Maximum principle for optimal control of McKean‐Vlasov FBSDEs with Lévy process via the differentiability with respect to probability law

Abstract

Summary In this paper, we study stochastic optimal control problem for general McKean‐Vlasov–type forward‐backward differential equations driven by Teugels martingales, associated with some Lévy process having moments of all orders, and an independent Brownian motion. The coefficients of the system depend on the state of the solution process as well as of its probability law and the control variable. We establish a set of necessary conditions in the form of Pontryagin maximum principle for the optimal control. We also give additional conditions, under which the necessary optimality conditions turn out to be sufficient. The proof of our main result is based on the differentiability with respect to probability law and a corresponding Itô formula.

On partially observed optimal singular control of McKean–Vlasov stochastic systems: Maximum principle approach

“…Optimal control problems for McKean–Vlasov SDEs have been investigated by many authors, for example, Buckdahn et al 26 proved the necessary conditions for general mean‐field systems by applying second order derivatives with respect to measures. Maximum principle for optimal control of McKean–Vlasov FBSDEs with Lévy process via the differentiability with respect to probability law has been proved by Meherrem and Hafayed 27 . Necessary and sufficient optimality conditions of optimal singular control problem for general Mckean–Vlasov differential equations have been discussed by Hafayed et al 28 Maximum principle for stochastic continuous‐singular control of McKean–Vlasov‐type systems, where the control domain is not assumed convex has been proved by Guenane et al 29 Necessary conditions for optimal partially observed control problems of general controlled mean‐field differential systems have been established by Lakhdari et al 30 Necessary conditions for partially observed optimal control of general McKean–Vlasov dynamics with Poisson jumps have been studied in Miloudi et al 31 A necessary condition for mean‐field‐type SDEs with correlated state and observation noises has been obtained in Zhang 32 …”

“…Maximum principle for optimal control of McKean-Vlasov FBSDEs with Lévy process via the differentiability with respect to probability law has been proved by Meherrem and Hafayed. 27 Necessary and sufficient optimality conditions of optimal singular control problem for general Mckean-Vlasov differential equations have been discussed by Hafayed et al 28 Maximum principle for stochastic continuous-singular control of McKean-Vlasov-type systems, where the control domain is not assumed convex has been proved by Guenane et al 29 Necessary conditions for optimal partially observed control problems of general controlled mean-field differential systems have been established by Lakhdari et al 30 Necessary conditions for partially observed optimal control of general McKean-Vlasov dynamics with Poisson jumps have been studied in Miloudi et al 31 A necessary condition for mean-field-type SDEs with correlated state and observation noises has been obtained in Zhang. 32 Our main goal in this paper is to establish a set of necessary conditions in the form of stochastic maximum principle for partially observed optimal singular control problems of McKean-Vlasov type.…”

Extended mean-field control problem with partial observation

“…To the best of our knowledge, the control problems for partially observed forward-backward stochastic differential equations (FBSDEs) of mean-field type is quite a new topic, and only some special cases have been solved. For example, Li and Liu [29] considered an optimal control problem for fully coupled FBSDEs of mean-filed type but without partial observation; Meherrem and Hafayed [32] studied stochastic optimal control problem for general McKean-Vlasov-type FBSDEs driven by Teugels martingales, associated with some Lévy process having moments of all orders, and an independent Brownian motion. Ma and Liu [31] introduced a linear quadratic optimal control problems for partially observed FBSDEs of mean-filed type; Wang, Xiao and Xing [43] investigated an optimal control problem derived by mean-field FBSDE with noisy observation, where the drift coefficients of the state equation and the observation equation are linear with respect to the state and its expectation.…”

McKean-Vlasov multivalued stochastic differential equations with oblique subgradients and related stochastic control problems

“…(see e.g. [2,8,20,24,14,21,22,14,5,6,9]). Motivated by the above articles, we shall study the following Mckean-Vlasov multivalued stochastic differential equations with oblique subgradients (MVMSDEswOS):…”