In this paper, we study the following nonlinear backward stochastic integral partial differential equation with jumps begin{equation*} left{ begin{split} -d V(t,x) =&displaystyleinf_{uin U}bigg{H(t,x,u, DV(t,x),D Phi(t,x), D^2 V(t,x),int_E left(mathcal I V(t,e,x,u)+Psi(t,x+g(t,e,x,u))right)l(t,e) u(de)) &+displaystyleint_{E}big[mathcal I V(t,e,x,u)-displaystyle (g(t, e,x,u), D V(t,x))big] u(d e)+int_{E}big[mathcal I Psi(t,e,x,u)big] u(d e)bigg}dt &-Phi(t,x)dW(t)-displaystyleint_{E} Psi (t, e,x)tildemu(d e,dt), V(T,x)=& h(x), end{split} right. end{equation*} where $tilde mu$ is a Poisson random martingale measure, $W$ is a Brownian motion, and $mathcal I$ is a non-local operator to be specified later. The function $H$ is a given random mapping, which arises from a corresponding non-Markovian optimal control problem. This equation appears as the stochastic Hamilton-Jacobi-Bellman equation, which characterizes the value function of the optimal control problem with a recursive utility cost functional. The solution to the equation is a predictable triplet of random fields $(V,Phi,Psi)$. We show that the value function, under some regularity assumptions, is the solution to the stochastic HJB equation; and a classical solution to this equation is the value function and gives the optimal control. With some additional assumptions on the coefficients, an existence and uniqueness result in the sense of Sobolev space is shown by recasting the backward stochastic partial integral differential equation with jumps as a backward stochastic evolution equation in Hilbert spaces with Poisson jumps.
We examine existence and uniqueness of strong solutions of multi-dimensional mean-field stochastic differential equations with irregular drift coefficients. Furthermore, we establish Malliavin differentiability of the solution and show regularity properties such as Sobolev differentiability in the initial data as well as Holder continuity in time and the initial data. Using the Malliavin and Sobolev differentiability we formulate a Bismut-Elworthy-Li type formula for mean-field stochastic differential equations, i.e. a probabilistic representation of the first order derivative of an expectation functional with respect to the initial condition.
The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations (SEs) in $cR^p$ ($pin [1, infty)$) and backward stochastic differential equations (BSDEs) in $cR^ptimes cH^p$ ($pin (1, infty)$) and in $cR^inftytimes bar{cH^infty}^{BMO}$, with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Feffermans inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear case for SDEs and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse Holder inequality for some suitable exponent $pge 1$. Finally, we establish some relations between Kazamakis quadratic critical exponent $b(M)$ of a BMO martingale $M$ and the spectral radius of the solution operator for the $M$-driven SDE, which lead to a characterization of Kazamakis quadratic critical exponent of BMO martingales being infinite.