Mean-Field SDE Driven by a Fractional Brownian Motion and Related Stochastic Control Problem

Abstract

Abstract. We study a class of mean-field stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) and a related stochastic control problem. We derive a Pontryagin type maximum principle and the associated adjoint mean-field backward stochastic differential equation driven by a classical Brownian motion, and we prove that under certain assumptions, which generalise the classical ones, the necessary condition for the optimality of an admissible control is also sufficient.

Mean-field anticipated BSDEs driven by fractional Brownian motion and related stochastic control problem

“…In this paper, we assume that X exists and belongs to L 2 (Ω × [0, T ]). For recent works about fractional stochastic differential equation, we refer the reader to Ferrante and Rovira [13], Buckdahn et al [6], Buckdahn and Jing [7], etc. For other examples of stochastic optimal control problems with delay driven by fBm, the reader may consult Agram, Douissi and Hilbert [10].…”

“…Then Maticiuc and Nie [18] obtained some general results of fractional BSDEs through a rigorous approach. Buckdahn and Jing [7] studied fractional mean-field stochastic differential equations (SDEs, for short) with H > 1/2 and a stochastic control problem. Some other recent developments of fractional BSDEs can be found in Bender [1], Borkowska [3], Maticiuc and Nie [18], Wen and Shi [23,24], etc., among theory and applications.…”

Mean-field optimal control problem of SDDES driven by fractional Brownian Motion

“…Stochastic control problems driven by fractional Brownian motion (fBm) were also studied by many authors: see, e.g., [2], [4], [14], [19]. However, compared with the papers on stochastic control problems driven by the classical Brownian motion, little has been done because classical methods to solve control problems cannot be used directly, since the fractional Brownian motion is not a semi-martingale or a Markov process.…”

“…Our work is also inspired by the recent paper of Buckdahn and Jing [4] where the system has a past-dependence feature. In [4] the dynamic of the adjoint process is driven by a standard Brownian motion; here the anticipated BSDE is driven by a fBM.…”

“…A lot of authors studied both the case where the stochastic systems are driven by a classical Brownian motion as well as where there is jumps, see,e.g., [1][2][3][4]. Stochastic control problems driven by fractional Brownian motion (fBm) were also studied by many authors, see,e.g., [5][6][7][8]. However, compared with the papers on stochastic control problems driven by the classical Brownian motion, few has been done because classical methods to solve control problems can not be used direclty, since the fractional Brownian motion is not a semi-martingale and not a Markov process.…”