Optimal control for uncertain stochastic dynamic systems with jump and application to an advertising model

Abstract Randomness is an objective indeterminacy, while uncertainty is a subjective indeterminacy. As an effective methodology, chance theory is applicable for disposing of indeterminacy composing of both uncertainty and randomness. Based on chance theory, the optimal control for uncertain stochastic dynamic systems described by both a stochastic differential equation driven by the standard Wiener process and an uncertain differential equation driven by the Liu process and V − n jumps process is considered. Then the principle of optimality is presented by drawing on the dynamic programming method. Particularly, the equation of optimality is established to solve the proposed problem. Furthermore, the optimal control problems with linear and quadratic objective functions are discussed by using the obtained equation. As an application, an advertising problem is analyzed, the corresponding optimal pricing policies and advertising strategies are provided.