Backward stochastic differential equations associated to jump Markov processes and applications

In this paper we study backward stochastic differential equations (BSDEs) driven by the compensated random measure associated to a given pure jump Markov process X on a general state space K. We apply these results to prove well-posedness of a class of nonlinear parabolic differential equations on K, that generalize the Kolmogorov equation of X. Finally we formulate and solve optimal control problems for Markov jump processes, relating the value function and the optimal control law to an appropriate BSDE that also allows to construct probabilistically the unique solution to the Hamilton-Jacobi-Bellman equation and to identify it with the value function

Backward stochastic differential equations and optimal control of marked point processes

We study a class of backward stochastic differential equations (BSDEs) driven by a random measure or, equivalently, by a marked point process. Under appropriate assumptions we prove well-posedness and continuous dependence of the solution on the data. We next address optimal control problems for point processes of general non-markovian type and show that BSDEs can be used to prove existence of an optimal control and to represent the value function. Finally we introduce a Hamilton-Jacobi-Bellman equation, also stochastic and of backward type, for this class of control problems: when the state space is finite or countable we show that it admits a unique solution which identifies the (random) value function and can be represented by means of the BSDEs introduced above

Dual and backward SDE representation for optimal control of non-Markovian SDEs

We study optimal stochastic control problem for non-Markovian stochastic differential equations (SDEs) where the drift, diffusion coefficients, and gain functionals are path-dependent, and importantly we do not make any ellipticity assumption on the SDE. We develop a controls randomization approach, and prove that the value function can be reformulated under a family of dominated measures on an enlarged filtered probability space. This value function is then characterized by a backward SDE with nonpositive jumps under a single probability measure, which can be viewed as a path-dependent version of the Hamilton-Jacobi-Bellman equation, and an extension to $G$ expectation

Stochastic maximum principle for optimal control of a class of nonlinear SPDEs with dissipative drift

We prove a version of the stochastic maximum principle, in the sense of Pontryagin, for the finite horizon optimal control of a stochastic partial differential equation driven by an infinite dimensional additive noise. In particular we treat the case in which the non-linear term is of Nemytskii type, dissipative and with polynomial growth. The performance functional to be optimized is fairly general and may depend on point evaluation of the controlled equation. The results can be applied to a large class of non-linear parabolic equations such as reaction-diffusion equations

Backward stochastic differential equation driven by a marked point process: An elementary approach with an application to optimal control

We address a class of backward stochastic differential equations on a bounded interval, where the driving noise is a marked, or multivariate, point process. Assuming that the jump times are totally inaccessible and a technical condition holds (see Assumption (A) below), we prove existence and uniqueness results under Lipschitz conditions on the coefficients. Some counter-examples show that our assumptions are indeed needed. We use a novel approach that allows reduction to a (finite or infinite) system of deterministic differential equations, thus avoiding the use of martingale representation theorems and allowing potential use of standard numerical methods. Finally, we apply the main results to solve an optimal control problem for a marked point process, formulated in a classical way.Comment: Published at http://dx.doi.org/10.1214/15-AAP1132 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org