Abstract
This paper deals with the general optimal control problem for fully coupled forward-backward stochastic differential equations with random jumps (FBSDEJs). The control domain is not assumed to be convex, and the control variable appears in both diffusion and jump coefficients of the forward equation. Necessary conditions of Pontraygin's type for the optimal controls are derived by means of spike variation technique and Ekeland variational principle. A linear quadratic stochastic optimal control problem is discussed as an illustrating example.
Highlights
Basic NotationsThroughout this paper, we denote by Rn the space of n-dimensional Euclidean space, by Rn×d the space of n × d matrices, and by Sn the space of n × n symmetric matrices. ·, · and | · | denote the scalar product and norm in the Euclidean space, respectively. appearing in the superscripts denotes the transpose of a matrix
Let Ω, F, {Ft}t≥0, P be a complete filtered probability space satisfying the usual conditions, where the filtration {Ft}t≥0 is generated by the following two mutually independent processes: i a one-dimensional standard Brownian motion {W t }t≥0; ii a Poisson random measure N on E × R, where E ⊂ R is a nonempty open set equipped with its Borel field B E, with compensator N dedt π de dt, such that N A × 0, t N − N A × 0, t t≥0 is a martingale for all A ∈ B E satisfying
For any u · ∈ U 0, T and a ∈ Rn, we refer to Θ · ≡ x ·, y ·, z ·, c ·, · as the state process corresponding to the admissible control u · if FBSDEJ 1.4 admits a unique adapted solution
Summary
Throughout this paper, we denote by Rn the space of n-dimensional Euclidean space, by Rn×d the space of n × d matrices, and by Sn the space of n × n symmetric matrices. ·, · and | · | denote the scalar product and norm in the Euclidean space, respectively. appearing in the superscripts denotes the transpose of a matrix. ·, · and | · | denote the scalar product and norm in the Euclidean space, respectively. Let Ω, F, {Ft}t≥0, P be a complete filtered probability space satisfying the usual conditions, where the filtration {Ft}t≥0 is generated by the following two mutually independent processes:. Let U 0, T be the set of all Ft-predictable processes u : 0, T × Ω → U such that sup0≤t≤T E|u t |i < ∞, for all i 1, 2,. We denote by L2 E, B E , π; Rm or L2 the set of integrable functions c : E → Rm with norm ce. Any process in M2 0, T is denoted by Θ · ≡ x · , y · , z · , c ·, · , whose norm is defined by.
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