# Minimizing the Probability of Lifetime Exponential Parisian Ruin

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## Abstract

We find the optimal investment strategy in a Black–Scholes market to minimize the probability of so-called *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 *control* the probability of Parisian ruin.

## Keywords

Exponential Parisian ruin Optimal investment Stochastic control## Mathematics Subject Classification

93E20 91B30 49K10 49L20## Notes

### Acknowledgements

X. Liang thanks the National Natural Science Foundation of China (11701139, 11571189) and the Natural Science Foundation of Hebei Province (A2018202057) for financial support of her research. V.R. Young thanks the Cecil J. and Ethel M. Nesbitt Professorship for financial support of her research.

## References

- 1.Milevsky, M.A., Robinson, C.: Self-annuitization and ruin in retirement, with discussion. N. Am. Actuar. J.
**4**(4), 112–129 (2000)MathSciNetCrossRefGoogle Scholar - 2.Young, V.R.: Optimal investment strategy to minimize the probability of lifetime ruin. N. Am. Actuar. J.
**8**(4), 105–126 (2004)MathSciNetCrossRefGoogle Scholar - 3.Bayraktar, E., Young, V.R.: Minimizing the probability of lifetime ruin under borrowing constraints. Insur. Math. Econ.
**41**(1), 196–221 (2007)MathSciNetCrossRefGoogle Scholar - 4.Bayraktar, E., Zhang, Y.: Minimizing the probability of lifetime ruin under ambiguity aversion. SIAM J. Control Optim.
**53**(1), 58–90 (2015)MathSciNetCrossRefGoogle Scholar - 5.Bayraktar, E., Hu, X., Young, V.R.: Minimizing the probability of lifetime ruin under stochastic volatility. Insur. Math. Econ.
**49**(2), 194–206 (2011)MathSciNetCrossRefGoogle Scholar - 6.Bayraktar, E., Zhang, Y.: Stochastic Perron’s method for the probability of lifetime ruin problem under transaction costs. SIAM J. Control Optim.
**53**(1), 91–113 (2015)MathSciNetCrossRefGoogle Scholar - 7.Liang, X., Young, V.R.: Minimizing the probability of ruin: two riskless assets with transaction costs and proportional reinsurance. Stat. Probab. Lett.
**140**, 167–175 (2018)MathSciNetCrossRefGoogle Scholar - 8.Bayraktar, E., Young, V.R.: Optimal investment strategy to minimize occupation time. Ann. Oper. Res.
**176**(1), 389–408 (2010)MathSciNetCrossRefGoogle Scholar - 9.Chesney, M., Jeanblanc-Picqué, M., Yor, M.: Brownian excursions and Parisian barrier options. Adv. Appl. Probab.
**29**(1), 165–184 (1997)MathSciNetCrossRefGoogle Scholar - 10.Dassios, A., Wu, S.: Parisian ruin with exponential claims. Working paper, London School of Economics (2008)Google Scholar
- 11.Czarna, I., Palmowski, Z.: Ruin probability with Parisian delay for a spectrally negative Lévy risk process. J. Appl. Probab.
**48**(4), 984–1002 (2011)MathSciNetCrossRefGoogle Scholar - 12.Loeffen, R., Czarna, I., Palmowski, Z.: Parisian ruin probability of spectrally negative Lévy processes. Bernoulli
**19**(2), 599–609 (2013)MathSciNetCrossRefGoogle Scholar - 13.Landriault, D., Renaud, J.-F., Zhou, X.: Occupation times of spectrally negative Lévy processes with applications. Stoch. Proc. Appl.
**121**, 2629–2641 (2011)CrossRefGoogle Scholar - 14.Landriault, D., Renaud, J.-F., Zhou, X.: An insurance risk model with Parisian implementation delays. Methodol. Comput. Appl. Probab.
**16**, 583–607 (2014)MathSciNetCrossRefGoogle Scholar - 15.Guérin, H., Renaud, J.-F.: On the distribution of cumulative Parisian ruin. Insur. Math. Econ.
**73**, 116–123 (2017)MathSciNetCrossRefGoogle Scholar - 16.Moore, K.S., Young, V.R.: Optimal and simple, nearly-optimal rules for minimizing the probability of financial ruin in retirement. N. Am. Actuar. J.
**10**(4), 145–161 (2006)MathSciNetCrossRefGoogle Scholar - 17.Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991)zbMATHGoogle Scholar
- 18.Walter, W.: Differential and Integral Inequalities. Springer, New York (1970)CrossRefGoogle Scholar
- 19.Liang, X., Young, V.R.: Minimizing the discounted probability of exponential Parisian ruin via reinsurance. Working paper, Department of Mathematics, University of Michigan (2019)Google Scholar