We denote by Athe set of such strategies. For a= (θ_j, α_j)_j∈^N∈ A, we introduceN^a the (random) number of switches before T:

N^a= #{k∈N^∗ : θ_k≤T}. (2.6)

To any switching strategy a= (θ_j, α_j)_j∈N ∈ A, we associate the current state process (a_t)_t∈[0,T] and the compound cost process (A^a_t)_t∈[0,T] defined respectively by

at := α01{0≤t<θ0}+

N^a

X

j=1

αj−11{θj−1≤t<θj} and A^a_t :=

N^a

X

j=1

C_θ^αj−1αj

j 1{θj≤t≤T},

for 0≤t≤T. For (t, i)∈[0, T]× I, the setA^t,i of admissible strategies starting fromi at timetis defined by

At,i ={a= (θ_j, α_j)_j ∈ A |θ₀=t, α₀=i, E

|A^a_T|²

<∞},

similarly we introduce A^ℜt,i the restriction toℜ−admissible strategies A^ℜt,i := {a= (θ_j, α_j)_j∈N∈ At,i | θ_j ∈ ℜ , ∀j ≤N^a } ,

and denoteA^ℜ :=S

i≤dA^ℜ0,i.

For(t, i)∈[0, T]× I,anda∈ A^ℜt,i, we consider as in [16] the associated one dimensional switched BSDE defined by

U_u^a=ξ^aT + Z T

u

F^as(s, U_s^a, V_s^a)ds− Z T

u

V_s^adWs−A^a_T +A^a_u, t≤u≤T . (2.7) Theorem 3.1 in [16] interprets each component of the solution to the continuously re- flected BSDE (1.1) as the Snell envelop associated to switched processes of the form (2.7), where the switching strategiesa are not restricted to lie in the reflection gridℜ. The next theorem is a new version of this Snell envelop representation adapted to the consideration of discretely obliquely reflected BSDE (2.5).

Theorem 2.1. Assume that(C₂)is in force. For anyi∈ I andt∈[0, T], the following holds:

(i) The processYe dominates any ℜ-switched BSDE, i.e.

U_t^a ≤ Ye_tⁱ P−a.s., for any a∈ A^ℜi,t . (2.8) (ii) Define the strategy a^∗ = (θ_j^∗, α^∗_j)j≥0 recursively by (θ₀^∗, α^∗₀) := (t, i) and, forj≥1,

θ_j^∗ := inf (

s∈[θ^∗_j−1, T]∩ ℜ eY^α

∗

s j−1 ≤ max

k6=α^∗_j−1

nYe_s^k−C^α

∗ j−1k s

o) ,

α^∗_j := min (

ℓ6=α^∗_j−1 eY_θ^ℓ∗

j −C^α

∗ j−1ℓ

θ_j^∗ = max

k6=α^∗_j−1

nYe_s^k−C^α

∗ j−1k θ_j^∗

o) .

Then, we have a^∗ ∈ A^ℜt,i and

Ye_tⁱ =U_t^a∗ P−a.s.. (2.9) (iii) The following “Snell envelop” representation holds:

Ye_tⁱ = ess sup

a∈A^ℜ_t,i

U_t^a, P−a.s.. (2.10)

Proof. Observe first that Assertion(iii) is a direct consequence of (i)and (ii).

Step 1. We first prove(i).

Fix t ∈[0, T] and i∈ I. Set a= (θ_k, α_k)_k≥0 ∈ A^ℜt,i and the process (Ye^a, Z^a) defined, for s∈[t, T], by

Ye_s^a:=X

k≥0

Ye_s^αk1{θk≤s<θk+1}+ξ^aT1{s=T} and Z_s^a:=X

k≥0

Z_s^αk1{θk≤s<θk+1}. (2.11) Observe that these processes jump between the components of the discretely reflected BSDE (3.5) according to the strategya, and, between two jumps, we have

Ye_θ^a_k = Ye_θ^αk

k = Y_θ^αk

k+1+ Z θk+1

θk

F^αk(s,Ye_s^αk, Z_s^αk)ds− Z θk+1

θk

Z_s^αkdWs+Ke_θ^αk

k+1−−Ke_θ^αk

k

= Ye_θ^a

k+1+ Z θk+1

θk

F^as(s,Ye_s^a, Z_s^a)ds− Z θk+1

θk

Z_s^adW_s+Ke_θ^αk

k+1−−Ke_θ^αk

k

+(Y_θ^αk

k+1−Ye_θ^αk+1

k+1 ), k≥0. (2.12)

Introducing Ke_s^a:=

NX^a−1

k=0

"Z

(θk∧s,θk+1∧s)

dKe_u^αk+1{θk+1≤s}(Y_θ^αk

k+1−Ye_θ^αk+1

k+1 +C_θ^αkαk+1

k+1 )

# ,

for s∈[t, T], and summing up (2.12) overk, we get, for t≤u≤T, Ye_u^a=ξ^aT +

Z T

u

F^as(s,Ye_s^a, Z_s^a)ds− Z T

u

Z_s^adW_s−A^a_T +A^a_u+Ke_T^a−Ke_u^a.

Using the relation Y_θk =Pθk(Ye_θk) for allk∈ {0, . . . , N^a}, we check thatKe^a is increas- ing. Since U^a solves (2.7), we deduce by a comparison argument (see Theorem 1.3 in [22]) thatU_t^a≤Ye_t^a. Since ais arbitrary inA^ℜt,i, we deduce (2.8).

Step 2. We now prove (ii). Consider the strategy a^∗ given above as well as the associated process(Ye^a∗, Z^a∗) defined as in (2.11). By definition of a^∗, we have

Y^α

∗ k

θ_k+1^∗ =

Pθ^∗_k+1(Ye_θ^∗

k+1)α^∗_k

= Ye^α

∗ k+1

θ^∗_k+1 −C^α

∗ kα^∗_k+1

θ^∗_k+1 , k≥0, which gives

Z

(θ_k^∗,θ^∗_k+1)

dKe^α

∗

sk = 0 and Y^α

∗ k

θ_k+1^∗ −Ye^α

∗ k

θ_k+1^∗ +C^α

∗ kα^∗_k+1

θ_k+1^∗ = 0, (2.13) for all k∈ {0, . . . , N^a∗−1}. We deduce from (2.2) that

Ye_u^a∗ = ξ^a∗T + Z T

u

F^a∗s(s,Ye_s^a∗, Z_s^a∗)ds− Z T

u

Z_s^a∗dWs−A^a_T^∗+A^a_u^∗, t≤u≤T .

Hence(Ye^a∗, Z^a∗)and (U^a∗, V^a∗) are solutions of the same BSDE and satisfyYe_tⁱ =U_t^a∗. To complete the proof, we only need to check that a^∗ ∈ A^ℜ, i.e. E|A^a_T^∗|² < ∞. By definition ofa^∗ on [t, T]and the structure condition on the cost (2.2), we have |A^a_t^∗| ≤ maxk6=i|C_t^i,k|which givesE[|A^a_t^∗|²]≤CL. Combining

A^a_T^∗ = Ye_T^a∗−Ye_t^a∗+ Z _T

t

F^a∗s(s,Ye_s^a∗, Z_s^a∗)ds− Z _T

t

Z_s^a∗dW_s+A^a_t^∗ ,

with the Lipschitz property ofFand the fact that(Y , Z)e ∈(S²×H²)^I, recall Proposition 2.1, we get the square integrability ofA^a_T^∗ and conclude the proof. ✷ Remark 2.2. Although the optimal strategy a^∗ depends on the initial parameters t and i, we omit the script (t, i) for ease of notation.

Combining the previous representation with the a priori estimates of Proposition 2.1 and the structure condition (2.2), we deduce the following estimates, whose proof is postponed to Section (6.1) in the Appendix.

Proposition 2.2. Assume that(C_p)holds for some given p≥2, then

E

"

sup

s∈[t,T]|U_s^a∗|^p+ Z T

t |V_u^a∗|²du ^p₂

+|A^a_T^∗|^p+|N^a∗|^p

#

≤C_L^p ,

for the optimal strategy a^∗∈ A^ℜt,i, (t, i)∈[0, T]× I.

2.3 Stability of obliquely reflected BSDEs

We now study the dependence on the solution with respect to the parameters of the BSDE. In the ‘abstract’ setting considered, we obtain precious estimates for the analysis of the regularity of the solution to the discretely obliquely reflected BSDE as well as the convergence of the discrete-time scheme.

We consider two discretely reflected BSDEs, with the same reflection gridℜbut different parameters. Forℓ∈ {1,2}, we consider anFT-measurable random terminal condition^ℓξ, a randomL-lipschitz continuous map(y, z)7→^ℓF(., y, z), satisfying(HF), and a matrix of continuous cost processes (^ℓC^ij)_1≤i,j≤d satisfying the structural condition (2.2).

We suppose that the coefficients satisfy the integrablity condition (C₄). Forℓ∈ {1,2}, we denote by (^ℓY,^ℓY ,e ^ℓZ) ∈ (S²× S²× H²)^I the solution of the obliquely discretely reflected BSDE D(ℜ,^ℓC,^ℓF,^ℓξ).

Defining δY =¹Y −²Y,δYe =¹Ye −²Ye,δZ =¹Z−²Z,δξ :=¹ξ−²ξ together with

|δC_s|∞:= max

i,j∈I|¹C^ij −²C^ij|(s), |δF_s|∞:= max

i∈I sup

y,z∈R^d×M^d,q|¹Fⁱ−²Fⁱ|(s, y, z), for s∈[0, T], we prove the following stability result.

Proposition 2.3. Assume that(C₄)holds. Then, we have, for any t∈[0, T],

E

|δY_t|² +E

h

|δYe_t|²i +1

κE Z T

t |δZ_s|²ds

≤C_L E

Z T

t |δF_s|²∞ds+|δξ|²

+E

sup

r∈ℜ|δC_r|⁴∞

¹₂ , Proof. The proof divides in three steps and relies heavilly on the reinterpretation in terms of switching problems. We first introduce a convenient dominating process, and then provide successively the controls on theδY andδZ terms.

Step 1. Introduction of an auxiliary BSDE.

Let us define F := ¹F ∨²F, ξ := ¹ξ∨²ξ and C by C^ij := ¹C^ij ∧²C^ij. Observe that F satisfies(HF), C satisfies the structure condition (2.2) and that (C₄) holds for the data(C, F, ξ). We denote by (Y,Y , Z)e the solution of the discretely obliquely reflected BSDE D(ℜ, C, F, ξ), recalling (2.4).

Using(HF), the definition ofF and the monotonicity property ofP, see Remark 2.1 (i), we easily obtain by a comparison argument on each interval[r_k, r_k+1),k∈ {0, . . . , κ−1}, that

Ye ¹Ye ∨²Y .e (2.14)

Recalling Theorem 2.1, we introduce the switched BSDEs associated to ¹Y, ²Y and Y and denote by aˇ = (ˇθ_j,ˇa_j)_j≥0 the optimal strategy related to Y starting from a fixed (i, t)∈ I ×[0, T]. Therefore, we have

Ye_tⁱ=U_t^aˇ=ξ^ˇaT + Z T

t

F^ˇas(s, U_s^aˇ, V_s^aˇ)ds− Z T

t

V_s^ˇadW_s−A^ˇa_T +A^ˇa_t . (2.15) Step 2. Stability of theY component.

Since ˇa∈ A^ℜt,i, we deduce from Theorem 2.1 (iii) that

ℓYe_tⁱ ≥ ^ℓU_t^ˇas = ^ℓξ^ˇaT + Z T

t

ℓF^ˇas(s,^ℓU_s^aˇ,^ℓV_s^aˇ)ds− Z T

t

ℓV_s^ˇadWs−^ℓA^ˇa_T +^ℓA^ˇa_t , ℓ∈ {1,2},

where ^ℓA^ˇa is the process of cumulated costs (^ℓC^ij)i,j∈I associated to the strategy ˇa.

Combining this estimate with (2.14) and (2.15), we derive

|¹Ye_tⁱ−²Ye_tⁱ| ≤ |U_t^aˇ−¹U_t^ˇa|+|U_t^aˇ−²U_t^ˇa|. (2.16) Since both terms on the right hand side of (2.16) are treated similarly, we focus on the first one and introduce the continuous processesΓ^aˇ :=U^ˇa+A^ˇa and ¹Γ^ˇa:=¹U^ˇa+¹A^ˇa. Applying Ito’s formula, we compute, for allt≤u≤T,

E_t

|Γ^ˇa_u−¹Γ^a_u^ˇ|²+ Z T

u |V_s^ˇa−¹V_s^ˇa|²ds

≤ (2.17)

E_t

|Γ^ˇa_T −¹Γ^a_T^ˇ|²+ 2 Z _T

u

(Γ^ˇa_s−¹Γ^a_s^ˇ)[F^ˇas(s, U_s^aˇ,¹V_s^aˇ)−¹F^ˇas(s,¹U_s^ˇa,¹V_s^ˇa)]ds

.

Since F =¹F ∨²F and¹F is Lipschitz continuous, we also get

|F^ˇas(s, U_s^ˇa,¹V_s^aˇ)−¹F^ˇas(s,¹U_s^ˇa,¹V_s^ˇa)| ≤

|δF_s|∞+L(|Γ^ˇa_s−¹Γ^ˇa_s|+|A^ˇa_s−¹A^ˇa_s|+|V_s^ˇa−¹V_s^ˇa|), 0≤s≤T .

Using classical arguments, we then deduce from the last inequality and (2.17) that

|Γ^ˇa_t −¹Γ^a_t^ˇ|² ≤ CL E_t

|δξ^ˇaT|² Z T

t |δFs|²∞ds

+ sup

t≤s≤T

E_t

|A^ˇa_s−¹A^ˇa_s|²!

(2.18) Moreover, using the inequality |x∨y−y| ≤ |x−y|for x, y ∈ Rand the convexity of the function x7→x², we compute

E_t

|A^a_s^ˇ−¹A^ˇa_s|²

=E_t





N^aˇ

X

k=1

₂

C^{αˇk−1αˇk}∧¹C^{αˇk−1αˇk}−¹C^{αˇk−1αˇk}

(ˇθ_k)1_{θˇk≤s}

2



≤E_t

|N^ˇa|sup

r∈ℜ|δC_r|²∞

, t≤s≤T . (2.19)

Plugging in (2.18) and recalling the definition of Γ^ˇa and ¹Γ^ˇa, we get

|U_t^ˇa−¹U_t^ˇa|² ≤ C_LE_t

|N^ˇa|sup

r∈ℜ|δC_r|²∞+ Z _T

t |δF_s|²∞ds+|δξ|²

.

The exact same reasoning leads to the same estimate for |U_t^ˇa−²U_t^ˇa|². Therefore, we deduce from (2.16) and Cauchy Schwartz inequality that

E

h|²Ye_tⁱ−¹Ye_tⁱ|²i

≤ C_L E

|N^ˇa|²¹₂ E

sup

r∈ℜ|δC_r|⁴∞

¹₂ +E

Z T

t |δF_s|²∞ds+|δξ|² . (2.20) Using Proposition 2.2, we compute, sincei is arbitrary,

E h

|²Ye_t−¹Ye_t|²i

≤ C_L E

Z _T

t |δF_s|²∞ds+|δξ|²

+E

sup

r∈ℜ|δC_r|⁴∞

¹₂

. (2.21) Step 3. Stability of theZ component.

Applying Ito’s formula to the càdlàg process|δYe|², we obtain

E



|δYe_t²|+ Z T

t |δZs|²ds+ X

t<r≤T

|∆δK˜r|²



=E

|δYe_T|²+ 2 Z T

t

δYsδFsds+ 2 Z T

t

δYrdδK˜r

,

where we used the fact that |δYe|² − |δY|²−2δY(δYe −δY) = |∆δK˜|². Since δKe is a pure jump process, we compute

E Z _T

t

δY_rdδK˜_r

≤ E



α X

t<r≤T,r∈ℜ

|δY_r|²+ 1 α

X

t<r≤T

|∆δK˜_r|²



 , α >0,

which, for α large enough and using standard arguments, leads to E



 Z _T

t |δZ_s|²ds+X

t<r≤T

|∆δK˜_r|²



≤C_L



E

|δξ|² +E



 Z _T

t |δF_s|²∞ds+ X

t<r≤T,r∈ℜ

|δY_r|²







 .

Since (2.21) holds true for anyt∈[0, T], we deduce E



 Z _T

t |δZ_s|²ds+X

t<r≤T

|∆δK˜_r|²



≤C_Lκ E

|δξ|² +E

Z _T

t |δF_s|²∞ds

+E

sup

r∈ℜ|δC_r|⁴∞

¹₂! ,

which concludes the proof of the proposition. ✷

3 Regularity of discretely obliquely reflected BSDEs

This section is dedicated to the derivation of regularity properties for the solution of discretely reflected BSDEs. These results will be obtained in a Markovian diffusion setting. This means that the randomness of the parameter (C, F, ξ), is due to a state process X, which is the solution of a Stochastic Differential Equation (SDE). In this framework, we focus on the H²-regularity of the Z component of the solution of the BSDEs. The main results are retrieved by means of kernel regularization and Malliavin differentiation arguments. Finally, we extend this result to the case where the diffusion X is replaced by its Euler scheme.

3.1 A diffusion setting for discretely RBSDEs

LetX be the solution on [0, T]to the following SDE:

Xt=X0+ Z t

0

b(Xs)ds+ Z t

0

σ(Xs)dWs, 0≤t≤T , (3.1) whereX₀∈R^m and (b, σ) :R^m →R^m× M^m,q(R)are L-Lipschitz functions.

Under the above assumption, the following estimates are well known (see e.g. [18]) E

"

sup

t∈[0,T]|X_t|^p

#

≤C_L^p and sup

s∈[0,T]

E

"

sup

u∈[0,T],|u−s|≤h|X_s−X_u|^p

#!¹_p

≤C_L^p √

h , (3.2)

for any p > 0. In the sequel, we shall denote byβ^X a positive random variable, which may change from line to line, but which depends only on sup_t∈[0,T]|X_t| and which satisfiesE

|β^X|^p

≤C_L^p for allp >0. Importantly, β^X does not depend on ℜ.

Remark 3.1. Observe that, as in [1, 7] and contrary to [19], we make no uniform ellipticity condition onσ. This allows us to treat the case of non homogenous diffusion by setting e.g. X_t¹=t,t∈[0, T].

In this context, we are given a matrix valued mapsc:= (c^ij) wherec^ij :R^m →R⁺, are L-Lipschitz continuous and satisfy:









cⁱⁱ(.) = 0, for 1≤i≤d;

inf_x∈^Rmc^ij(x)>0, for 1≤i, j≤d , withi6=j ; inf_x∈Rm{c^ij(x)+c^jl(x)−c^il(x)}>0, for 1≤i, j, l≤d, withi6=j, j 6=l.

(3.3)

We then introduce a family (Q(x))_x∈^Rm of closed convex domains:

Q(x) :=

y∈R^d|yⁱ ≥max

j∈I(y^j−c^ij(x)), ∀i∈ I

, where I:={1, . . . , d}.(3.4) We introduce the oblique projection operator P(x, .) ontoQ(x)defined by

P : (x, y)∈R^m×R^d 7→

maxj∈I

y^j−c^ij(x)

1≤i≤d

.

Finally, we are given

- an L-Lipschitz functiong:R^m →R^d such thatg(x)∈ Q(x)for all x∈R^m, - a generator function, i.e. an L-lipschitz mapf :R^m×R^d× M^d,q→R^d. From now on, we shall appeal to the following assumption:

(Hf) the componentiof f(t, y, z) depends only on the componentiof the vector y and on the column iof the matrixz i.e. fⁱ(t, y, z) =fⁱ(t, yⁱ, zⁱ).

We denote by(Y,Y , Z)e the solution of the discretely reflected BSDED(ℜ, c(X), f(X, ., .), g(X)) which reads on each interval[r_j, r_j+1), forj < κ :

( Yet = Yrj+1+Rrj+1

t f(Xu,Yeu, Zu)du−Rrj+1

t ZudWu ,

Y_t = Ye_t1{t /∈ℜ}+P(X_t,Ye_t)1{t∈ℜ}. (3.5) Or equivalently on[0, T]as

( Ye_t = g(X_T) +RT

t f(X_u,Ye_u, Z_u)du−RT

t Z_udW_u+ (Ke_T −Ke_t), 0≤t≤T , Ket := P

r∈ℜ\{0}∆Ker1{r≤t} and ∆Ket=Yt−Yet=−(Yet−Yet−), 0≤t≤T. (3.6)

From (3.2), it follows that the data(c(X), f(X, ., .), g(X))satisfies the integrability con- dition(C_p)for allp≥2. We thus deduce from the proof of Proposition 2.1 and Propo- sition 2.2, the following estimate on (Y,Y , Z)e and their associated optimal switched BSDEs, recalling Theorem 2.1.

Proposition 3.1. There exists a unique solution (Y , Y, Z)e to (3.5) and it satisfies E_t

"

sup

s∈[t,T]|Ye_s|^p+ ( Z _T

t |Z_s|²ds)^p2 +|K_T −K_t|^p

#

≤β^X , ∀t≤T . (3.7) Moreover, for all (t, i)∈[0, T]× I, the optimal strategya^∗ ∈ A^ℜt,i satisfies

E_t

"

sup

s∈[t,T]|U_s^a∗|^p+ ( Z _T

t |V_s^a∗|²ds)^p2 +|A^a_T^∗|^p+|N^a∗|^p

#

≤β^X . (3.8)

3.2 Malliavin differentiability of (X, Y,Y , Z)e

We shall sometimes use the following regularity assumption on the coefficients:

(Hr) The coefficientsb,σ,g f, and(c^ij)_i,j areC^1,b in all their variables, with the Lipschitz constants dominated by L.

We denote byID^1,2the set of random variablesGwhich are differentiable in the Malliavin sense and such that kGk²_ID^1,2 := kGk²_L² +R_T

0 kD_tGk²_L²dt <∞, where D_tG denotes the Malliavin derivative ofGat time t≤T. After possibly passing to a suitable version, an adapted process belongs to the subspace L^1,2a of H² whenever Vs ∈ ID^1,2 for alls ≤T and kVk²_L^1,2

a :=kVk²_H² +R_T

0 kD_tVk²_H²dt <∞. For a general presentation on Malliavin calculus for stochastic differential equations, the reader may refer to [20].

Remark 3.2. Under(Hr), the solution of (3.1) is Malliavin differentiable and its deriva- tive satisfies

ksup

s≤T|D_sX|k^Sp <∞, (3.9)

and we have sup

s≤ukD_sX_t−D_sX_ukLp+k sup

t≤s≤T|D_tX_s−D_uX_s| kLp ≤ C_L^p|t−u|^1/2, (3.10) for any 0 ≤u ≤ t≤T. Let G∈ ID^1,2(R^d). Since X belongs to L^1,2a under(Hr), and P is L-lipschitz continuous, we deduce that P(X_t, G)∈ID^1,2(R^d). Using Lemma 5.1 in