Uncertain Stochastic Optimal Control with Jump and Its Application in a Portfolio Game

This article describes a class of jump-uncertain stochastic control systems, and derives an Ito-Liu formula with jump. We characterize an optimal control law, that satisfies the Hamilton-Jacobi-Bellman equation with jump. Then, this paper deduces the optimal portfolio game under uncertain stochastic financial markets with jump. The information of players is symmetrical. The financial market is constituted of a risk-free asset and a risky asset whose price process is subjected to the jump-uncertain stochastic Black-Scholes model. The game is formulated by two utility maximization problems, each investor tries to maximize his relative utility, which is the weighted average of terminal wealth difference between his terminal wealth and that of his competitor. Finally, the explicit expressions of equilibrium investment strategies and value functions for the constant absolute risk-averse and constant relative risk-averse utility function are derived by using the dynamic programming principle.