Wealth Optimization and Dual Problems for Jump Stock Dynamics with Stochastic Factor

An incomplete financial market is considered with a risky asset and a bond. The risky asset price is a pure jump process whose dynamics depend on a jump-diffusion stochastic factor describing the activity of other markets, macroeconomics factors or microstructure rules that drive the market. With a stochastic control approach, maximization of the expected utility of terminal wealth is discussed for utility functions of Constant Relative Risk Aversion (CRRA) type. Under suitable assumptions, closed form solutions for the value functions and for the optimal strategy are provided and Verification results are discussed. Moreover, the solution to the dual problems associated to the utility maximization problems are derived.