We introduce a suitable backward stochastic differential equation (BSDE) to represent the value of an optimal control problem with partial observation for a controlled stochastic equation driven by Brownian motion. Our model is general enough to include cases with latent factors in Mathematical Finance. By a standard reformulation based on the reference probability method, it also includes the classical model where the observation process is affected by a Brownian motion (even in presence of a correlated noise), a case where a BSDE representation of the value was not available so far. This approach based on BSDEs allows for greater generality beyond the Markovian case, in particular our model may include path-dependence in the coefficients (both with respect to the state and the control), and does not require any non-degeneracy condition on the controlled equation. We also discuss the issue of numerical treatment of the proposed BSDE.We use a randomization method, previously adopted only for cases of full observation, and consisting, in a first step, in replacing the control by an exogenous process independent of the driving noise and in formulating an auxiliary ("randomized") control problem where optimization is performed over changes of equivalent probability measures affecting the characteristics of the exogenous process. Our first main result is to prove the equivalence between the original partially observed control problem and the randomized problem. In a second step we prove that the latter can be associated by duality to a BSDE, which then characterizes the value of the original problem as well.

This paper develops systematically the stochastic calculus via regularization in the case of jump processes. In particular one continues the analysis of real-valued càdlàg weak Dirichlet processes with respect to a given filtration. Such a process is the sum of a local martingale and an adapted process A such that [N, A] = 0, for any continuous local martingale N . Given a function u : [0, T ] × R → R, which is of class C 0,1 (or sometimes less), we provide a chain rule type expansion for u(t, X t ) which stands in applications for a chain Itô type rule.

We study the following backward stochastic differential equation on finite time horizon driven by an integer-valued random measure µ on R + ×E, where E is a Lusin space, with compensator ν(dt, dx) = dA t φ t (dx):The generator f satisfies, as usual, a uniform Lipschitz condition with respect to its last two arguments. In the literature, the existence and uniqueness for the above equation in the present general setting has only been established when A is continuous or deterministic. The general case, i.e. A is a right-continuous nondecreasing predictable process, is addressed in this paper. These results are relevant, for example, in the study of control problems related to Piecewise Deterministic Markov Processes (PDMPs). Indeed, when µ is the jump measure of a PDMP, then A is predictable (but not deterministic) and discontinuous, with jumps of size equal to 1.

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in [4], we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton-Jacobi-Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation, as usually done for the separated problem. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear quadratic model.The coefficients b and σ = (σ V σ W ) are deterministic measurable functions from R n × A into R n and R n×(m+d) , and assumed to satisfy the following standing assumptions.(H1) There exists some positive constant C 1 such that for all x, x ′ ∈ R n , a ∈ A,Under (H1), it is shown by standard arguments that there exists a unique strong solution X t,ξ,α to (1.1), which isF-adapted, and satisfiesfor some positive constant C, whereĒ denotes the expectation with respect toP. The aim is to maximize, over all admissible control processes α ∈ A W , the gain functional:T t f (X t,ξ,α s , α s )ds + g(X t,ξ,α T ) ,where f : R n × A → R, and g : R n → R are continuous functions satisfying the quadratic growth condition:

We consider an infinite horizon discounted optimal control problem for piecewise deterministic Markov processes, where a piecewise open-loop control acts continuously on the jump dynamics and on the deterministic flow. For this class of control problems, the value function can in general be characterized as the unique viscosity solution to the corresponding HamiltonJacobi-Bellman equation. We prove that the value function can be represented by means of a backward stochastic differential equation (BSDE) on infinite horizon, driven by a random measure and with a sign constraint on its martingale part, for which we give existence and uniqueness results. This probabilistic representation is known as nonlinear Feynman-Kac formula. Finally we show that the constrained BSDE is related to an auxiliary dominated control problem, whose value function coincides with the value function of the original non-dominated control problem.MSC 2010: 93E20, 60H10, 60J25.