Keywords: jump diffusions, marked point processes, minimal entropy measure, utility maximisation, risky asset prices, Hamilton-Jacobi-Bellman equation, HJB equation, optimal trading strategy, financial risk
An HJB approach to exponential utility maximisation for jump processes
This paper deals with the problem of exponential utility maximisation in a model where the risky asset price S is a geometric marked point process whose dynamics depend on another process X, referred to as the stochastic factor. The process X is modelled as a jump diffusion process which may have common jump times with S. The classical dynamic programming approach leads us to characterise the value function as a solution of the Hamilton-Jacobi-Bellman equation. The solution, together with the optimal trading strategy, can be computed under suitable assumptions. Moreover, an explicit representation of the density of the minimal entropy measure (MEMM) and a duality result, which gives a relationship between the utility maximisation problem and the MEMM, are given. This duality result is obtained for a class of strategies greater than those usually considered in literature. A discussion on the pricing of a European claim by the utility indifference approach and its asymptotic variant is performed.