On optimal singular control problem for general Mckean‐Vlasov differential equations: Necessary and sufficient optimality conditions

Abstract

Summary In this paper, we derive the necessary and sufficient conditions for optimal singular control for systems governed by general controlled McKean‐Vlasov differential equations, in which the coefficients depend on the state of the solution process as well as of its law and control. The control domain is assumed to be convex. The control variable has 2 components, ie, the first being absolutely continuous and the second being singular. The proof of our result is based on the derivative of the solution process with respect to the probability law and a corresponding Itô formula. Finally, an example is given to illustrate the theoretical results.

On optimal solutions of general continuous‐singular stochastic control problem of McKean‐Vlasov type

“…Optimal control problems for McKean‐Vlasov‐type stochastic differential equations (SDEs) have been studied by many authors; see, for example, previous studies. Peng's type necessary conditions in the form of maximum principle for SDEs of mean‐field type have proved by Buckdahn et al The necessary optimality conditions for SDEs have been established by Wang et al Stochastic optimal control of mean‐field jump‐diffusion systems with delay has been studied by Meng and Shen . The necessary and sufficient conditions for mean‐field SDEs governed by Teugels martingales associated to Lévy process have been studied in previous studies .…”

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

“…The history of mean‐field–type SDEs, also known as McKean‐Vlasov systems, can be traced back to the works by Kac in 1956 and McKean in 1966 on stochastic systems with a large number of interacting particles. Optimal control problems for McKean‐Vlasov–type SDEs have been studied by many authors; see, for example, other works . The partial‐information maximum principle of optimality for SDEs has been established by Wang et al Stochastic optimal control of mean‐field jump‐diffusion systems with delay has been studied by Meng and Shen .…”

“…Controlled McKean‐Vlasov–type FBSDEs have been studied by Carmona and Delarue . Singular optimal control problem for general controlled nonlinear SDEs, in which the coefficients depend on the state of the solution process as well as of its law and control, has been investigated by Hafayed et al…”

“…Optimal control problems for McKean-Vlasov-type SDEs have been studied by many authors; see, for example, other works. [8][9][10][11][12][13][14][15][16][17] The partial-information maximum principle of optimality for SDEs has been established by Wang et al 8 Stochastic optimal control of mean-field jump-diffusion systems with delay has been studied by Meng and Shen. 9 The necessary and sufficient conditions for mean-field SDEs governed by Teugels martingales associated to Lévy process have been studied by Hafayed et al 10 Maximum principle for McKean-Vlasov FBSDEs of mean-field type has been studied by Hafayed et al 11 The mean-field maximum principle for SDEs has been established in Buckdahn et al 13 A general mathematical modeling approach for high-dimensional systems corresponding to a large number of particles has been introduced by Lasry and Lions.…”

“…Controlled McKean-Vlasov-type FBSDEs have been studied by Carmona and Delarue. 16 Singular optimal control problem for general controlled nonlinear SDEs, in which the coefficients depend on the state of the solution process as well as of its law and control, has been investigated by Hafayed et al 17 As it is well known, the theory of BSDEs and FBSDEs is based essentially on the martingale representation theorem. Pardoux and Peng 18 have introduced BSDEs driven by a Brownian motion, by applying Itô's representation theorem for Brownian martingales.…”