Optimal investment and consumption with downside risk constraint in jump-diffusion models

TL;DR

This paper extends existing models of optimal investment with downside risk constraints to jump-diffusion settings, providing explicit strategies under positive jumps and recommending stricter constraints when negative jumps are possible.

Contribution

It introduces a jump-diffusion framework for downside risk constrained investment, deriving explicit solutions for positive jumps and proposing adaptive constraints for negative jumps.

Findings

Explicit optimal strategies for positive jump scenarios.

Stricter risk constraints recommended when negative jumps are likely.

Extension of previous diffusion models to jump-diffusion context.

Abstract

This paper extends the results of the article [C. Kl\"{u}ppelberg and S. M. Pergamenchtchikov. Optimal consumption and investment with bounded downside risk for power utility functions. In Optimality and Risk: {\it Modern Trends in Mathematical Finance. The Kabanov Festschrift}, pages 133-169, 2009] to a jump-diffusion setting. We show that under the assumption that only positive jumps in the asset prices are allowed, the explicit optimal strategy can be found in the subset of admissible strategies satisfying the same risk constraint as in the pure diffusion setting. When negative jumps probably happen, the regulator should be more conservative. In that case, we suggest to impose on the investor's portfolio a stricter constraint which depends on the probability of having negative jumps in the assets during the whole considered horizon.