A global maximum principle for optimal control of general mean-field forward-backward stochastic systems with jumps

Abstract

The purpose of this paper is to explore the necessary conditions for optimality of meanfield forward-backward delay control systems. A new estimate is proved, which is a powerful tool to deal with the optimal control problems of mean-field type with delay. Different from the classical situation, in our case the first-order adjoint system is an anticipated mean-field backward stochastic differential equation, and the second-order adjoint system is a system of matrix-valued process, not mean-field type. With the help of two adjoint systems, the second-order expansion of the variation of the state Y is proved, and therewith the Peng's stochastic maximum principle. As an illustrative example, we apply our result to the meanfield game in Finance. Although we just investigate the case of one pointwise delay for convenience, but our method is adequate for analysing the case of pointwise delay.

The Global Maximum Principle for Progressive Optimal Control of Partially Observed Forward-Backward Stochastic Systems with Random Jumps

“…Combining the two aspects above with the partially observed feature, in this paper, we aim to study a global maximum principle for partially observed controlled forward-backward stochastic system with random jumps, where all the coefficients contain the control variable. We note that, very recently, Hao and Meng [6] obtained a global maximum principle for optimal control of the general mean-field forwardbackward stochastic system with jumps. However, the jump term in the SDEP of their paper does not contain the control variable and the cost functional is a special recursive case.…”

“…By introducing a state decomposition and backward separation approach, Wang et al [42] studied an linear-quadratic (LQ for short) stochastic control problem of FBSDEs, where the drift coefficient of the observation equation is linear with respect to state x, and the observation noise is correlated with the state noise. Some recent progress for partially observed problems, has been made in mean-field type controls (see [19], [44], [6]), differential games (see [45], [10], [51], [57], [63]), random jumps (see [20], [37], [52], [53], [54], [62]), time delay (see [16]) and applications in finance (see [56], [58]). The partially observed stochastic control problem is usually associated with the state filtering and estimation technique, and we only list some main references such as [18], [2], [55], [39], [54], [43], [34].…”

Second‐order necessary optimality conditions for discrete‐time stochastic systems

“…Using the classical variational analysis approach, Frankowska, Zhang & Zhang 5,6 derived the second‐order necessary optimality conditions for stochastic optimal controls under nonconvex control domain. Some representative works on the necessary optimality conditions in continuous‐time framework, to name a few, include References 7–17.…”

The Maximum Principle for Discounted Optimal Control of Partially Observed Forward-Backward Stochastic Systems with Jumps on Infinite Horizon

“…Recently, global stochastic maximum principle for recursive utilities and fully coupled forward-backward stochastic differential equations (FBSDEs for short) were obtained in Hu [12] and Hu et al [13], respectively. Hao and Meng [10] proved a maximum principle of optimal control problem for a class of general mean-field forward-backward stochastic systems with jumps where the diffusion coefficients depend on control, but the coefficients of jump terms are independent of control. Song et al [32] obtained a maximum principle for progressive optimal control of SDEPs by introducing a new method of variation which is a correction of [33].…”