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We explore martingale and convex duality techniques to study optimalinvestment strategies that maximize expected risk-averse utility fromconsumption and terminal wealth. We consider a market model with jumps drivenby (multivariate) marked point processes and so-called non-linear wealthdynamics which allows to take account of relaxed assumptions such asdifferential borrowing and lending interest rates or short positions with cashcollateral and negative rebate rates. We give suffcient conditions forexistence of optimal policies for agents with logarithmic and CRRA powerutility. We find closed-form solutions for the optimal value function in thecase of pure-jump models with jump-size distributions modulated by a two-stateMarkov chain.

Utility maximization in pure-jump models driven by marked point processes and nonlinear wealth dynamics