  We explore martingale and convex duality techniques to study optimal
investment strategies that maximize expected risk-averse utility from
consumption and terminal wealth. We consider a market model with jumps driven
by (multivariate) marked point processes and so-called non-linear wealth
dynamics which allows to take account of relaxed assumptions such as
differential borrowing and lending interest rates or short positions with cash
collateral and negative rebate rates. We give suffcient conditions for
existence of optimal policies for agents with logarithmic and CRRA power
utility. We find closed-form solutions for the optimal value function in the
case of pure-jump models with jump-size distributions modulated by a two-state
Markov chain.
