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Risk Sensitive Control of the Lifetime Ruin Problem

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Abstract

We study a risk sensitive control version of the lifetime ruin probability problem. We consider a sequence of investments problems in Black–Scholes market that includes a risky asset and a riskless asset. We present a differential game that governs the limit behavior. We solve it explicitly and use it in order to find an asymptotically optimal policy.

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Notes

  1. Although we are not using explicitly large deviation arguments in the paper, for intuition reasons we still choose to define the cost by using the rate function instead of simply using only \(\tfrac{1}{2}\int _0^t\dot{\psi }^2(s)ds.\)

  2. We use the convention that \(\inf \emptyset =\infty .\) Also, hereafter, in case that \(b=\infty \) then by the notation \((x,\,b]\) and \([x,\,b]\) mean \((x,\,\infty )\) and \([x,\,\infty ),\) respectively.

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Acknowledgments

We thank the two anonymous referees, the AE and Huyên Pham for insightful comments, which helped us improve our paper. We are also grateful to Virginia Young for many discussions that we had on the subject. This research is supported in part by the National Science Foundation through the DMS-1613170 Grant.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, 48109, USA

    Erhan Bayraktar & Asaf Cohen

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  1. Erhan Bayraktar
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  2. Asaf Cohen
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Correspondence to Erhan Bayraktar.

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Bayraktar, E., Cohen, A. Risk Sensitive Control of the Lifetime Ruin Problem. Appl Math Optim 77, 229–252 (2018). https://doi.org/10.1007/s00245-016-9372-2

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