STOCHASTIC MAXIMUM PRINCIPLE FOR SUBDIFFUSIONS AND ITS APPLICATIONS

In this paper, we study optimal stochastic control problems for stochastic systems driven by non-Markov subdiffusion BLt , which have mixed features of deterministic and stochastic controls. Here Bt is the standard Brownian motion on R, and Lt := inf\{ r > 0 : Sr > t\} , t ≥ 0, is the inverse of a subordinator St with drift κ > 0 that is independent of Bt. 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 BLt , 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.