Probabilistic representation of HJB equations for optimal control of jump processes, EDSR (stochastic backward differential equations) and stochastic calculus.

Summary In this paper, three different topics related to stochastic control and computation are discussed, based on the notion of a random measure directed stochastic backward differential equation (SRDE). The first three chapters of the thesis deal with optimal control problems for different classes of non-diffusive Markovian processes, with finite or infinite horizon. In each case, the value function, which is the unique solution of a Hamilton-Jacobi-Bellman (HJB) integro-differential equation, is represented as the unique solution of an appropriate RLS. In the first chapter, we control a class of finite-horizon semi-Markovian processes. The second chapter is devoted to the optimal control of pure jump Markovian processes, while in the third chapter we consider the case of infinite-horizon piecewise deterministic Markovian processes (PDMPs). In the second and third chapters the HJB equations associated with optimal control are completely nonlinear. This situation arises when the laws of the controlled processes are not absolutely continuous with respect to the law of a given process. Given this completely nonlinear character, these equations cannot be represented by classical EDSRs. In this framework, we obtained nonlinear Feynman-Kac formulas by generalizing the control randomization method introduced by Kharroubi and Pham (2015) for diffusions. These techniques allow us to relate the value function of the control problem to a random measure directed SDE, one component of whose solution is sign constrained. Furthermore, we show that the value function of the original non-dominated control problem coincides with the value function of an auxiliary dominated control problem, expressed in terms of changes in equivalent probability measures. In the fourth chapter, we study a finite horizon backward stochastic differential equation directed by an integer-valued random measure on $R_+ times E$, where $E$ is a Lusinian space, with compensator of the form $nu(dt, dx) = dA_t phi_t(dx)$. The generator of this equation satisfies a uniform Lipschitz condition with respect to the unknowns. In the literature, the existence and uniqueness for EDSRs in this framework have been established only when $A$ is continuous or deterministic. We provide an existence and uniqueness theorem even when $A$ is a predictable, nondecreasing, right-hand continuous process. This result applies, for example, to the case of control related to PDMPs. Indeed, when $mu$ is the jump measure of a PDMP on a bounded domain, $A$ is predictable and discontinuous. Finally, in the last two chapters of the thesis we deal with stochastic computation for general discontinuous processes. In the fifth chapter, we develop the stochastic calculus via regularizations of jump processes which are not necessarily semimartingales. In particular we continue the study of so-called weak Dirichlet processes in the discontinuous framework. Such a process $X$ is the sum of a local martingale and an adapted process $A$ such that $[N, A] = 0$, for any continuous local martingale $N$. For a function $u: [0, T] times R rightarrow R$ of class $C^{0,1}$ (or sometimes less), we express a development of $u(t, X_t)$, in the spirit of a generalization of the Itô lemma, which holds when $u$ is of class $C^{1,2}$. The computation is applied in the sixth chapter to the theory of EDSRs directed by random measures. In many situations, when the underlying process $X$ is a special semimartingale, or more generally, a special weak Dirichlet process, we identify the solutions of the considered EDSRs via the process $X$ and the solution $u$ of an associated integro-differential PDE.