This paper is devoted to real valued backward stochastic differential equations (BSDEs for short) with generators which satisfy a stochastic Lipschitz condition involving BMO martingales. This framework arises naturally when looking at the BSDE satisfied by the gradient of the solution to a BSDE with quadratic growth in Z. We first prove an existence and uniqueness result from which we deduce the differentiability with respect to parameters of solutions to quadratic BSDEs. Finally, we apply these results to prove the existence and uniqueness of a mild solution to a parabolic partial differential equation in Hilbert space with nonlinearity having quadratic growth in the gradient of the solution

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.

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. and the conditional probability that the process is in A immediately after a jump at time T 1 = s issee below for precise statements. We denote by F the natural filtration of the process X. Denoting by T n the jump times of X, we consider the marked point process (T n , X Tn ) and the associated random measure p(dt dy) = n δ (Tn,XT n ) on (0, ∞) × K, where δ denotes the Dirac measure. In the markovian case the dual predictable projectionp of p (shortly, the compensator) has the following explicit expressioñ p(dt dy) = ν(t, X t− , dy) dt.In the first part of the paper we introduce a class of BSDEs driven by the compensated random measure q(dt dy) := p(dt dy) −p(dt dy) and having the following formfor given generator f and terminal condition g. Here Y is real-valued, while Z is indexed by y ∈ K, i.e. it is a random field on K, with appropriate measurability conditions, and the generator depends on Z(1.4) By the results above there exists a unique solution (Y t,x s , Z t,x s ) s∈[t,T ] and previous estimates on the BSDEs are used to prove well-posedness of (1.3). As a by-product we also obtain the representation formulae, which are sometimes called, at least in the diffusive case, non linear Feynman-Kac formulae.The second application, that we present in section 5 is an optimal control problem. This is formulated in a classical way by means of a change of probability measure, see e.g. [11], [12], [4]. For every fixed (t, x) ∈ [0, T ] × K, we define a class A t of admissible control processes u, and the cost to be minimized and the corresponding value function are J(t, x, u(·)) = E t,x u T t l(s, X s , u s ) ds + g(X T ) , v(t, x) = inf u(·)∈A t J(t, x, u(·)),

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.

In this paper, we study a class of stochastic optimal control problems, where the drift term of the equation has a linear growth on the control variable, the cost functional has a quadratic growth, and the control process belongs to the class of square integrable, adapted processes with no bound assumed on it.