Optimal investment problems with marked point stock dynamics

Optimal investment problems in an incomplete financial market with pure jump stock dynamics are studied. An investor with Constant Relative Risk Aversion (CRRA) preferences, including the logarithmic utility, wants to maximize her/his expected utility of terminal wealth by investing in a bond and in a risky asset. The risky asset price is modeled as a geometric marked point process, whose dynamics is driven by two independent Poisson processes, describing upwards and downwards jumps. A stochastic control approach allows us to provide optimal investment strategies and closed formulas for the value functions associated to the utility optimization problems. Moreover, the solution to the dual problems associated to the utility maximization problems are derived. The case when intermediate consumption is allowed is also discussed.