Optimal Control Problems Of Some Forward-Backward Stochastic Pantograph Systems

As we all know, the stochastic optimal control problem is one of the basic problems in the field of stochastic control, and plays a very important role in modern control theory. The stochastic maximum principle is an important method for solving stochastic optimal control problem, and it is also one of core contents of stochastic optimal control. In this thesis, we are dedicated to study stochastic optimal problems of some forward-backward stochastic pantograph systems which arise from finance and other areas. Using the duality theory, we establish the stochastic Pontryagin’s maximum principle of the optimal control for these problems and deduce the necessary and sufficient conditions for these optimal control problems.In Chapter 2, we consider the stochastic optimal control problem of the fol-lowing stochastic delayed system described by the anticipated forward-backward stochastic pantograph equations: The objective function isFirst, we establish the existence and uniqueness of the solutions for the anticipated backward stochastic pantograph equations. Next, we study the convergence of the solutions for the anticipated forward-backward stochastic pantograph equations. Then, we make an investigation into the stochastic optimal control problem of this kind of forward-backward stochastic panto-graph system in which the domain of the control is convex with the aid of classical variational approach, duality method and the anticipated anticipated backward stochastic pantograph equations, establishing the stochastic Pon-tryagin’s maximum principle of for this optimal control problem, deducing the necessary conditions for the stochastic optimal control. In the end, we de-duce the sufficient conditions for the stochastic optimal control under some additional concavity assumptions.In Chapter 3, we consider stochastic optimal control problem of the follow-ing stochastic delayed system described by the anticipated forward-backward stochastic pantograph equations with random jumps: The objective function isFirst, we establish the existence and uniqueness of the solutions for the anticipated backward stochastic pantograph equations with random jump-s. Next, we study the convergence of the solutions for the the anticipated forward-backward stochastic pantograph equations with random jumps. Then, we make an investigation into the stochastic optimal control problem of this kind of forward-backward stochastic pantograph system with random jumps in which the domain of the control is convex with the aid of classical variation-al approach, duality method and the anticipated backward stochastic panto-graph equations with random jumps, establishing the stochastic Pontryagin’s maximum principle of for this optimal control problem, deducing the neces-sary conditions for the stochastic optimal control. In the end, we deduce the sufficient conditions for the stochastic optimal control under some additional convexity assumptions.In Chapter 4, we consider stochastic optimal control problem of the follow-ing stochastic delayed system described by the anticipated forward-backward stochastic pantograph equations with Markov chains:The objective function isFirst, we establish the existence and uniqueness of the solutions for the anticipated backward stochastic pantograph equations with Markov chain-s. Next, we study the convergence of the solutions for the the anticipated forward-backward stochastic pantograph equations with Markov chains. Then, we make an investigation into the stochastic optimal control problem of this kind of forward-backward stochastic pantograph system with random jumps in which the domain of the control is convex with the aid of classical vari-ational approach, duality method and the anticipated backward stochastic pantograph equations with regime switching, establishing the stochastic Pon-tryagin’s maximum principle of for this optimal control problem, deducing the necessary conditions for the stochastic optimal control. In the end, we de-duce the sufficient conditions for the stochastic optimal control under some additional concavity assumptions.