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

Approximation schemes for mixed optimal stopping and control problems with nonlinear expectations and jumps

We propose a class of numerical schemes for mixed optimal stopping and control of processes with infinite activity jumps and where the objective is evaluated by a nonlinear expectation. Exploiting an approximation by switching systems, piecewise constant policy timestepping reduces the problem to nonlocal semi-linear equations with different control parameters, uncoupled over individual time steps, which we solve by fully implicit monotone approximations to the controlled diffusion and the nonlocal term, and specifically the Lax-Friedrichs scheme for the nonlinearity in the gradient. We establish a comparison principle for the switching system and demonstrate the convergence of the schemes, which subsequently gives a constructive proof for the existence of a solution to the switching system. Numerical experiments are presented for a recursive utility maximization problem to demonstrate the effectiveness of the new schemes

Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems

International audienceWe study the optimal stopping problem for a monotonous dynamic riskmeasure induced by a Backward Stochastic Differential Equation with jumps in theMarkovian case.We show that the value function is a viscosity solution of an obstacleproblem for a partial integro-differential variational inequality and we provide anuniqueness result for this obstacle problem