Rate of convergence for the discrete-time approximation of reflected BSDEs arising in switching problems

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Rate of convergence for the discrete-time approximation

of reflected BSDEs arising in switching problems

Jean-François Chassagneux, Adrien Richou

To cite this version:

Jean-François Chassagneux, Adrien Richou. Rate of convergence for the discrete-time approxima-tion of reflected BSDEs arising in switching problems. Stochastic Processes and their Applicaapproxima-tions, Elsevier, 2019, 129 (11), �10.1016/j.spa.2018.12.009�. �hal-01264959�

Rate of convergence for the discrete-time approximation of

reflected BSDEs arising in switching problems

Jean-Fran¸cois Chassagneux

Laboratoire de Probabilit´es et Mod`eles Al´eatoires CNRS, UMR 7599, Universit´e Paris Diderot jean-francois.chassagneux@univ-paris-diderot.fr

Adrien Richou

Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France. adrien.richou@math.univ-bordeaux.fr

January 29, 2016

Abstract

In this paper, we prove new convergence results improving the ones by Chas-sagneux, ´Elie and Kharroubi [Ann. Appl. Probab. 22 (2012) 971–1007] for the discrete-time approximation of multidimensional obliquely reflected BSDEs. These BSDEs, arising in the study of switching problems, were considered by Hu and Tang [Probab. Theory Related Fields 147 (2010) 89–121] and generalized by Hamad`ene and Zhang [Stochastic Process. Appl. 120 (2010) 403–426] and Chassagneux, ´Elie and Kharroubi [Electron. Commun. Probab. 16 (2011) 120–128]. Our main re-sult is a rate of convergence obtained in the Lipschitz setting and under the same structural conditions on the generator as the one required for the existence and uniqueness of a solution to the obliquely reflected BSDE.

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

1

Introduction

In this paper, we study the discrete-time approximation of the following system of reflected backward stochastic differential equations

$ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % Yt“ gpXTq ` żT t f<sub>pX</sub>s, Ys, Zsq ds ´ żT t ZsdWs` KT ´ Kt, 0ď t ď T, Y_tℓ _{ě max} jPI tY j t ´ cℓjpXtqu, 0ď t ď T, ℓ P I, żT 0 ” Y_tℓ_{´ max} jPIztℓutY j t ´ cℓjpXtqu ı dKtℓ“ 0, ℓP I, (1.1) where I :_{“ t1, . . . , du, f , g and pc}ij_q

i,jPI are Lipschitz functions and X is solution to

the following forward stochastic differential equation (SDE) with Lipschitz coefficients

Xt“ x ` żt 0 b<sub>pX</sub>sqds ` żt 0 σ_pXsqdWs. (1.2)

An important motivation for this study comes from economics applications, espe-cially to energy markets. Indeed, it has been shown that the solution to the above equa-tions allows to compute the solution of optimal switching problems which are linked to real option pricing (see e.g. [3]). This motivated a huge literature on switching problems both on the financial economics and applied mathematics sides, as pointed out in the introduction of [16]. The theoretical study of equation (1.1) has started in dimension 2 in the paper [14] and was latter extended in higher dimension in [9, 3, 22]. These studies are related to optimal switching problem and, in terms of existence and uniqueness result to (1.1), impose really strong conditions on the driver f of the BSDEs. These conditions were then weakened successively in [17, 16, 7]. It is quite important to notice that contrary to normally reflected BSDEs [13], the best existence and unique-ness result available in the literature requires structural conditions, see below, both on the driver f and the function c. To the best of our knowledge, it can be found in the paper [15].

The numerical study of (1.1) by probabilistic methods has attracted much less attention [22, 11, 8]. The first rate of convergence for a numerical scheme associated to (1.1) was proved in [7] but under quite restrictive condition on the driver f . The main goal of our work is actually to prove a rate of convergence for a discrete-time scheme to obliquely reflected BSDEs under the same conditions on f required to have existence and uniqueness to (1.1) and minimal Lipschitz condition on the function c.

As in [1, 19, 8], we first introduce a discretely reflected version of (1.1), where the reflection occurs only on a deterministic grid ℜ“ tr0 :“ 0, . . . , rκ :“ T u: YTℜ “ rY

ℜ T :“

g_pXTq P QpXTq, and, for j ď κ ´ 1 and t P rrj, rj`1q,

$ & % r Y_tℜ_{“ Yr}ℜ_j`1` żrj`1 t f<sub>pX</sub>u, rYuℜ, Z ℜ uq du ´ żrj`1 t Z_uℜdWu, Y_tℜ_{“ r}Y_tℜ1_{ttRℜu` PpX}t, rYtℜq1ttPℜu, (1.3)

where Ppx, .q is the oblique projection operator on the closed convex domain Qpxq :“ " y_{P R}d_|yi _{ě max} jPI py j ´ cijpxqq, @i P I * , defined by P : px, yq P Rdˆ RdÞÑ ˆ max jPIty j ´ cijpxqu ˙ 1_ďiďd . We denote _{|ℜ| the modulus of ℜ given by |ℜ| :“ max}0ďiďκ´1|ri`1´ ri|.

An important step in our study is to prove that these discretely reflected BSDEs are a good approximation of the continuously reflected ones (1.1). In section 4, we are able to control the error in terms of_{|ℜ| under minimal Lipschitz condition for the cost} functions c, which is new in the literature, improving, in particular, the results of [8].

We then consider a Euler type approximation scheme associated to the BSDE (1.3) defined on a grid π_{“ tt}0, . . . , tnu by Ynℜ,π :“ gpX_Tπq and, for i P tn ´ 1, . . . , 0u,

$ ’ ’ & ’ ’ % Z_iℜ,π :_{“ ErYi`1</sub>ℜ,πHi| Ftis, r Y<sub>i</sub>ℜ,π:<sub>“ ErYi`1}ℜ,π _{| F}tis ` hifpX π ti, rY ℜ,π i , Z ℜ,π i q, Y<sub>i</sub>ℜ,π:<sub>“ r</sub>Y<sub>i</sub>ℜ,π1<sub>ttiRℜu` PpX</sub>π ti, rY ℜ,π i q1ttiPℜu, (1.4)

where Xπ _{is the Euler scheme associated to X, h}

i:“ ti`1´ ti and weightspHiq0_ďiďn´1

are matrices in M1,d _{given by}

pHiqℓ“ ´R hi _ W_tℓ_{i`1´ Wt}ℓ_i hi ^ R hi , 1_{ď ℓ ď d,}

with R a positive parameter. We denote|π| the modulus of π given by |π| :“ max0ďiďn´1hi

and we assume that we always have ℜ_{Ă π.}

To obtain our convergence results, we work, throughout this paper, under the fol-lowing assumption:

pHfq

(i) The functions σ : Rd_{Ñ M}d,d_{and b : R}d_{Ñ R}d_{are Lipschitz-continuous functions.}

(ii) The functions f : Rd_{ˆ R}d_{ˆ M}d,d_{Ñ R}d_{, g : R}d_{Ñ R}d _{and pc}ij _{: R}d_{Ñ Rq}

i,jPI are

Lipschitz-continuous functions and fj_{px, y, zq “ f}j_{px, y, z}j._{q. We denote by L}Y

and LZ _{the Lipschitz constants of f with respect to y and z.}

(iii) g_{pxq P Qpxq, for all x P R}d_.

(iv) The cost functions_pcij_q

i,jPI satisfy the following structure condition

$ ’ & ’ % cii_{“ 0,} for 1ď i ď d;

inf_xPRdcijpxq ą 0, for 1ď i, j ď d with i ‰ j;

inf_xPRdtcijpxq ` cjlpxq ´ cilpxqu ą 0, for 1 ď i, j ď d with i ‰ j, j ‰ l.

Let us emphasize here the fact that our results are obtained without any assumption on the non-degeneracy of the volatility matrix σ. We also point out that _{pHf qpiiq is} the best condition – up to now – for existence and uniqueness to (1.1) to hold.

A fundamental result to obtain convergence for continuously reflected BSDEs is first to prove that the scheme given in (1.4) approximates efficiently discretely reflected BSDEs. This result is interesting in itself if one is only interested in the approximation of Bermudan switching problem (i.e. when the switching times are restricted to lie in the grid ℜ). It is discussed in section 3 below and requires, in particular, the use of a new representation result for the scheme (1.4).

Combining the fact that discretely reflected BSDEs are a good approximation of continuously reflected BSDEs and that the scheme (1.4) is also a good approximation of (1.3), we obtain our new convergence result, which is the main result of this paper and is summarised in the following Theorem.

Theorem 1.1. Let us assume that _{pHf q is in force. Set R such that L}Z_{Rď 1, π such}

that LY_{|π| ă 1 and define αp|π|q “ logp2T {|π|q. Then the following holds, for some}

positive constant C: (i) Taking_{|ℜ| „ |π|}1_{2 , we have sup 0_ďiďn E ” |Yti ´ rY ℜ,π i | 2 ` |Yti´ Y ℜ,π i | 2ı ď C|π|1{2α<sub>p|π|q.</sub> (ii) Taking<sub>|ℜ| „ |π|</sub>1<sub>{3</sub> , we have sup 0ďiďn E ” |Yti ´ rY ℜ,π i | 2 ` |Yti´ Y ℜ,π i | 2ı ď C|π|1{3α_p|π|q, and E «_n´1 ÿ i“0 żti`1 ti |Zs´ Z_iℜ,π|2ds ff ď C|π|1{6aα_p|π|q.

Moreover, if the cost functions c are constant, then the previous estimates remain true with α_{p|π|q :“ 1.}

It is important to compare the previous result with Theorem 5.4 in [8] which gives also rates of convergence for the discrete-time approximation of obliquely reflected BS-DEs. Up to a slight modification of the scheme (introducing the truncation of the Brownian increments), we see that we are able to obtain the convergence rate 1_{4, when the previous result, under_{pHfq, were only predicting a logarithmic convergence.} Also, we are able to work under a minimal Lipschitz condition for the cost functions, which was not the case before.

The rest of the paper is organised as follows. In Section 2, we present preliminary results that will be useful in the rest of the paper. We discuss the representation prop-erty of obliquely reflected BSDEs in terms of auxiliary one-dimensional BSDEs. We

also give new regularity results for the discretely reflected BSDEs which are key tools to obtain our convergence results. Section 3 is devoted to the study of the numerical scheme, in particular its fundamental stability property. Using this stability property and the regularity results given in Section 2, we prove a control of the error between the scheme and the discretely reflected BSDEs. Section 4 is concerned with the approxi-mation of continuously reflected BSDEs by the discretely reflected ones. A convergence rate is obtained that allows to prove, using the result of Section 3, our main result, Theorem 1.1 above. For the reader convenience, some technical proofs are postponed in an Appendix Section.

Notations Throughout this paper we are given a finite time horizon T and a proba-bility space_{pΩ, F, Pq endowed with a d-dimensional standard Brownian motion pW}tqtě0.

The filtrationpFtqtďT is the Brownian filtration. P denotes the σ-algebra onr0, T s ˆ Ω

generated by progressively measurable processes. Any element x_{P R}n_{will be identified}

to a column vector with ith component xi _{and Euclidean norm|x|. For x, y P R}n_{, x.y}

denotes the scalar product of x and y. We denote by ď the component-wise partial ordering relation on vectors. Mn,m_{denotes the set of real matrices with n lines and m}

columns. For a matrix M _{P M}n,m_{, M}ij _{is the component at row i and column j, M}i.

is the ith row and M.j _{the jth column.}

We denote by Ck,b _{the set of functions with continuous and bounded derivatives up}

to order k. For a function f : Rn_{Ñ R, x ÞÑ f pxq, we denote by B}

xf “ pBx1f, . . . ,B_xnfq.

If f : Rn_{ˆ R}d _{Ñ R, px, yq ÞÑ f px, yq we denote B}

xf (resp. Byf) the derivatives with

respect to the variable x (resp. y). For g : Rn _{ÞÑ R}d_{, x Ñ gpxq, B}

xg is a matrix and

pBxgqi.“ Bxgi.

For ease of notation, we will sometimes write Etr.s instead of Er.|Fts, t P r0, T s.

Finally, for any p_{ě 1, we introduce the following:}

- Lp the set of FT-measurable random variables G satisfying |G|Lp :“ Er|G|ps 1 p ă

`8,

- Sp the set of c`adl`ag adapted processes U satisfying

|U|Sp:“ E « sup tPr0,T s|U t|p ff1 p ă 8,

and Scp the subset of continuous processes in Sp,

- Hp _{the set of progressively measurable processes V satisfying}

|V |Hp:“ E «ˆżT 0 |V t|2dt ˙p 2ff 1 p ă 8,

- Kp _{the set of continuous non-decreasing processes in S}p_,

In the sequel, we denote by C a constant whose value may change from line to line but which never depends on_{|π| nor |ℜ|. The notation C}α is used to stress the fact that

the constant depends on some parameter α.

2

Preliminary results

In this section, we present key properties of continuously and discretely reflected BSDEs. We start by recalling the representation property in terms of ”switched” BSDEs of the multidimensional systems of reflected BSDEs (1.1) or (1.3).

In a second part, we study the regularity properties of the solution to discretely reflected BSDEs in a Markovian setting. These results are key tools to obtain a con-vergence rate for the numerical approximation. They are new in the framework of this paper but their proofs rely on arguments that are now quite well understood.

2.1 Representation of obliquely reflected BSDEs

As mentioned in the introduction, the motivation to work on the above class of obliquely reflected BSDEs comes from the study of ”switching problems” in the financial eco-nomics literature. Indeed, RBSDEs provide a characterization of the solution to these switching problems. Interestingly, the interpretation of the RBSDE in term of the solu-tion of a ”switching problem” is a key tool in our work. We now recall the link between the two objects, which takes the form of a representation theorem for the solution of the RBSDEs in terms of ”switched BSDEs”. This link has been established before, see e.g. [17]. We state it here in a generic framework as this will be useful latter on.

We consider a matrix valued process C_{“ pC}ij_q

1ďi,jďn such that Cij belongs to S2

for i, j_{P I and satisfies the structure condition} $ ’ & ’ % Cii t “ 0, for 1ď i ď d and 0 ď t ď T ;

inf_{tPr0,T s}C_tij _{ě ε ą 0,} for 1_{ď i, j ď d with i ‰ j;} inftPr0,T stCtij` C

jl

t ´ Ctilu ą 0, for 1 ď i, j ď d with i ‰ j, j ‰ l.

(2.1)

We introduce a random closed convex set family associated to C:

Q_t:_“ " y_{P R}d_|yi _{ě max} jPI py j ´ Ctijq, 1 ď i ď d * , 0_{ď t ď T,}

and the oblique projection operator onto Qt, denoted Pt and defined by

P_t: y_{P R}d_ÞÑ ˆ max jPIty j ´ Ctiju ˙ 1ďiďd . (2.2)

Remark 2.1. It follows from the structure condition (2.1) that Pt is increasing with

A switching strategy a is a nondecreasing sequence of stopping times pθjqjPN ,

combined with a sequence of random variables _pαjqjPN valued in I, such that αj is

F_{θj´measurable, for any j P N. We denote by A the set of such strategies. For} a_{“ pθ}j, αjqjPNP A , we introduce Na the (random) number of switches before T :

Na

“ #tk P N˚ : θkď T u . (2.3)

To any switching strategy a_{“ pθ}j, αjqjPN P A , we associate the current state process

patqtPr0,T s and the cumulative cost process pAatqtPr0,T s defined respectively by

at :“ α01_t0ďtăθ0u` Na ÿ j“1 αj´11tθj´1ďtăθju and A a t :“ Na ÿ j“1 Cαj´1αj θj 1tθjďtďT u, for 0_{ď t ď T .}

Remark 2.2. (i) The sequence of stopping times is only supposed to be non-decreasing, but the assumptions on the cost processes (2.1) imply that any reasonable strategy uses a sequence of increasing stopping times. This is specially the case for the optimal strategies.

(ii) Note that the cumulative cost process will keep track of all the switching times, even the instantaneous ones; whereas the state process will keep track of the last state when instantaneous switches occur.

For _{pt, iq P r0, T s ˆ I, the set A}t,i of admissible strategies starting from state i at

time t is defined by

A_{t,i “ ta “ pθj, αjqj P A |θ0 “ t, α0 “ i, E}“_|Aa

T| 2‰

ă 8u , similarly we introduce A_t,iℜ the restriction to ℜ_{´admissible strategies}

A_t,iℜ _{:“ t a “ pθj, αjqjPN P At,i | θj P ℜ , @j ď N}a _{u ,} and denote Aℜ :_“Ť_iďdAℜ

0,i.

For a strategy a _{P A}t,ℓ, we introduce the one-dimensional switched BSDE whose

solution _pUa_{, V}a_{q satisfies} Ua t “ ξaT ` żT t Fas ps, Vsaqds ´ żT t Va sdWs´ AaT ` Aat (2.4)

where the terminal condition ξ, the random costs process and the random driver F satisfies following assumptions, for some p_{ě 2:}

pHFpq

(i) F : Ω_{ˆ r0, T s ˆ M}d,d_{Ñ R}d _{is Pb BpR}d_{q b BpM}d,d_{q-measurable,}

(iii) |F ps, zq ´ F ps, z1_{q| ď C|z ´ z}1_{| for all s P r0, T s, z, z}1 _{P M}d,d_,

(iv) ξ is FT-measurable and is valued in QT,

(v) E”_|ξ|p_`şT0 |F ps, 0q| p_dsı

ď Cp.

We now define multidimensional processes ¯Y and ¯Yℜ

as follows, for ℓ_{P t1, . . . , du} p ¯Y_tqℓ _: “ ess sup aPAt,ℓ Ua t and p ¯Y ℜ t qℓ :“ ess sup aPAℜ t,ℓ Ua t .

The process Y represents the optimal value that can be obtained from the switched BSDEs following strategies in A . The process ¯Yℜcan be seen as a ”Bermudan” version of it i.e. when the switching times are restricted to lie in ℜ. Both processes enjoy a representation in terms of reflected BSDEs, the main difference lying into the reflecting process that for the latter will be a pure jump process with jump times in ℜ.

Let_{pY, Z, Kq be the solution to the following BSDE} $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % Y_tℓ_{“ ξ}ℓ<sub>`</sub> żT t Fℓ<sub>ps, Z</sub>sq ds ´ żT t Z<sub>s</sub>ℓdWs` KℓT ´ Kℓt, 0ď t ď T, ℓ P I, Y_tℓ_{ě max} jPI tY j t ´ C ℓj t u, 0ď t ď T, ℓ P I, żT 0 ” Y_tℓ_{´ max} jPIztℓutY j t ´ C ℓj t u ı dKℓ_{t“ 0,} ℓ_{P I,} (2.5) and _{p ˜}Yℜ_{, Y}ℜ_{, Z}ℜ_{, K}ℜ q with Ytℜ “ ˜Y ℜ

t´, tP p0, T s be the solution of following discretely

reflected BSDEs, $ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ % ˜ Y_tℜ<sub>“ ξ `</sub> żT t F<sub>ps, Zs</sub>ℜ<sub>q ds ´</sub> żT t Z<sub>s</sub>ℜdWs` K ℜ T ´ K ℜ t, 0ď t ď T, Y_rℜ_{P Q}r, rP ℜ, żT 0 ” pYtℜqℓ´ max jPIztℓutpY ℜ t qℓ´ C ℓj t u ı d_pKℜ_tqℓ _{“ 0,} ℓ_{P I,} (2.6)

Existence and uniqueness of a solution for equation (2.5) has been addressed in [17, 16] and in [8] (Proposition 2.1) for equation (2.6). For the reader convenience we recall here these results.

Proposition 2.1. Assume that pHFpq holds for some p ě 2. There exists a unique

solution _{pY, Z, Kq P S}2 c ˆ H

2

ˆ K 2

to (2.5) and a unique solution _{p ˜}Yℜ, Yℜ, Zℜ, Kℜ_q with _{p ˜}Yℜ, Zℜ, Kℜ_{q P S}2_{ˆ H}2_{ˆ K} ℜ,2 to (2.6). They also satisfy

|Y|Sp` |Z|<sub>H</sub>p` |K_T|_Lp ď C_p and | ˜Yℜ|_Sp` |Zℜ|<sub>H</sub>p` |Kℜ_T|_Lpď C_p.

Gathering Proposition 3.2 in [7] and Theorem 2.1 in [8], we have the following key representation result.

Proposition 2.2. Assume that pHF2q is in force. The following hold:

(i) for all ℓ_{P t1, . . . , du, t P r0, T s,} pYtqℓ “ p ¯Ytqℓ “ U ¯ a t and p ˜Y ℜ t qℓ “ p ¯Y ℜ t qℓ“ U ¯ aℜ t

for some ¯a_{P A}t,ℓ and ¯aℜ P A_t,ℓℜ.

(ii) The strategy ¯a_{“ p¯θ}j,α¯jqjě0 can be defined recursively byp¯θ0,α¯0q :“ pt, ℓq and, for

j_{ě 1,} ¯ θj :“ inf " s_{P r¯θ}j´1, Ts ˇ ˇ ˇ ˇpY˜sqα¯j´1 ď max k‰ ¯αj´1tp ˜ Y_sqk ´ Cα¯j´1k s u * , ¯ αj :“ min " ℓ_{‰ ¯}αj´1 ˇ ˇ ˇ ˇpY˜α¯jq ℓ ´ Cα¯j´1ℓ ¯ θj “ max_{k‰ ¯αj´1}tp ˜Yθ¯jq k ´ Cα¯j´1k ¯ θj u * .

(iii) The strategy ¯aℜ_{“ p¯θj}ℜ,α¯_jℜ_qjě0 can be defined recursively by p¯θℜ0,α¯ ℜ 0q :“ pt, ℓq and, for j _{ě 1,} ¯ θℜ_j :_{“ inf} # s_{P r¯θj´1}ℜ , T_{s X ℜ} ˇ ˇ ˇ ˇpY˜sℜq ¯ αℜ j´1 ď max k‰ ¯αℜ j´1 tp ˜Y_sℜ_qk_{´ C}α¯ ℜ j´1k s u + , ¯ αℜ_j :_{“ min} # ℓ_{‰ ¯}αℜ_j´1 ˇ ˇ ˇ ˇpY˜_θℜ¯ℜ jq ℓ ´ Cα¯ ℜ j´1ℓ ¯ θℜ j “ max_{k‰ ¯α}ℜ j´1 tp ˜Y_θℜ¯ℜ jq k ´ Cα¯ ℜ j´1k ¯ θℜ j u + . Remark 2.3. If ˜Yℓ

t R Qt then there is an instantaneous jump, i.e. ¯θ1 “ t. In the same

way, if t_{P ℜ and p ˜}Y_tℜ_qℓ_{R Q}

t then ¯θ1ℜ“ t.

2.2 Discretely obliquely reflected BSDEs in a Markovian setting

We will now study the discretely obliquely reflected BSDEs (2.6) in a Markovian setting, namely the solution to (1.3). We will in particular prove regularity results for this process. The main difference with Section 3 in [8] comes from the assumption on f , in particular the full dependence in the y-variable, recall _{pHf q(ii).}

Let us recall that under assumption _{pHf q(i), there exists a unique strong solution to} the SDE (1.2) which satisfies

E_t « sup sPrt,T s|X s|p ff ď Cpp1 ` |Xt|pq , for all pě 2 , t P r0, T s . (2.7)

2.2.1 Basic properties

The following proposition gives some usefull estimates on the solution to (1.3). Its proof is postponed to the Appendix.

Proposition 2.3. Assume that _{pHfq is in force. There exists a unique solution} p ˜Yℜ, Yℜ, Zℜ_{q P S}2

ˆ S2

ˆ H2

to (1.3) and it satisfies, for all p_{ě 2,} | ˜Yℜ_|Sp` |Zℜ|Hp` |K_Tℜ|Lpď C_p.

We now precise the results of Proposition 2.2, in the setting of this section. In particular, we describe the optimal strategy and some of its properties that will be useful in the sequel.

Corollary 2.1. (i) The following equalities hold, for all ℓ_{P t1, . . . , du, t P r0, T s,}

p ˜Y_tℜ_qℓ_{“ ess sup}

aPAℜ t,ℓ

U_tℜ,a_{“ Ut}ℜ,¯aℜ for some ¯aℜ _{P At,ℓ}ℜ,

where _pUℜ,a, Vℜ,a, Nℜ,a_{q is solution of the switched BSDE (2.4) with random driver} F_{ps, zq :“ f ps, X}s, ˜Ysℜ, zq for ps, zq P r0, T s ˆ Md,d, terminal condition ξ :“ gpXTq and

costs Csij “ cijpXsq.

(ii) The optimal strategy ¯aℜ _{“ pθ}j, αjqjě0 can be defined recursively by pθ0, α0q :“

pt, ℓq and, for j ě 1, θj:“ inf " s_{P rθ}j´1, Ts X ℜ ˇ ˇ ˇ p rY_sℜ_qαj´1 _{ď max} k‰αj´1 ! p rY_sℜ_qk_{´ c}αj´1k_pX sq )* , αj :“ min " q_{‰ α}j´1 ˇ ˇ ˇ p rY_θℜ jq q ´ cαj´1q_pX θjq “ max k‰αj´1 ! p rY_θℜ jq k ´ cαj´1k_pX θjq )* .

(iii) Moreover, for all ℓ _{P t1, . . . , du, t P r0, T s, the optimal strategy ¯a}ℜ _{P At,ℓ}ℜ satisfies E_t « sup sPrt,T s ˇ ˇ ˇUsℜ,¯aℜ ˇ ˇ ˇ p ` ˆżT t ˇ ˇ ˇVsℜ,¯aℜ ˇ ˇ ˇ 2 ds ˙p{2 `ˇˇˇAℜ,¯a ℜ T ˇ ˇ ˇ p `ˇˇˇNℜ,¯aℜ ˇ ˇ ˇ pff ď Cpp1 ` |Xt|pq. (2.8)

Proof. Thanks to Proposition 2.3, we can apply Proposition 2.2 with the random driver F, the terminal condition ξ and costs Csij defined above, which gives us the

represen-tation result. The first estimate in (2.8) is a direct application of this represenrepresen-tation result and Proposition 2.3. Other estimates in (2.8) are obtained by using standard arguments for BSDEs combined with the estimate (A.1), see proof of Proposition 2.2

2.2.2 Fine estimates on pYℜ_{, ˜Y}ℜ_{, Z}ℜ

q

In this section, we prove regularity results on the solution_pYℜ, ˜Yℜ, Zℜ_{q of the discretely} reflected BSDEs. To do that, we will use techniques already exposed in [1, 5, 8], based essentially on a representation of Zℜ, obtained by using Malliavin Calculus. For a general presentation of Malliavin Calculus, we refer to [20]. We now introduce some notations and recall some known results on Malliavin differentiability of SDEs solution.

We will work under the following assumption. pHrq The coefficients b, σ, g f , and pcij_q

i,j are C1,b in all their variables, with the

Lipschitz constants dominated by L.

This assumption is classically relieved using a kernel regularisation argument, see e.g. the proofs of Proposition 4.2 in [5] or Proposition 3.3 in [1].

We denote by D1,2 the set of random variables G which are differentiable in the Malliavin sense and such that _}G}2

D1,2 :“ }G} 2 L2 ` şT 0 }DtG} 2 L2dt ă 8, where DtG

denotes the Malliavin derivative of G at time t_{ď T . After possibly passing to a suitable} version, an adapted process belongs to the subspace L1,2a of H2 whenever VsP D1,2 for

all s_{ď T and }V }}2 L1,2_a :“ }V } 2 H2 ` şT 0 }DtV} 2 H2dtă 8.

Remark 2.4. Under _{pHrq, the solution of (1.2) is Malliavin differentiable and its} derivative satisfies } sup sďT|Ds X_|}S p ă 8 and Er „ sup rďsďT|Du Xs|p  ď Cp1 ` |Xr|pq , uď r ď T . (2.9) Moreover, we have sup sďu}Ds Xt´ DsXu}L p ` } sup tďsďT|Dt Xs´ DuXs| }L p ď C p L|t ´ u| 1_{2 , (2.10) for any 0_{ď u ď t ď T .}

Malliavin derivatives of _pYℜ, ˜Yℜ, Zℜ_{q. We now study the Malliavin} differentia-bility of _pYℜ, ˜Yℜ, Zℜ_{q. The techniques used are classical by now, see [1, 5]. In this} paragraph, we will follow the presentation of [8]. Once again, the main difference with this paper is the assumption_{pHfq made on the driver f . In the setting of [8], f has to} satisfy fi_{px, y, zq “ f}i_{px, y}i_{, z}i_{q whereas pHf q does not impose such restriction on the y}

variable. This implies that the representation of Z, see Corollary 2.2 below, is slightly more complicated. Namely, it contains the term D ˜Y, compare to Proposition 3.2 in [8]. To obtain the regularity results on _pYℜ, ˜Yℜ, Zℜ_{q, we need thus to prove estimates on} D ˜Y, which is the main result of the next Proposition.

deriva-tive satisfies, for all rP r0, T s, u ď r, i P I, Dup rYrℜqi “ Er „ BxgaTpXTqDuXT ` żT rB xfaspΘℜsqDuXsds ` żT rB yfaspΘ ℜ sqDuYr ℜ s ds` żT r d ÿ ℓ“1 BzasℓfaspΘ ℜ sqDupZ ℜ sqasℓds ´ Na ÿ j“1 Bxcαj´1αjpXθjqDuXθj ff (2.11) where a :“ ¯aℜ

is the optimal strategy associated with the representation in terms of switched BSDEs, recall Corollary 2.1, and Θℜ:_{“ pX, ˜}Yℜ, Zℜ_{q. Moreover, the following} estimates hold true: for all r_{P r0, T s, 0 ď u ď r, 0 ď v ď r,}

|DuY˜r|2 ď CLp1 ` |Xr|2q (2.12) and |DuY˜r´ DvY˜r| 2 ď CLp1 ` |Xr|qEr „ sup rďsďT|Du Xs´ DvXs| 4 1 2 . (2.13) Proof.

Let G _{P D}1,2_pRd_{q. Since X belongs to L}1,2

a under pHrq, and P is a Lipschitz

continuous function, we deduce that P_pXt, Gq P D1,2pRdq. Using Lemma 5.1 in [1], we

compute DspPpXt, Gqqi “ (2.14) d ÿ j“1 pDsGj´DscijpXtqq1Gj_´cij_pX

tqąmaxℓăjpGℓ´ciℓpXtqq1Gj´cijpXtqěmaxℓąjpGℓ´ciℓpXtqq.

Combining (2.14), Proposition 5.3 in [10] and an induction argument, we obtain that _pYℜ, rYℜ, Zℜ_{q is Malliavin differentiable and that a version of pD}uYrℜ, DuZℜq is

given by, for all iP I, t P r0, T s, 0 ď u ď t,

Dup rYtℜqi “DupYrℜj`1q i ´ d ÿ k“1 żrj`1 t DupZsℜqikdWsk` żrj`1 t B xfipΘℜsqDuXsds ` żrj`1 t B yfipΘℜsqDuYrsℜds` żrj`1 t d ÿ ℓ“1 BziℓfipΘℜ_sqD_upZℜqiℓ_sds (2.15) recall _{pHf q.}

Now, we consider the optimal strategy a :_{“ ¯a}ℜ defined in Corollary 2.1 (ii) above and fix j _{ă κ. Observing that the process a is constant on the interval rθ}j, θj`1q, we

deduce from (2.15) Dup rYtℜqαj “ DupYθℜj`1q αj<sub>´</sub> d ÿ k“1 żθj`1 t DupZsℜqαjkdWsk` żθj`1 t B xfαjpΘℜsqDuXsds (2.16) ` żθj`1 t B yfαjpΘℜsqDuYrsℜds` żθj`1 t d ÿ ℓ“1 Bzαj ℓf αj_pΘℜ sqDupZℜq αjℓ s ds

for t _{P rθ}j, θj`1s and 0 ď u ď t. Combining (2.14) and the definition of a given in

Corollary 2.1 (ii), we compute, for u_{ď θ}j`1 and j ă κ,

DupYθℜj`1q

αj_{“ D}

up rYθℜj`1q

αj`1_{´ B}

xcαjαj`1pXθj`1qDuXθj`1.

Inserting the previous equality into (2.16) and summing up over j we obtain, for all t_{ď r ď T ,} Dup rYrℜqi “BxgaTpXTqDuXT ´ żT r d ÿ k“1 DupZsℜqaskdWs` żT rB xfaspΘℜsqDuXsds ` żT rB yfaspΘℜsqDuYrsℜds` żT r d ÿ ℓ“1 BzasℓfaspΘℜsqDupZsℜqasℓds ´ Na ÿ j“1 Bxcαj´1αjpXθjqpDuXqθj. (2.17)

Taking conditional expectation on both sides of the previous equality proves (2.11). Moreover, we are in the framework of section A.2 in the Appendix by setting Y_{“ D}uY

and X_{“ D}uX. Condition (A.2) is satisfied here by N¯a

ℜ

with β :_{“ C}Lp1 ` |X|q, recall

(2.8). Using Proposition A.1 and (2.9), we then obtain (2.12).

From equation (2.17), we easily deduce the dynamics of Dup rYℜq ´ Dvp rYℜq, which

leads, using again Proposition A.1, to (2.13). l The representation result for Zℜ is then an easy consequence of the previous propo-sition.

Corollary 2.2. Under _{pHf q-pHrq the following representation holds true,}

Z_tℜ _{“ E}tr BxgaTpXTqΛat,TDtXT ´ Na ÿ j“1 Bxcαj´1αjpXθjqΛ a t,θjDtXθj ` żT t ´ BxfaspΘℜsqΛat,sDtXs` ByfaspΘℜsqΛat,sDtYrsℜ ¯ ds  , (2.18)

where a :_{“ ¯a}ℜ is the optimal strategy associated with the representation in terms of switched BSDEs, recall Corollary 2.1, and for ℓP I,

Λa_t,s :_{“ exp} ˆżs t B zar.farpΘℜ_uqdWr´ 1 2 żs t |B zar.farpΘℜ_uq|2dr ˙ . (2.19)

Moreover, underpHfq, we have ˇ ˇ ˇZtℜ ˇ ˇ ˇ ď ¯L_{p1 ` |X}t|q, for all tP r0, T s, (2.20)

for some positive constant ¯L that does not depend on the grid ℜ.

Proof. 1. A version of Zℜis given by_pDtY˜tℜq0_ďtďT. The expression of DtY is obtained

directly by applying Itˆo’s formula, recall (2.11).

2. Under_{pHfq-pHrq the estimate (2.20) follows from (2.12) and (2.9). Under pHfq,} we can obtain the result by a standard kernel regularisation argument. l

Regularity of _pYℜ, Zℜ_{q. With the above results at hand, the study of the regularity} ofpYℜ_{, Z}ℜ

q follows from ”classical” arguments, see e.g. [5, 8]. For sake of completeness, we reproduce them below.

We consider a grid π :_{“ tt}0 “ 0, . . . , tn “ T u on the time interval r0, T s, with

modulus|π| :“ max0ďiďn´1|ti`1´ ti|, such that ℜ Ă π.

We need to control the following quantities, representing the H2

-regularity of _{p r}Y , Z_q: E „żT 0 | r Y_tℜ_{´ r}Y_πptqℜ _|2dt  and E „żT 0 |Z ℜ t ´ ¯Z ℜ πptq| 2 dt  , (2.21) where π_{ptq :“ suptt}i P π ; ti ď tu is defined on r0, T s as the projection to the closest

previous grid point of π and ¯ Z_tℜ_i :_“ 1 ti`1´ ti E „żti`1 ti Z_sℜds_{| F}ti  , i_{P t0, . . . , n ´ 1u.} (2.22)

Remark 2.5. Observe that_{p ¯}Z_sℜ_qsďT :“ p ¯Z_πpsqℜ qsďT interprets as the best H2-approximation

of the process Zℜ by adapted processes which are constant on each interval _rti,ti`1q,

for all i_{ă n.}

The first result is the regularity of the Y -component, which is a direct consequence of the bound (2.20).

Proposition 2.5. Under _{pHfq, the following holds} sup

tPr0,T s

E ”

| ˜Y_tℜ_{´ ˜}Y_πptqℜ _|2ı_{ď C}L|π| .

Proof. We first observe that, for all 0_{ď t ď T ,}

E ” | rY_tℜ_{´ r}Y_πptqℜ _|2ı_{ď E} » – ˇ ˇ ˇ ˇ ˇ żt πptq f_pXs, rYsℜ, Z ℜ s qds ` żt πptq Z<sub>s</sub>ℜdWs ˇ ˇ ˇ ˇ ˇ 2fi fl , ď CLE «żt πptq ´ 1<sub>` |X</sub>s|2` | ˜Ysℜ| 2 ` | ˜Z_sℜ_|2¯ds ff . ď CLE « |π| ` żt πptq| ˜ Z_sℜ_|2ds ff (2.23)

where we used (2.7) and Proposition 2.3. From (2.20), we easily get E”şt_πptqˇˇZ_tℜˇˇ2dtı_ď CL|π|. Inserting the previous inequality into (2.23) concludes the proof of this

Propo-sition. l

The following Proposition gives us the regularity of Zℜ. Its proof is postponed to the Appendix.

Proposition 2.6. Under _{pHfq, the following holds}

E „żT 0 |Z ℜ t ´ ¯Z ℜ t | 2 dt  ď CL ´ |π|12 ` κ|π| ¯ .

3

Study of the discrete-time approximation

The aim of this section is to obtain a control on the error between the obliquely reflected backward scheme (1.4) and the discretely obliquely reflected BSDE (1.3). This is the purpose of Theorem 3.1 in subsection 3.4 below. In order to prove this key result, we start by interpreting the scheme in terms of the solution of a switching problem in subsection 3.2. We then use this representation to obtain a general stability property for the scheme in subsection 3.3. Subsection 3.1 is devoted to preliminary definition and propositions.

3.1 Definition and first estimates

Given a grid π of the interval r0, T s, we first consider an obliquely reflected backward scheme with a random generator and a random cost process Cπ. For t _{P r0, T s, we} denote by Qπ

t the random closed convex set associated to Ctπ and Ptπ the projection

onto Qπ

t, recall (2.2) . The scheme is defined as follows.

Definition 3.1.

(i) The terminal condition Ynℜ,π is given by a random variable ξπ P L2pFTq valued in

Qπ T (ii) for 0_{ď i ă n, $} ’ ’ & ’ ’ % r Y_iℜ,π :<sub>“ ErYi`1</sub>ℜ,π <sub>| F</sub>tis ` hiF π i pZ ℜ,π i q, Z_iℜ,π :_{“ ErYi`1</sub>ℜ,πHi| Ftis, Y<sub>i</sub>ℜ,π :<sub>“ r</sub>Y<sub>i</sub>ℜ,π1<sub>ttiRℜu` P}π tip rY ℜ,π i q1ttiPℜu, (3.1)

with_pHiq0_ďiăn some R1ˆd independent random vectors such that, for all 0ď i ă n, Hi

is Fti`1-measurable, EtirHis “ 0, λiIdˆd“ hiErHiJHis “ hiEtirH J i His, (3.2) and λ d ď λi ď Λ d, (3.3)

Remark 3.1. Let us remark that (3.2) and (3.3) imply that λ_{ď h}iEr|Hi| 2 s “ hiEtir|Hi| 2 s ď Λ. (3.4)

In this section we use following assumptions. _{pHF d}pq

(i) For all i P t0, ..., n ´ 1u, Fπ

i : Ωˆ Md,d Ñ Rd is a Fti b BpM

d,d_q-measurable

function,

(ii) the random cost process Cπ _{satisfies the structure condition (2.1),}

(iii) F_iπ,j_{pzq “ Fi}π,j_pzj._{q for all j P I and all 0 ď i ď n ´ 1,}

(iv) _|Fπ i pzq ´ Fiπpz1q| ď LZ|z ´ z1| for all z, z1 P Md,d, (v) E”_|ξπ_|2 `řn´1i“0 |Fiπp0q| 2 hi` suptiPℜ|C π ti| pı_{ď C} p,

(vi) sup0ďiďn´1hi|Hi| LZ ď 1.

Remark 3.2. i) UnderpHF d2q, it is clear that the general scheme (3.1) has a unique

solution.

ii) The weights_pHiq0_ďiăn depend also on the grid π but we omit the script π for ease

of notation.

We observe that this obliquely reflected backward scheme can be rewritten equiva-lently for i_{P J0, nK as} # r Y_iℜ,π _{“ ξ}π<sub>`</sub>řn´1 k“i FkπpZ ℜ,π k qhk´ řn´1 k“i hkλ´1k Z ℜ,π k HkJ´ řn´1 k“i ∆Mk` pKnℜ,π´ K_iℜ,πq Kℜ,π_k :_“řk_r“1∆Krℜ,π with ∆Kℜ,πr :“ Yrℜ,π´ rYrℜ,π, (3.5) where _pλkq are given by (3.2) and, for all k P J0, n ´ 1K, ∆Mk is an Ftk`1-measurable

random vector satisfying E_t

kr∆Mks “ 0, Etkr|∆Mk|

2

s ă 8 and Etkr∆MkHks “ 0. (3.6)

Following Corollary 2.5 in [4], we know that assumption_{pHF d}pq(v) is an essential

ingredient to obtain a comparison result for classical time-discretized BSDE schemes. We are able to adapt this comparison result in the context of obliquely reflected back-ward scheme in the following proposition.

Proposition 3.1. Let us consider two obliquely reflected backward schemes solutions p1_Yrℜ,π_,1_Yℜ,π_,1_Zℜ,π_{q and p}2_Yrℜ,π_,2_Yℜ,π_,2_Zℜ,π_{q, associated to generators p}1_Fπ

. q, p 2_Fπ

. q,

terminal conditions 1

ξπ,2

ξπ and random cost processes _p1_Cπ_{q, p}2

Cπ_{q such that pHF d}2q is in force. If 1 ξ ď2ξ, 1Fip 2 Z_iℜ,π_{q ď}2Fip 2 Z_iℜ,π_q, for all 0_{ď i ď n ´ 1,}

and p1C_tπ_iqjk_{ě p}2C_tπ_iqjk, for all j, kP I, tiP ℜ,

then we have

1

Y_iℜ,π ď2Y_iℜ,π and 1Yr_iℜ,π ď2Yr_iℜ,π, for all 0_{ď i ď n.}

Moreover, this comparison result stays true if these obliquely reflected backward schemes have two different reflection grids ℜ1

and ℜ2

with ℜ1

Ă ℜ2

. In particular, we are allowed to have no projection for the first scheme, i.e. ℜ1 _{“ H.}

Proof. We just have to use the comparison theorem for backward schemes (Corollary 2.5 in [4]) and the monotonicity properties of P (see Remark 2.1). l Proposition 3.2. Assume thatpHF d2q is in force. The unique solution p rYℜ,π, Yℜ,π, Zℜ,πq

to (3.1) satisfies E „ sup 0_ďiďn ˇ ˇ ˇ rY_iℜ,π ˇ ˇ ˇ 2 ` sup 0<sub>ďiďn</sub> ˇ ˇ ˇYiℜ,π ˇ ˇ ˇ 2 ` E «_n´1 ÿ i“0 hi ˇ ˇ ˇZiℜ,π ˇ ˇ ˇ 2ff ` E”|Kℜ,πn | 2ı ď C.

Proof. The proof of uniform estimates (with respect to n and κ) divides, as usual, in two steps controlling separately_{p r}Yℜ,π, Yℜ,π_{q and pZ}ℜ,π, Kℜ,π_{q. It consists in transposing} continuous time arguments, see e.g. proof of Theorem 2.4 in [16], in the discrete-time setting.

Step 1. Control of rYℜ,π _{and Y}ℜ,π_{. We consider two non-reflected backward schemes}

bounding rYℜ,π.

Define the Rd_{-valued random variable ˘ξ and random maps p ˘F}

iq0ďiďn´1 by p˘ξqj :“ řd k“1 ˇ ˇpξπ_qkˇ_{ˇ and p ˘F} iqjpzq :“řd_k“1 ˇ ˇpFπ i qkpzq ˇ

ˇ for 1 ď j ď d and 0 ď i ď n ´ 1. We then denote by _{p ˘}Y , ˘Z_{q the unique solution of the following non-reflected backward scheme:}

$ ’ & ’ % ˘ Yn“ ˘ξ ˘ Zi“ Er ˘Yi`1Hi| Ftis, ˘ Yi“ Er ˘Yi`1 | Ftis ` hiF˘ip ˘Ziq.

Since all the components of ˘Y are similar, ˘Y _{P Q}π: Thus the above backward scheme is an obliquely reflected backward scheme with same switching costs as in (3.1). We also introduce _p˚Y , ˚Z_{q the solution of the following non-reflected backward scheme}

$ ’ & ’ % ˚ Yn“ ξπ ˚ Zi“ Er˚Yi`1Hi| Ftis, ˚ Yi “ Er˚Yi`1 | Ftis ` hiF π i p˚Ziq.

Using the comparison result given by Proposition 3.1, we straightforwardly deduce that p˚Y_qj _{ď p r}Yℜ,π_qj _{ď pY}ℜ,π_qj _{ď p ˘Yq}j_{, for all j P I. Since p˚Y , ˚Zq and p ˘Y , ˘Zq are solutions}

to standard backward schemes, classical estimates andpHF d2q lead to E_{r sup} 0_ďiďn| r Y_iℜ,π_|2 ` sup 0<sub>ďiďn</sub>|Y ℜ,π i | 2 s ď Er sup 0<sub>ďiďn</sub>|˚ Yi|2` sup 0_ďiďn| ˘ Yi|2s ď CE « |ξπ|2` ˜_n´1 ÿ i“0 |Fiπp0q| 2 hi ¸ff (3.7) ď C.

Step 2. Control of _pZℜ,π, Kℜ,π_{q. Let us rewrite (3.5) for Y}ℜ,π between k and k_{` 1} with k_{P J0, n ´ 1K:}

Y_kℜ,π _{“ Yk`1</sub>ℜ,π <sub>` Fk}π_pZkℜ,π_{qhk´ hk}λ´1_k Z_kℜ,πH_kJ_{´ ∆M}k` ∆Kℜ,πk .

Using the identity _|y|2

“ |x|2

` 2xpy ´ xq ` |x ´ y|2

, we obtain, setting x_{“ Yk}ℜ,π and y<sub>“ Yk`1</sub>ℜ,π, |Yk`1ℜ,π| 2 “|Ykℜ,π| 2 ` 2Ykℜ,π ´ ´FkπpZ ℜ,π k qhk` hkλ´1k Z ℜ,π k HkJ` ∆Mk´ ∆Kℜ,πk ¯ ` ˇ ˇ ˇFkπpZ ℜ,π k qhk´ hkλ ´1 k Z ℜ,π k H J k ´ ∆Mk` ∆Kℜ,π_k ˇ ˇ ˇ 2 .

Taking the expectation in the previous inequality, we get, combining _{pHF d}2q with

(3.2)-(3.3) and (3.6), E_r|Yℜ,π k`1| 2 s ě Er|Ykℜ,π| 2 s ´ 2E”Y<sub>k</sub>ℜ,π´F<sub>k</sub>π<sub>pZk</sub>ℜ,π<sub>qh</sub>k` ∆Kℜ,π_k ¯ı ` E„ˇˇˇhkλ´1<sub>k</sub> Z<sub>k</sub>ℜ,πHkJ ˇ ˇ ˇ 2 ` Er|∆Mk|2s ě Er|Ykℜ,π| 2 s ´ CE”|Ykℜ,π| ´ |Fkπp0q|hk` |Zkℜ,π|hk ¯ı ´ 2E”Y<sub>k</sub>ℜ,π∆Kℜ,π<sub>k</sub> ı ` E » –h2 kλ´2k Etk „ ÿ i,jPJ1,dK ppZkℜ,πq J_Zℜ,π k q ij_pH kq1ipHkq1j fi fl ` Er|∆Mk|2s ě Er|Ykℜ,π| 2 s ´ CE”|Ykℜ,π| ´ |Fkπp0q|hk` |Z_kℜ,π|hk ¯ı ´ 2E”Y_kℜ,π∆Kℜ,π_k ı `<sub>Λ</sub>dE ” hk|Z<sub>k</sub>ℜ,π|2 ı ` Er|∆Mk|2s.

Then we sum over kP J0, n ´ 1K and we compute, using Young inequality with ε ą 0, n´1_ÿ k“0 E ” hk|Z_kℜ,π|2 ı ` n´1<sub>ÿ</sub> k“0 E<sub>r|∆Mk|</sub>2<sub>s ď Cε</sub>E « sup 0<sub>ďkďn</sub>|Y ℜ,π k | 2 ` n´1_ÿ k“0 |Fπ kp0q| 2 hk ff ` ε n´1<sub>ÿ</sub> k“0 E ” hk|Zkℜ,π| 2ı ` 2E „ sup 0_ďkďn|Y ℜ,π k ||Kℜ,πn |  ď CεE « sup 0_ďkďn|Y ℜ,π k | 2 ` n´1<sub>ÿ</sub> k“0 |Fkπp0q| 2 hk ff ` ε n´1_ÿ k“0 E ” hk|Z_kℜ,π|2 ı ` εE”|Kℜ,πn | 2ı . (3.8)

Moreover, we get from (3.5)

E ” |Knℜ,π| 2ı ď CE « sup 0ďkďn| r Y_kℜ,π_|2<sub>`</sub> n´1<sub>ÿ</sub> k“0 |Fkπp0q| 2 hk ff ` C n´1_ÿ k“0 E ” hk|Z_kℜ,π|2 ı ` C n´1_ÿ k“0 E_r|∆Mk|2_{s. (3.9)}

Combining (3.8) with (3.9), and using _{pHF d}2q and (3.7), classical calculations yield,

for ε small enough,

n´1_ÿ k“0 E ” hk|Zkℜ,π| 2ı ` n´1_ÿ k“0 E_r|∆Mk|2_{s ď C.}

Finally we can insert this last inequality into (3.9) and use once again_{pHF d}2q and (3.7)

to conclude the proof. l

3.2 Optimal switching problem representation

We now introduce a discrete-time version of the switching problem, which will allow us to give a new representation of the scheme given in Definition 3.1. To simplify notations, we start by adapting the definition of switching strategies to the discrete-time setting: A switching strategy a is now a nondecreasing sequence of stopping times_pθrqrPNvalued

in N, combined with a sequence of random variables pαrqrPN valued in I, such that αr

is Ft_θr-measurable for any rP N.

Then by mimicking Section 2.1, we define classical objects related to switching strategies. For a switching strategy a “ pθr, αrqrPN, we introduce Na the (random)

number of switches before n: Na

To any switching strategy a “ pθr, αrqrPN, we associate the current state process

paiqiPJ0,nK and the cumulative cost processpAaiqiPJ0,nK defined respectively by

ai :“ α01t0ďiăθ 0u` Na ÿ r“1 αr´11tθ r´1ďiăθru and A a i :“ Na ÿ r“1 pCπ t_θrqαr´1αr1tθ rďiďnu,

for 0_{ď i ď n. We denote by A}ℜ,π the set of ℜ-admissible strategies: Aℜ,π _{“ ta “ pθr, αrqrPN switching strategy | tθ} r P ℜ @r P J1, N a_{K, E}“ |Aan| 2‰ ă 8u .

For_{pi, jq P J0, nK ˆ I, the set Ai,j}ℜ,π of admissible strategies starting from j at time ti is

defined by

Aℜ,π

i,j “ ta “ pθr, αrqrPN P A ℜ,π

|θ0“ i, α0 “ ju .

For a strategy aP Ai,jℜ,π we define the one dimensional ℜ-switched backward scheme

whose solution _pUℜ,π,a, Vℜ,π,a_{q satisfies} $ ’ ’ & ’ ’ % Unℜ,π,a“ ξπ,an

V_kℜ,π,a_{“ ErUk`1}ℜ,π,aH_kJ_{| F}tks,

U_kℜ,π,a_{“ ErUk`1</sub>ℜ,π,a<sub>| Ftks ` hk}Fπ,ak

k pV ℜ,π,a k q ´ řNa j“1pCtπ_θjqαj´1αj1θ jďk, iď k ă n. (3.11) Similarly to equation (3.1), we observe that this obliquely reflected backward scheme can be rewritten equivalently for k_{P Ji, nK as}

U_kℜ,π,a_“ξπ,an<sub>`</sub> n´1<sub>ÿ</sub> m“k Fπ,am m pVmℜ,π,aqhm´ n´1<sub>ÿ</sub> m“k hmλ´1m Vmℜ,π,aHmJ´ n´1<sub>ÿ</sub> m“k ∆Ma<sub>m</sub> ´ Aan` Aak (3.12)

where _pλkq are given by (3.2) and, for all k P J0, n ´ 1K, ∆Mak is an Ftk`1-measurable

random variable satisfying E_t kr∆M a ks “ 0, Etkr|∆M a k| 2 s ă 8 and Etkr∆M a kHks “ 0. (3.13)

The next theorem is a Snell envelope representation of the obliquely reflected back-ward scheme.

Proposition 3.3. For any j _{P I and 0 ď i ď n, the following hold:}

(i) The discrete process Yℜ,π dominates any ℜ-switched backward scheme, that is, U_iℜ,π,a_{ď p r}Y_iℜ,π_qj, P_{-a.s. for any aP A}ℜ,π

(ii) Define the strategy ¯aℜ,π _{“ p¯θ}r,α¯rqrě0 recursively by p¯θ0,α¯0q :“ pi, jq and, for r _{ě 1,} ¯ θr :“ inf kP J¯θr´1, nK ˇ ˇ tkP ℜ, p rYkℜ,πq ¯ αr´1 _{ď max} m‰ ¯αr´1tp r Y_kℜ,π_qm_{´ C}α¯r´1m tk u ( , ¯ αr :“ min ℓ‰ ¯αr´1 ˇ ˇ p rY_θℜ,π r q ℓ_{´ C}α¯r´1ℓ t_θr “ max m‰ ¯αr´1tp r Y_θℜ,π r q m_{´ C}α¯r´1m t_θr u (

Then we have ¯aℜ,π _{P Ai,j}ℜ,π and

p rY_iℜ,π_qj _{“ Ui}ℜ,π,¯aℜ,π P_{-a.s. (3.15)} (iii) The following “Snell envelope” representation holds:

p rY_iℜ,π_qj _{“ ess sup}

aPA_i,jℜ,π

U_iℜ,π,a P_{-a.s. (3.16)}

Proof. We will adapt the proof of Theorem 2.1 in [8] to the discrete time setting. Observe first that assertion (iii) is a direct consequence of (i) and (ii).

Let us fix i_{P J0, nK and j P I.}

Step 1. We first prove (i).

Set a_{“ pθ}r, αrqrě0 P Ai,jℜ,π and the processp rYa, Zaq defined, for k P Ji, nK, by

# r Ya k :“ ř rě0p rY ℜ,π k qαr1θ rďkăθr`1` ξ π,an 1k“n Za k :“ ř rě0pZ ℜ,π k q αr 1θ rďkăθr`1. (3.17)

Observe that these processes jump between the components of the obliquely reflected backward scheme (3.5) according to the strategy a, and, between two jumps, we have

r Y_θa r “ pY ℜ,π θr`1q αr<sub>`</sub> θr`1<sub>ÿ</sub>´1 k“θr Fπ,αr k ppZ ℜ,π k q αr<sub>qh</sub> k´ θr`1_ÿ´1 k“θr hkλ´1_k pZ_kℜ,πqαrHkJ ´ θr`1<sub>ÿ</sub>´1 k“θr ∆<sub>pM</sub>kqαr ` pKℜ,π<sub>pθr`1´1q_θr</sub>qαr ´ pKℜ,πθr q αr “ rYa θr`1` θr`1_ÿ´1 k“θr Fπ,ak k pZ a kqhk´ θr`1<sub>ÿ</sub>´1 k“θr hkλ´1<sub>k</sub> ZkaHkJ´ θr`1_ÿ´1 k“θr ∆_pMkqαr ` pKℜ,πpθr`1´1q_θrq αr _{´ pK}ℜ,π θr q αr <sub>` ppY</sub>ℜ,π θr`1q αr _{´ p rY}ℜ,π θr`1q αr`1_{q, rě 0. (3.18)} Introducing Ka_k:_“ Na_´1 ÿ r“0 „pθr`1´1q_θ<sub>ÿ</sub> r^k m“θr^k p∆Kℜ,πm qαr `1θ r`1ďk ´ pYθℜ,πr`1q αr ´ p rY_θℜ,π r`1q αr`1_{` pC</sub>π t<sub>θr`1}qαrαr`1 ¯ 

for kP Ji, nK, and summing up (3.18) over r, we get, for k P Ji, nK, r Y_ka_“ξπ,an<sub>`</sub> n´1<sub>ÿ</sub> m“k Fπ,am m pZmaqhm´ n´1<sub>ÿ</sub> m“k hmλ´1m ZmaHmJ´ Na<sub>´1</sub> ÿ r“0 pθr`1_ÿ´1q_k m“θr_k ∆_pMmqαr ´ Aan` Aak` Kan´ Kak.

Using the relation Y_θℜ,π

r “ Pθkp rY

ℜ,π

θk q for all r P J0, N

a_{K, we easily check that K}a _is

an increasing process. Since Uℜ,π,a solves (3.12), we deduce by a comparison argument (see Corollary 2.5 in [4]) that U_iℜ,π,a _{ď r}Ya

i. Since a is arbitrary in A ℜ,π

i,j , we deduce

(3.14).

Step 2. We now prove (ii).

Consider the strategy ¯aℜ,πgiven above as well as the associated process_{p r}Y¯aℜ,π

, Z¯aℜ,π_q defined as in (3.17). By definition of ¯aℜ,π, we have

pY_θℜ,π¯ r`1q ¯ αr <sub>“ pP</sub>π ¯ θr`1p rY ℜ,π ¯ θr`1qq ¯ αr <sub>“ p rY</sub>ℜ,π ¯ θr`1q ¯ αr`1<sub>´ pC</sub>π t<sub>αr`1</sub>¯ q ¯ αrα¯r`1_{, r ě 0,}

which gives that K¯aℜ,π

“ 0 and then, for all k P Ji, nK,

r Y_k¯aℜ,π _“ξπ,¯aℜ,πn <sub>`</sub> n´1<sub>ÿ</sub> m“k Fπ,¯aℜ,πm m pZ ¯ aℜ,π m qhm´ n´1<sub>ÿ</sub> m“k hmλ´1m Z ¯ aℜ,π m HmJ ´ N¯aℜ,π<sub>´1</sub> ÿ r“0 p¯θr`1_ÿ´1q_k m“¯θr_k p∆Mmqα¯r ´ A¯a ℜ,π n ` A ¯ aℜ,π k . (3.19) Hence,_{p r}Y¯aℜ,π

, Z¯aℜ,π_{q and pU}¯aℜ,π, Va¯ℜ,π_{q are solutions of the same backward scheme and} p rY_iℜ,π_qj _{“ U}¯aℜ,π

i . To complete the proof, we only need to check that ¯aℜ,π P Aℜ,π, that

is E_r|A¯aℜ,π

n | 2

s ă 8. By definition of ¯aℜ,π on Ji, nK and the structure condition on costs (2.1), we have _|A¯aℜ,π

i | ď maxk‰j|Ctjki | which gives Er|A

¯ aℜ,π

i | 2

s ď C. Combining (3.19) with the Lipschitz property of Fπ _{and estimates in Proposition 3.2, we get the square}

integrability of A¯anℜ,π and the proof is complete. l

Proposition 3.4. Assume than _{pHF d}2q is in force. For all 0 ď i ď n, j P I, we have

E « sup iďkďn ˇ ˇ ˇUkℜ,π,¯aℜ,π ˇ ˇ ˇ 2 ` n´1<sub>ÿ</sub> k“i hk ˇ ˇ ˇVkℜ,π,¯aℜ,π ˇ ˇ ˇ 2 `ˇˇˇA¯anℜ,π ˇ ˇ ˇ 2 `ˇˇˇN¯aℜ,π ˇ ˇ ˇ 2ff ď C,

for the optimal strategy ¯aℜ,π _{P Ai,j}ℜ.

Proof. Fix _{pi, jq P J0, nK ˆ I. According to the identification of pU}ℜ,π,¯aℜ,π, Vℜ,π,¯aℜ,π_q withp rYa¯ℜ,π

, Za¯ℜ,π_{q obtained in the proof of Proposition 3.3, we deduce from Proposition} 3.2 expected controls on Uℜ,π,¯aℜ,π and Vℜ,π,¯aℜ,π.

By taking conditional expectation in (3.19), we have E_t irA ¯ aℜ,π n s “Eti « ξπ,¯aℜ,πn _{´ rY}¯aℜ,π i ` n´1<sub>ÿ</sub> m“i Fπ,¯aℜ,πm m pZ ¯ aℜ,π m qhm` A¯a ℜ,π i ff .

Thus, using standard inequalities and the growth of Fπ_{, we easily obtain}

E_r|A¯a_nℜ,π_|2_{s ďCE} « sup iďkďn| r Y¯aℜ,π k | 2 ` n´1<sub>ÿ</sub> m“i |Zm¯aℜ,π| 2 hm` |A¯a ℜ,π i | 2 ff .

We have already noticed in the proof of Proposition 3.3 that we have |Aa¯ℜ,π

i | ď

maxk‰j|Cijk|, which inserted into the previous inequality leads to Er|A ¯ aℜ,π

n | 2

s ď C. We finally complete the proof, observing from the structure condition (2.1) that

E_r|N¯aℜ,π_|2_{s ď CEr|A}¯a_nℜ,π_|2_s.

l

3.3 Stability of obliquely reflected backward schemes

We now consider two obliquely reflected backward schemes, with different parameters but the same reflection grid ℜ. For ℓ _{P t1, 2u, we consider an F}T-measurable random

terminal condition ℓ_{ξ, a random generator z ÞÑ} ℓ_{Fp., zq and random cost processes}

pℓ_Cij_q

1ďi,jďd satisfying the structural condition (2.1). As in Subsection 3.2, terminal

conditions, generators and cost processes are allowed to depend on π but we omit the script π for reading convenience. We denote by_pℓ_Yrℜ,π_,ℓ_Yℜ,π_,ℓ_Zℜ,π_{q the solution of the}

associated obliquely reflected backward scheme.

Defining δYℜ,π :_“1_Yℜ,π_´2_Yℜ,π_{, δ rY}ℜ,π _:“1_Yrℜ,π_´2_Yrℜ,π_{, δZ}ℜ,π _:“1_Zℜ,π_´2_Zℜ,π_, δξ:_“1ξ_´2ξ together with |δCt|₈:“ max i,jPI ˇ ˇ ˇ1C_tij_´2C_tij ˇ ˇ ˇ , |δFk|8:“ max_iPI sup

zPMd,d

ˇ ˇ1

F_ki_´2F_kiˇˇ pzq, for 0_{ď k ď n ´ 1, we prove the following stability result.}

Proposition 3.5. Assume that _{pHF d}pq is in force for some given p ě 2. Then we

have, for any i_{P J0, nK,}

sup iďkďn E „ˇ_ˇ ˇδYkℜ,π ˇ ˇ ˇ 2 ` ˇ ˇ ˇδ rY<sub>k</sub>ℜ,πˇˇ<sub>ˇ</sub> 2 `_κ1E «_n´1 ÿ k“i hk ˇ ˇ ˇδZkℜ,π ˇ ˇ ˇ 2ff ď CE «_n´1 ÿ k“i |δFk| 2 8hk` |δξ| 2 ff ` Cpκ4{pE « sup 0_{ďkďn,tkPℜ}|δC tk| p 8 ff2_{p .

Proof. We adapt to our setting the proof of Proposition 2.3 in [8]. The proof is divided into three steps and relies heavily on the reinterpretation in terms of switching problems. We first introduce a convenient dominating process and then provide successively the controls on δYℜ,π and δZℜ,π terms.

Step 1. Introduction of an auxiliary backward scheme. Let us define F :“ 1_F _2_F , ξ :_“1_ξ _2_ξ and C by Cij _:“1_Cij_{_}2_Cij_{. ObservepHF d}

pq holds for the data

pC, F, ξq and C satisfies the structure condition (2.1). We denote by p rYℜ,π, Yℜ,π, Zℜ,π_q the solution of the discretely obliquely reflected backward scheme with generator F , terminal condition ξ, reflection grid ℜ and cost process C.

Using Proposition 3.1 and the definition of F , ξ and C, we obtain that r

Yℜ,π ě1Yrℜ,π_{_}2Yrℜ,π. (3.20) Using Proposition 3.3, we introduce switched backward schemes associated to 1Yℜ,π_, 2_Yℜ,π _{and Y}ℜ,π _{and denote by ˇa “ pˇθ}

r,αˇrqrě0 the optimal strategy related to Yℜ,π

starting from a fixed _{pi, jq P J0, nK ˆ I. therefore, we have}

p rY_iℜ,π_qj _{“ Ui}ℜ,π,ˇa_“ξˇan<sub>`</sub> n´1<sub>ÿ</sub> k“i Fˇak k pV ℜ,π,ˇa k qhk´ n´1<sub>ÿ</sub> k“i hkλ´1k V ℜ,π,ˇa k HkJ´ n´1<sub>ÿ</sub> k“i ∆Mˇa<sub>k</sub> ´ Aanˇ ` A ˇ a i (3.21)

Step 2. Stability of the Y component. Since ˇa_{P Ai,j}ℜ,π, we deduce from Proposi-tion 3.3 (i) that

pℓYr_iℜ,π_qj _ěℓU_iℜ,π,ˇa_“ℓξaˇn_{`</sub> n´1<sub>ÿ</sub> k“i ℓ<sub>F</sub>ˇak k p ℓ<sub>V</sub>ℜ,π,ˇa k qhk´ n´1<sub>ÿ</sub> k“i hkλ´1k ℓ<sub>V</sub>ℜ,π,ˇa k HkJ´ n´1<sub>ÿ</sub> k“i ∆ℓMˇa<sub>k</sub> ´ℓAˇa<sub>n`}ℓAˇa_i, ℓ_{P t1, 2u,} (3.22) where ℓ_Aˇa _{is the process of cumulated costs p}ℓ_Cij_q

i,jPI associated to the strategy ˇa.

Combining this estimate with (3.20) and (3.21), we derive

|p1Yr_iℜ,π_qj_{´ p}2Yr_iℜ,π_qj_{| ď |Ui}ℜ,π,ˇa_´1U_iℜ,π,ˇa_{| ` |Ui</sub>ℜ,π,ˇa<sub>´</sub>2U<sub>i</sub>ℜ,π,ˇa<sub>|.</sub> (3.23) Since both terms on the right-hand side of (3.23) are treated similarly, we focus on the first one and introduce discrete processes Γˇa<sub>:“ U</sub>ℜ,π,ˇa<sub>` A}ˇa _and 1

Γˇa_:“1_Uℜ,π,ˇa_`1_Aˇa_.

Rewriting (3.21) and (3.22) between k and k` 1 for k P Ji, n ´ 1K, we get Γˇa<sub>k´</sub>1Γˇa<sub>k “ Γk`1</sub>aˇ _´1Γˇa_{k`1` rF}ˇak k pV ℜ,π,ˇa k q ´ 1 Faˇk k p 1_Vℜ,π,ˇa k qshk ´ hkλ´1k rV ℜ,π,ˇa k ´ 1 V_kℜ,π,ˇa_sHkJ_{´ r∆Mk}aˇ_{´ ∆}1Mˇa_ks. Using the identity _|y|2

“ |x|2 ` 2xpy ´ xq ` |x ´ y|2 , we obtain, E_t kr|Γ ˇ a k`1´ 1 Γˇa<sub>k`1|</sub>2_s “ |Γˇak´ 1 Γˇa_k|2_{´ 2pΓ}ˇa_k´1Γˇa_kqpFˇak k pV ℜ,π,ˇa k q ´ 1 Fˇak k p 1 V_kℜ,π,ˇa_qqhk ` Etkr|rF ˇ ak k pV ℜ,π,ˇa k q ´ 1 Fˇak k p 1 V_kℜ,π,ˇa_{qshk´ hk}λ_k´1_rVkℜ,π,ˇa_´1V_kℜ,π,ˇa_sHkJ_{´ r∆M}ˇa_{k´ ∆}1Ma_kˇ_s|2_s.

Then, by the same reasoning as in the step 2 of the proof of Proposition 3.2, previous equality becomes E_t kr|Γ ˇ a k`1´ 1 Γˇa<sub>k`1</sub>|2 s ě |Γˇa k´ 1 Γˇak| 2 ´ 2pΓˇa k´ 1 ΓˇakqpF ˇ ak k pV ℜ,π,ˇa k q ´ 1 Fˇak k p 1_Vℜ,π,ˇa k qqhk ` _Λdhk|Vkℜ,π,ˇa´ 1 V_kℜ,π,ˇa_|2,

and we obtain, by summing over k and taking expectation,

E « |Γˇai ´ 1 Γa_iˇ_|2<sub>`</sub> n´1<sub>ÿ</sub> k“i hk|V<sub>k</sub>ℜ,π,ˇa´1V<sub>k</sub>ℜ,π,ˇa|2 ff ď CE „ |Γˇa n´ 1 Γˇa n| 2 ` n´1_ÿ k“i |Γˇa k´ 1 Γˇa k||F ˇ ak k pV ℜ,π,ˇa k q ´ 1 Fˇak k p 1_Vℜ,π,ˇa k q|hk  . Since F _“1_F _2_F and 1_F

is a Lipschitz function, we also get |Fˇak k pV ℜ,π,ˇa k q ´ 1 Fˇak k p 1_Vℜ,π,ˇa k q| ď |δFk|8` C|V ℜ,π,ˇa k ´ 1_Vℜ,π,ˇa k |,

and then, by using Young’s inequality and discrete Gronwall’s lemma, we deduce from the last and the penultimate inequalities that

E ” |Uiℜ,π,ˇa´ 1_Uℜ,π,ˇa i | 2ı ď C ˜ E_r|δξ|2<sub>`</sub> n´1<sub>ÿ</sub> k“i |δFk| 2 8hk` Er|Aˇan´ 1_Aˇa n| 2 ` |Aˇai ´ 1_Aˇa i| 2 s ¸ . (3.24) Moreover we compute, for all k_{P Ji, nK,}

E_r|Aˇa_k´1Aˇa k| 2 s ď Er|Nˇa_|2 sup 0_{ďmďn,tmPℜ}|δCtm| 2 8s. If p_{“ 2, then N}ˇa_{ď κ yields} E_r|Aˇa_k´1Aˇa k| 2 s ď κ2E_{r sup} 0ďmďn,tmPℜ |δCtm| 2 8s.

Otherwise, from Proposition 3.4, H¨older inequality and the fact that Nˇa_{ď κ, we deduce}

E_r|Aˇa_k´1A_kaˇ_|2_{s ď E}”_|Nˇa_| 2p p´2 ıp´2 p E « sup 0_{ďmďn,tmPℜ}|δCtm| p 8 ff2_{p ď E”κ 2p p´2´2_|Nˇa_|2 ıp´2 p E « sup 0ďmďn,tmPℜ |δCtm| p 8 ff2{p ď Cpκ4{pE « sup 0ďmďn,tmPℜ |δCtm| p 8 ff2{p .

Inserting the last estimate into (3.24), we get E ” |Uiℜ,π,ˇa´ 1 U_iℜ,π,ˇa_|2ı_{ď C} ˜ E_r|δξ|2<sub>`</sub> n´1<sub>ÿ</sub> k“i |δFk| 2 8hk ¸ ` Cpκ4{pE « sup 0_{ďmďn,tmPℜ}|δC tm| p 8 ff2{p .

By symmetry, we have the same estimate for E”_|Uiℜ,π,ˇa_´2_Uℜ,π,ˇa i |

2ı

. Therefore, from (3.23) and the fact that j is arbitrary, we deduce the wanted estimate for E_{r|δ r}Y_iℜ,π_|2

s. The estimate for E_r|δYiℜ,π_|2

s is a direct corollary.

Step 3. Stability of the Z component. Observing that δZ_kℜ,π _{“ E}tkrpδY

ℜ,π k`1 ´ E<sub>t</sub> krδY ℜ,π k`1sqHkJs, one computes hk|δZ_kℜ,π|2 ď CEtk ” |δYk`1ℜ,π| 2 ´ |Etk ” δY<sub>k`1</sub>ℜ,πı_|2ı (3.25) From the scheme’s definition, we have

|Etk

” δY_k`1ℜ,π

ı

|2 ě |δ rY_kℜ,π_|2_{´ 2|δ r}Y_kℜ,π_r1Fkp1Z_kℜ,πq ´2Fkp2Z_kℜ,πqshk| .

Inserting the last estimate into (3.25) and using pHF dpq, we obtain, for some η ą 0,

hk|δZ_kℜ,π|2ď C ˆ E_t k ” |δYk`1ℜ,π| 2ı ´ |δ rY<sub>k</sub>ℜ,π<sub>|</sub>2 ` Chk ˆ 1<sub>`</sub> 1 η ˙ |δ rY<sub>k</sub>ℜ,π<sub>|</sub>2 ` ηhk|δZ_kℜ,π|2` hk|δFk|28 ˙ .

Taking expectation on both sides and summing over k with η_{“ 1{2, we get} 1 2 n´1_ÿ k“i hkE ” |δZkℜ,π| 2ı ď C ¨ ˚ ˝E ” |δYnℜ,π| 2ı ` n´1<sub>ÿ</sub> k“i tkPℜ E ” |δYkℜ,π| 2 ´ |δ rY<sub>k</sub>ℜ,π<sub>|</sub>2ı<sub>` max</sub> iďkďn´1 E ” |δ rY_kℜ,π_|2ı_{`</sub> n´1<sub>ÿ</sub> k“i hk|δFk| 2 8 ˛ ‹ ‚ , ď C ˜ E ” |δYnℜ,π| 2ı ` κ max iďkďn´1 E ” |δYkℜ,π| 2 ` |δ rY<sub>k</sub>ℜ,π<sub>|</sub>2ı<sub>`} n´1_ÿ k“i hk|δFk| 2 8 ¸ .

The proof is concluded using estimates on δ rYℜ,π _{and δY}ℜ,π _{already obtained in the first}

part of the proof. l

We will now use this general stability result on obliquely reflected backward schemes to obtain a L2-stability result for the scheme (1.4) (see [6] for a general definition of L2

-stability for backward schemes). Firstly, we introduce a perturbed version of the scheme given in (1.4).

Definition 3.2. (i) The terminal condition is given by a FT-measurable random

(ii) for 0ď i ă n, $ ’ ’ & ’ ’ % ¯ Z_iℜ,π :_{“ Er ¯}Y_{i`1</sub>ℜ,πHi | Ftis, r¯ Y<sub>i</sub>ℜ,π :<sub>“ Er ¯</sub>Y<sub>i`1}ℜ,π _{| F}tis ` hifpX π ti, rY¯ ℜ,π i , ¯Z ℜ,π i q ` ζ f i, ¯ Y_iℜ,π :_{“ r¯}Y_iℜ,π1ttiRℜu` ¯PtipX π ti, rY¯ ℜ,π i q1ttiPℜu, (3.26)

with ¯P the oblique projection ¯ P_ti :px, yq P Rdˆ RdÞÑ ˆ max jPIty j ´ ¯ckjtipxqu ˙ 1ďkďd , associated to costs ¯ctipxq :“ cpxq ` ζ c ti. Perturbations ζ Y i :“ pζ f i , ζtciq are Fti

-measurable and square integrable random variables. Moreover we assume that the random costs _p¯ctipXtiqq0ďiďn satisfy the structure conditions (2.1).

Setting δYi “ Yiℜ,π´ ¯Y ℜ,π i , δ rYi “ rYiℜ,π´ r¯Y ℜ,π i and δZi “ Ziℜ,π´ ¯Z ℜ,π i , we obtain

the following L2-stability result for the scheme (1.4).

Proposition 3.6. Assume that _{pHf q is in force and, for all p ě 2,}

E « | ¯Yn|2` n´1<sub>ÿ</sub> i“0 |ζif| 2 ` sup 0_ďiďn ˇ ˇζc ti ˇ ˇp ff ď C. (3.27)

We also assume that _|π|LY _{ă 1 and}

ˆ sup 0ďiďn´1 hi|Hi| ˙ LZ _{ď 1.} (3.28)

Then schemes (1.4) and (3.26) are well defined and the following L2-stability holds true, for all p_{ě 2,} sup 0ďiďn E ” |δYi|2` |δ rYi|2 ı `1_κ n´1_ÿ i“0 hiE “ |δZi|2 ‰ ď C ˜ E“_|δYn|2‰<sub>`</sub> n´1<sub>ÿ</sub> i“0 1 hi E ” |ζif| 2ı ¸ ` Cpκ 4_{p E » – sup 0ďiďn tiPℜ |ζic|p fi fl 2_{p . (3.29)

Proof. Since we have assumed _|π|LY _{ă 1, then a simple fixed point argument shows}

that schemes (1.4) and (3.26) are well defined, i.e. there exists a unique solution to each scheme.

For the L2

-stability, we want to apply Proposition 3.5 with 1_ξ

“ gpXπ Tq, 2_ξ “ ¯Yn, 1 Fipzq “ f pXtπi, rY ℜ,π i , zq, 2 Fipzq “ f pXtπi, rY¯ ℜ,π i , zq ` ζ f i, 1 Cti “ cpX π tiq and 2 Cti “ c<sub>pXt</sub>π iq ` ζ c

ti. To do this, we have to check that assumptionpHF dpq is fulfilled for these

two obliquely reflected backward schemes. Firstly, we have assumed ˆ sup 0ďiďn´1 hi|Hi| ˙ LZ _{ď 1.}

Moreover, hypothesis pHfq, assumption (3.27) and classical estimates for processes X and Xπ _{leads to} E « |1ξ_|2_{` |</sub>2ξ<sub>|</sub>2<sub>`} n´1_ÿ i“0 r|1Fip0q|2` |2Fip0q|2shi` sup 0_ďiďnr| 1 Cti| p ` |2Cti| p s ff ď Cp` CE „ sup 0_ďiďnr| r Y_iℜ,π_|2_{` | r¯}Y_iℜ,π_|2_s  .

To estimate quantities E”sup0_ďiďn| rY ℜ,π i |

2ı

and E”sup0_ďiďn| r¯Y ℜ,π i |

2ı

, we just have to rewrite slightly the first step of the proof of Proposition 3.2. The beginning of the proof stays true: (3.7) yields, for all i_{P J0, nK,}

E „ sup iďkďn ” | rY_kℜ,π_|2_{` | r¯</sub>Y<sub>k</sub>ℜ,π<sub>|</sub>2 ı ď CE « |1ξ<sub>|</sub>2<sub>` |}2ξ_|2<sub>`</sub> n´1<sub>ÿ</sub> k“i “ |1Fkp0q|2` |2Fkp0q|2 ‰ hk ff ď C ˜ 1_{`</sub> n´1<sub>ÿ</sub> k“i E „ sup kďmďn ” | rY<sub>m</sub>ℜ,π<sub>|</sub>2<sub>` | r¯}Y_mℜ,π_|2 ı hk ¸ .

Thus, the discrete Gronwall lemma allows to conclude that

E „ sup 0ďkďn ” | rY_kℜ,π_|2_{` | r¯}Y_kℜ,π_|2ı_{ď C}

and then assumption _{pHF d}pq is fulfilled. Proposition 3.5 and pHf q imply, for all i P

J0, nK, sup iďkďn E ” |δYk| 2 ` |δ rYk| 2ı `1_κ n´1_ÿ k“i hkE “ |δZk| 2‰ ď C ˜ E“_|δYn|2‰<sub>`</sub> n´1<sub>ÿ</sub> k“i |δFk|28 ¸ ` Cpκ4{pE » — – sup 0ďkďn tkPℜ |ζtck| p fi ffi fl 2_{p ď C ˜ E“_|δYn|2‰<sub>`</sub> n´1<sub>ÿ</sub> k“0 1 hk E ” |ζkf| 2ı ` n´1_ÿ k“i sup kďmďn E ” |δYm|2` |δ rYm|2 ı hk ¸ ` Cpκ4{pE » — – sup 0ďkďn tkPℜ |ζtck| p fi ffi fl 2{p .

Applying the discrete Gronwall lemma to the last inequality completes the proof. l

3.4 Convergence analysis of the discrete-time approximation

We will give now the main result of this section that provides an upper bound for the error between the obliquely reflected backward scheme (1.4) and the discretely obliquely reflected BSDE (1.3).

Theorem 3.1. Assume that pHf q is in force. We also assume that |π|LY _{ă 1 and}

weights _pHiq0_ďiďn´1 are given by

pHiqℓ“ ´R hi _ W_tℓ_{i`1´ Wt}ℓ_i hi ^ R hi , 1ď ℓ ď d, (3.30)

with R a positive parameter such that RLZ_{ď 1. Then the following holds:}

sup 0ďiďn E ” | rY_tℜ_{i ´ r}Y_iℜ,π_|2<sub>` |Yt</sub>ℜ<sub>i ´ Yi</sub>ℜ,π<sub>|</sub>2 ı `1_κE «_n´1 ÿ i“0 żti`1 ti |Zsℜ´ Z ℜ,π i | 2 ds ff ď CR ´ |π|1{2` κ|π|¯. Proof.

Step 1. Expression of the perturbing error. Since we want to apply Proposition 3.6, we first observe that_pYℜ, Zℜ_{q can be rewritten as a perturbed obliquely reflected} backward scheme. Namely, setting ¯Yi :“ Ytℜi and rY¯i :“ rY

ℜ

ti, for all iP J0, nK, we have

$ ’ ’ & ’ ’ % ¯ Z_i:_{“ Er ¯}Y_{i`1</sub>Hi| Ftis, r¯ Y<sub>i</sub> :<sub>“ Er ¯</sub>Y<sub>i`1| F}tis ` hifpX π ti, rY¯i, ¯Ziq ` ζ f i, ¯ Y_i :_{“ r¯}Y_i1<sub>ttiRℜu` ¯</sub>P<sub>tipX</sub>π ti, rY¯iq1ttiPℜu, (3.31) with ζ<sub>i</sub>f <sub>“ E</sub>ti „żti`1 ti ´ f_pXs, rYsℜ, Z ℜ sq ´ f pXtπi, rY ℜ ti, ¯Ziq ¯ ds  and ζ_tc_{i “ cpX}tiq ´ cpX π tiq.

Let us check that (3.27) is fulfilled for all p _{ě 2: using pHf q, Proposition 2.3 and} classical estimates for X and Xπ_{, we get}

E « | ¯Yn| 2 ` n´1<sub>ÿ</sub> i“0 |ζif| 2 ` sup 0ďiďn ˇ ˇζc ti ˇ ˇp ff ď CpE « 1_{` sup</sub> sPr0,T s|X s|p` sup iPJ0,nK|X π ti| p ` sup sPr0,T s| ˜ Y<sub>s</sub>ℜ<sub>|</sub>2<sub>`} żT 0 |Z ℜ s| 2 ds_{`</sub> n´1<sub>ÿ</sub> i“0 | ˜Y<sub>t</sub>ℜ i`1Hihi| 2 ff ď Cp ¨ ˝1 ` E » – sup sPr0,T s| ˜ Y<sub>s</sub>ℜ<sub>|</sub>4<sub>`} ˜_n´1 ÿ i“0 |Hihi|2 ¸2fi fl ˛ ‚.

Applying Burkholder-Davis-Gundy inequality, we have E„´řn´1_{i“0 |H}ihi|2

¯2

ď C and so (3.27) is fulfilled. Finally, we easily check that (3.30) implies (3.28).