Minimizing the Probability of Lifetime Exponential Parisian Ruin

TL;DR

This paper derives optimal investment strategies in a Black-Scholes market to minimize the probability of lifetime exponential Parisian ruin, revealing the impact of negative wealth and hazard rates on investment behavior.

Contribution

It introduces the first control framework for minimizing the probability of Parisian ruin, including asymptotic analysis for small hazard rates.

Findings

Leveraging risky assets is worse when wealth is negative for Parisian ruin.

Optimal investment in risky assets increases with hazard rate when wealth is negative.

The minimum probability is proportional to the hazard rate for small hazard rates.

Abstract

We find the optimal investment strategy in a Black-Scholes market to minimize the probability of so-called {\it lifetime exponential Parisian ruin}, that is, the probability that wealth exhibits an excursion below zero of an exponentially distributed time before the individual dies. We find that leveraging the risky asset is worse for negative wealth when minimizing the probability of lifetime exponential Parisian ruin than when minimizing the probability of lifetime ruin. Moreover, when wealth is negative, the optimal amount invested in the risky asset increases as the hazard rate of the exponential ``excursion clock'' increases. In view of the heavy leveraging when wealth is negative, we also compute the minimum probability of lifetime exponential Parisian ruin under a constraint on investment. Finally, we derive an asymptotic expansion of the minimum probability of lifetime exponential Parisian ruin for small values of the hazard rate of the excursion clock. It is interesting to find that, for small values of this hazard rate, the minimum probability of lifetime exponential Parisian ruin is proportional to the minimum occupation time studied in Bayraktar and Young, and the proportion equals the hazard rate. To the best of our knowledge, our work is the first to {\it control} the probability of Parisian ruin.