Stochastic Maximum Principle for Subdiffusions and Its Applications

SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 953-981, April 2024.
Abstract. In this paper, we study optimal stochastic control problems for stochastic systems driven by non-Markov subdiffusion [math], which have mixed features of deterministic and stochastic controls. Here [math] is the standard Brownian motion on [math], and [math] is the inverse of a subordinator [math] with drift [math] that is independent of [math]. We obtain stochastic maximum principles (SMPs) for these systems using both convex and spiking variational methods, depending on whether or not the domain is convex. To derive SMPs, we first establish a martingale representation theorem for subdiffusions [math], and then use it to derive the existence and uniqueness result for the solutions of backward stochastic differential equations (BSDEs) driven by subdiffusions, which may be of independent interest. We also derive sufficient SMPs. Application to a linear quadratic system is given to illustrate the main results of this paper.