Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles

We study a discrete time approximation scheme for the solution of a doubly reflected Backward Stochastic Differential Equation (DBBSDE in short) with jumps, driven by a Brownian motion and an independent compensated Poisson process. Moreover, we suppose that the obstacles are right continuous and left limited (RCLL) processes with predictable and totally inaccessible jumps and satisfy Mokobodski's condition. Our main contribution consists in the construction of an implementable numerical sheme, based on two random binomial trees and the penalization method, which is shown to converge to the solution of the DBBSDE. Finally, we illustrate the theoretical results with some numerical examples in the case of general jumps

Reflected scheme for doubly reflected BSDEs with jumps and RCLL obstacles

We introduce a discrete time reflected scheme to solve doubly reflected Backward Stochastic Differential Equations with jumps (in short DRBSDEs), driven by a Brownian motion and an independent compensated Poisson process. As in Dumitrescu-Labart (2014), we approximate the Brownian motion and the Poisson process by two random walks, but contrary to this paper, we discretize directly the DRBSDE, without using a penalization step. This gives us a fully implementable scheme, which only depends on one parameter of approximation: the number of time steps $n$ (contrary to the scheme proposed in Dumitrescu-Labart (2014), which also depends on the penalization parameter). We prove the convergence of the scheme, and give some numerical examples.Comment: arXiv admin note: text overlap with arXiv:1406.361

Mean-field games of optimal stopping: a relaxed solution approach

We consider the mean-field game where each agent determines the optimal time to exit the game by solving an optimal stopping problem with reward function depending on the density of the state processes of agents still present in the game. We place ourselves in the framework of relaxed optimal stopping, which amounts to looking for the optimal occupation measure of the stopper rather than the optimal stopping time. This framework allows us to prove the existence of the relaxed Nash equilibrium and the uniqueness of the associated value of the representative agent under mild assumptions. Further, we prove a rigorous relation between relaxed Nash equilibria and the notion of mixed solutions introduced in earlier works on the subject, and provide a criterion, under which the optimal strategies are pure strategies, that is, behave in a similar way to stopping times. Finally, we present a numerical method for computing the equilibrium in the case of potential games and show its convergence

Mixed generalized Dynkin game and stochastic control in a Markovian framework

We introduce a mixed {\em generalized} Dynkin game/stochastic control with ${\cal E}^f$-expectation in a Markovian framework. We study both the case when the terminal reward function is supposed to be Borelian only and when it is continuous. We first establish a weak dynamic programming principle by using some refined results recently provided in \cite{DQS} and some properties of doubly reflected BSDEs with jumps (DRBSDEs). We then show a stronger dynamic programming principle in the continuous case, which cannot be derived from the weak one. In particular, we have to prove that the value function of the problem is continuous with respect to time $t$, which requires some technical tools of stochastic analysis and some new results on DRBSDEs. We finally study the links between our mixed problem and generalized Hamilton Jacobi Bellman variational inequalities in both cases

Generalized Dynkin Games and Doubly Reflected BSDEs with Jumps

We introduce a generalized Dynkin game problem with non linear conditional expectation ${\cal E}$ induced by a Backward Stochastic Differential Equation (BSDE) with jumps. Let $\xi, \zeta$ be two RCLL adapted processes with $\xi \leq \zeta$. The criterium is given by \begin{equation*} {\cal J}_{\tau, \sigma}= {\cal E}_{0, \tau \wedge \sigma } \left(\xi_{\tau}\textbf{1}_{\{ \tau \leq \sigma\}}+\zeta_{\sigma}\textbf{1}_{\{\sigma<\tau\}}\right) \end{equation*} where $\tau$ and $\sigma$ are stopping times valued in $[0,T]$. Under Mokobodski's condition, we establish the existence of a value function for this game, i.e. $\inf_{\sigma}\sup_{\tau} {\cal J}_{\tau, \sigma} = \sup_{\tau} \inf_{\sigma} {\cal J}_{\tau, \sigma}$. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. When $\xi$ and $\zeta$ are left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping. We then address the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles

A Weak Dynamic Programming Principle for Combined Optimal Stopping and Stochastic Control with Ef\mathcal{E}^f- expectations

We study a combined optimal control/stopping problem under a nonlinear expectation ${\cal E}^f$ induced by a BSDE with jumps, in a Markovian framework. The terminal reward function is only supposed to be Borelian. The value function $u$ associated with this problem is generally irregular. We first establish a {\em sub- (resp. super-) optimality principle of dynamic programming} involving its {\em upper- (resp. lower-) semicontinuous envelope} $u^*$ (resp. $u_*$). This result, called {\em weak} dynamic programming principle (DPP), extends that obtained in \cite{BT} in the case of a classical expectation to the case of an ${\cal E}^f$-expectation and Borelian terminal reward function. Using this {\em weak} DPP, we then prove that $u^*$ (resp. $u_*$) is a {\em viscosity sub- (resp. super-) solution} of a nonlinear Hamilton-Jacobi-Bellman variational inequality