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Register a new userThe Global Maximum Principle for Progressive Optimal Control of Partially Observed Forward-Backward Stochastic Systems with Random Jumps
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In this paper, we study a partially observed progressive optimal control problem of forward-backward stochastic differential equations with random jumps, where the control domain is not necessarily convex, and the control variable enter into all the coefficients. In our model, the observation equation is not only driven by a Brownian motion but also a Poisson random measure, which also have correlated noises with the state equation. For preparation, we first derive the existence and uniqueness of the solutions to the fully coupled forward-backward stochastic system with random jumps in $L^2$-space and the decoupled forward-backward stochastic system with random jumps in $L^beta(beta>2)$-space, respectively, then we obtain the $L^beta(betageq2)$-estimation of solutions to the fully coupled forward-backward stochastic system, and the non-linear filtering equation of partially observed stochastic system with random jumps. Then we derive the partially observed global maximum principle with random jumps with a new hierarchical method. To show its applications, a partially observed linear quadratic progressive optimal control problem with random jumps is investigated, by the maximum principle and stochastic filtering. State estimate feedback representation of the optimal control is given in a more explicit form by introducing some ordinary differential equations.
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In this paper, the optimal control problem of neutral stochastic functional differential equation (NSFDE) is discussed. A class of so-called neutral backward stochastic functional equations of Volterra type (VNBSFEs) are introduced as the adjoint equation. The existence and uniqueness of VNBSFE is established. The Pontryagin maximum principle is constructed for controlled NSFDE with Lagrange type cost functional.
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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.
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