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

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.