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# A Note on Applications of Stochastic Ordering to Control Problems in Insurance and Finance

## Author

Listed:
• Nicole Bauerle
• Erhan Bayraktar

## Abstract

We consider a controlled diffusion process $(X_t)_{t\ge 0}$ where the controller is allowed to choose the drift $\mu_t$ and the volatility $\sigma_t$ from a set $\K(x) \subset \R\times (0,\infty)$ when $X_t=x$. By choosing the largest $\frac{\mu}{\sigma^2}$ at every point in time an extremal process is constructed which is under suitable time changes stochastically larger than any other admissible process. This observation immediately leads to a very simple solution of problems where ruin or hitting probabilities have to be minimized. Under further conditions this extremal process also minimizes "drawdown" probabilities.

## Suggested Citation

• Nicole Bauerle & Erhan Bayraktar, 2012. "A Note on Applications of Stochastic Ordering to Control Problems in Insurance and Finance," Papers 1210.3800, arXiv.org, revised Jul 2013.
• Handle: RePEc:arx:papers:1210.3800
as

File URL: http://arxiv.org/pdf/1210.3800

## References listed on IDEAS

as
1. Romuald Elie & Nizar Touzi, 2008. "Optimal lifetime consumption and investment under a drawdown constraint," Finance and Stochastics, Springer, vol. 12(3), pages 299-330, July.
Full references (including those not matched with items on IDEAS)

## Citations

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Cited by:

1. Erhan Bayraktar & Yuchong Zhang, 2014. "Minimizing the Probability of Lifetime Ruin Under Ambiguity Aversion," Papers 1402.1809, arXiv.org, revised Nov 2014.
2. Nicole Bäuerle & Ulrich Rieder, 2013. "Optimal Deterministic Investment Strategies for Insurers," Risks, MDPI, Open Access Journal, vol. 1(3), pages 1-18, November.
3. Bahman Angoshtari & Erhan Bayraktar & Virginia R. Young, 2015. "Optimal Investment to Minimize the Probability of Drawdown," Papers 1506.00166, arXiv.org, revised Feb 2016.

### NEP fields

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