yn j = E[yn j+1|Fn j ] + g(tj, E[yn j+1|Fn j], zn j, un j)δn+ an j − kn j an j ≥ 0, kn j ≤ 0, an jkn j = 0, ξn j ≤ yn j ≤ ζn j, (yn j − ξn j)an j = (yn j − ζn j)kn j = 0. (2.5.6)
Nous illustrons num´eriquement les r´esultats th´eoriques et nous montrons qu’ils co¨ıncident avec ceux obtenus en utilisant le sch´ema p´enalis´e (2.5.5) pour de grandes valeurs du param`etre de p´enalisation p.
2.6 Conclusions et perspectives
Dans cette th`ese, nous avons explor´e de nouveaux probl`emes en analyse stochastique, contrˆole stochastique, th´eorie de jeux et math´ematiques financi`eres, `a la fois d’un point de vue th´eorique et num´erique. Les r´esultats principaux sont les suivants:
• Dans le chapitre 3 (article [57]), nous introduisons une nouvelle classe d’EDSR avec condition terminale faible non lin´eaire, associ´ees `a la couverture approximative sous contraintes de mesures de risque dynamiques.
• Dans le chapitre 4 (article [61]), nous ´etudions un probl`eme d’arrˆet optimal pour des mesures de risque dynamiques induites par des EDSR acec sauts et nous montrons que la fonction valeur correspond `a l’unique solution de viscosit´e d’un probl`eme d’obstacle pour les ´equation partielles int´egro-diff´erentielles.
• Dans le chapitre 5 (article [62]), nous introduisons un nouveau probl`eme de jeu, qui g´en´eralise le jeu classique de Dynkin au cas d’esp´erance non lin´eaire, permettant d’obtenir une repr´esentation de la solution des EDSR doublement r´efl´echies non lin´eaires en termes de jeux stochastiques. • Dans le chapitre 6 (article [63]), nous ´etudions dans un cadre markovien un probl`eme de contrˆole stochastique et d’arrˆet optimal mixte dans le cas o`u l’esp´erance classique dans le crit`ere est remplac´ee par une esp´erance non lin´eaire induite par la solution d’une EDSR avec sauts et la fonction de profit terminal est seulement mesurable. Nous ´etablissons un principe de programmation dynamique faible et en d´eduisons les ´equations HJB non lin´eaires associ´ees.
• Dans le chapitre 7 (article [63]), nous introduisons une approximation num´erique pour la so-lution d’une EDSRDR avec sauts et obstacles irr´eguliers qui admet `a la fois des sauts totale-ment inacessibles et pr´evisibles. Nous proposons un sch´ema compl`etetotale-ment impl´etotale-mentable, reposant sur une m´ethode de p´enalisation et l’approximation du mouvement brownien et
du processus de Poisson par deux marches al´eatoires ind´ependantes dont on montre la con-vergence vers la solution de l’EDSRDR. Nous illustrons les r´esultats th´eoriques avec des exemples num´eriques.
• Dans le chapitre 8 (article [60]), nous introduisons un sch´ema compl`etement impl´ementable alternatif `a celui pr´esent´e dans le chapitre 6 afin d’approximer la solution de l’EDSR double-ment r´efl´echie avec sauts et obstacles irr´eguliers. Ce sch´ema est obtenu par une discr´etisation directe des EDSRDR et ne d´epend alors que du nombre de pas de temps n (et plus du param`etre de p´enalisation p). Nous obtenons la convergence du sch´ema et donnons des exemples num´eriques.
Concernant les perspectives, ils y a plusieurs orientations de recherche `a venir, `a la fois d’un point de vue th´eorique et num´erique. En collaboration avec R. Elie et D. Possamai nous sommes en train de finir un travail sur les EDSR avec r´eflexion faible ([58]) qui sont reli´ees `a la couverture approximative des options am´ericaines. Avec M.C. Quenez et A. Sulem, nous travaillons sur un nouveau probl`eme de jeux mixte avec controle stochastique et temps d’arrˆet dans le cadre markovien [64] et on ´etudie les liens entre les jeux de Dynkin G´en´eralis´es et le pricing non lin´eaire, dans des march´es complets et incomplets [65].
D’un pont de vue num´erique, il serait utile de proposer des sch´emas num´eriques pour la solu-tion des EDSRDR avec barri`eres c`adl`ag dans le cas d’une mesure de Poisson g´en´erale ainsi que des EDSR avec condition terminale faible.
Part I
Stochastic control and Optimal Stopping
with non linear expectations
Chapter 3
BSDEs with nonlinear weak terminal
condition
Abstract. In a recent paper, Bouchard, Elie and Reveillac [31] have introduced a new class of Backward Stochastic Differential Equations with weak terminal condition, in which the T -terminal value YT of the solution (Y, Z) is not fixed as a random variable, but only satisfies a constraint of the form E[Ψ(YT)] ≥ m. The aim of this paper is to study a more general class of BSDEs, with nonlinear expectation constraints on the terminal condition, induced by the solution of a Backward Stochastic Differential Equation. More precisely, the constraint takes the form E0,Tf [Ψ(YT)] ≥ m, where Ef represents the f -conditional expectation associated to a nonlinear driver f . These BSDEs are called BSDEs with nonlinear weak terminal solution. We carry out a similar analysis as in [31] of the value function corresponding to the minimal solutions Y of the BSDE with nonlinear weak terminal condition: we study the regularity, establish the main properties, in particular continuity and convexity with respect to the parameter m, and finally provide a dual representation in the case of concave constraints. From a financial point of view, our study is closely related to the approximative hedging of an European option under dynamic risk measures constraints. The nonlinearity f raises subtle difficulties, highlighted through out the paper, which cannot be handled by the arguments used in the case of classical expectations constraints studied in [31].
Key words : Backward stochastic differential equations, g-expectation, dynamic risk measures, optimal control, stochastic targets.
3.1 Introduction
Linear backward stochastic differential equations (BSDEs) were introduced by Bismut as the ad-joint equations associated with Pontryagin maximum principles in stochastic control theory. The general case of non-linear BSDEs was then studied by Pardoux and Peng [131]. They provided Feynman-Kac representations of solutions of non-linear parabolic partial differential equations. The solution of a BSDE consists in a pair of predictable processes (Y, Z) satisfying
−dYt = g(t, Yt, Zt)dt− ZtdWt; YT = ξ. (3.1.1) 53
These equations appear as an useful mathematical tool in various problems in finance, for example in the theory of derivatives pricing. In a complete market - when it is possible to construct a portfolio which attains as final wealth the payoff- the value of the replicating portfolio is given by Y and the hedging strategy by Z. Since in incomplete markets is not always possible to construct a portfolio which attains exactly as final wealth the amount ξ, it was suggested to replace the terminal condition into a weaker one of the form YT ≥ ξ. In this case, the minimal initial value Y0
defines the smallest initial investment which allows one to superhedge the contingent claim ξ. Recently, Bouchard, Elie and Reveillac [31] introduced a new class of BSDEs, the so called BSDEs with weak terminal condition, in which the T -terminal value YT only satisfies a weak constraint. More precisely, a couple of predictable processes (Y, Z) is said to be a solution of such a BSDE if it satisfies:
−dYt = g(t, Yt, Zt)dt− ZtdWt; (3.1.2)
E[Ψ(YT)]≥ m, (3.1.3)
where m is a given threshold and Ψ a non-decreasing map. The main question in [31] is the following:
Find the minimal Y0 such that (3.1.2) and (3.1.3) hold for some Z. (3.1.4)
From a financial point of view, this study is related to the hedging in quantile or more generally to the hedging with expected loss constraints. This problem was addressed in the literature for the first time by Follmer and Leukert [85] and then further studied in a Markovian framework in [32] and [118], using stochastic target techniques .
In [31], the key point of the analysis is the reformulation of the problem written in terms of BSDE with weak terminal condition into an optimization problem on a family of BSDEs with strong terminal condition, by using the martingale representation theorem. The main observation is that if Y0 and Z are such that (3.1.3) holds, then the martingale representation Theorem implies that it exists an element α ∈ A0, the set of predictable square integrable processes, such that:
Ψ(YT)≥ MTm,α = m + Z T
0
αsdWs, (3.1.5)
It is then shown that the initial problem (3.1.4) is equivalent to: inf{Yα
0 , α∈ A0}, (3.1.6)
where Yα
t corresponds to the solution at time t of the BSDE with (strong) terminal condition Φ(Mα
T), Φ representing the left-continuous inverse of Ψ.
The aim of this paper is to introduce a new class of BSDEs with weak nonlinear terminal condition . We extend the results of [31] to a more general class of constraints which take the form:
E0,Tf [Ψ(YT)]≥ m, (3.1.7)
where f is a nonlinear driver and E·,Tf [ξ] the solution of the BSDE with generator f and terminal condition ξ.
We can easily remark that the constraint (3.1.3) is a particular case of (3.1.7) for f = 0. The problem under study in this paper is the following:
inf{Y0 such that ∃Z : (3.1.2) and (3.1.7) hold}. (3.1.8) Following the key idea of [31], we rewrite our problem (3.1.8) into an equivalent one expressed in terms of BSDEs with strong terminal condition. The main difference with respect to [31] is given by the fact that in our case we have to introduce a new controlled diffusion process, which is an f−martingale, contrary to [31] where it is a classical martingale. Indeed, for a given Y0 and Z such that (3.1.2) and (3.1.7) are satisfied, appealing to the BSDE representation of Ψ(YT), we can find α ∈ A0 such that:
Ψ(YT)≥ Mm,αT = m− Z T 0 f (s,Mm,αs , αs)ds + Z T 0 αsdWs. (3.1.9)
Thanks to this observation, we show that Problem (3.1.8) is equivalent to (3.1.6), where, in our more general framework, Yα
t corresponds to the solution at time t of the BSDE with (strong) terminal condition Φ(Mα
T). We study the dynamical counterpart of (3.1.6):
Yα(τ ) := essinf{Yτα′, α′ ∈ A0 s.t.α′ = α on [[0, τ ]]}. (3.1.10) We carry out a similar analysis as in [31] of the family {Yα, α ∈ A0}. We start by studying the regularity of the family Yα and show that it can be aggregated into a RCLL process, proof which becomes considerably more technical in our context with respect to [31], because we have to deal with the nonlinearity f . We then provide a BSDE representation of Yα and show that, under a concavity assumption on the driver f , there exists an optimal control. We also study the main properties of the value function, as continuity and convexity with respect to the threshold m, and propose proofs specific to the nonlinear case. We finally get, in the case of concave constraints, a dual representation of the value function, related to a stochastic control problem in Meyer’s form. Besides the mathematical interest of our study, this work is also motivated by some financial applications, as it provides the approximative hedging under dynamic risk
measures contraints of an European option, when the shortfall risk is quantified in terms of dy-namic risk measures induced by BSDEs.
The paper is organized as follows. In Section 2 we introduce notation, assumptions and the BSDEs with nonlinear weak terminal condition. In Section 3, we study the regularity and the BSDE representation of the value function Yα. In Section 4, we establish the main properties of the value function and finally we provide a dual representation in Section 5.