Optimal variational principle for backward stochastic control systems associated with Lévy processes

Abstract

The paper is concerned with optimal control of backward stochastic differential equation (BSDE) driven by Teugel's martingales and an independent multidimensional Brownian motion, where Teugel's martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see Nualart and Schoutens [14]). We derive the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. As an application, the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria (called backward linear-quadratic problem, or BLQ problem for short) is discussed and characterized by stochastic Hamilton system.

Linear Quadratic Stochastic Optimal Control of Forward Backward Stochastic Control System Associated with Lévy Process

“…They got the existence and uniqueness theory of this GBSDE when the coefficient verifies some conditions of Lipschitz. More results about BSDE associated with Teugels martingale can be found in the theses of El Otmani [19], Ren and Fan [20], Tang and Zhang [21], and Huang and Wang [22]. On the basis of these results, in 2008, Mitsui and Tabata [23] studied a LQ regulation stochastic control problem with Lévy process and obtained the optimal control for the nonhomogeneous case.…”

On Mean-Field Partial Information Maximum Principle of Optimal Control for Stochastic Systems with Lévy Processes

“…To the best of our knowledge, the stochastic optimal control problems related to Teugels martingales have been investigated by many authors; see, for example, [1][2][3][4][5][6]. The general maximum principle for stochastic differential equations (SDEs) has been studied in [1].…”

“…The stochastic linear quadratic problem with Lévy processes has been derived in [3,4]. The optimality conditions for BSDEs and forward-backward stochastic differential equations (FBSDEs), driven by Teugels martingales, have been investigated in [5]. A new stochastic distribution control method for nonlinear SDEs with constraints has been proved in [7].…”

On partial-information optimal singular control problem for mean-field stochastic differential equations driven by Teugels martingales measures

“…These appear in various fields like mathematical finance, problem of optimal consumption, etc. A number of results have been obtained for these types of problems, see Meng & Tang (19); Meng, Zhang & Tang (20); Mitsui & Tabata (24); Tang & Zhang (28); Tang & Wu (29), and references therein. Under partial-information, the necessary and sufficient optimality conditions for stochastic differential equations (SDEs), driven by Teugels martingales and an independent multi-dimensional Brownian motion have been proved by using convex perturbation, see Meng & Tang (19).…”

“…Under partial-information, the necessary and sufficient optimality conditions for stochastic differential equations (SDEs), driven by Teugels martingales and an independent multi-dimensional Brownian motion have been proved by using convex perturbation, see Meng & Tang (19). Partial-information optimal control problems for backward stochastic differential equations (BSDEs), and for forward-backward stochastic differential equations (FBSDEs) associated with Lévy processes have been investigated in Meng,Zhang & Tang (20); Tang & Zhang (28). The stochastic linear-quadratic problems with Lévy processes have been studied by Mitsui & Tabata (24) and Tang & Wu (29).…”