## Abstract

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks.

Original language | English |
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Article number | 136 |

Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Risks |

Volume | 8 |

Issue number | 4 |

DOIs | |

Publication status | Published - 9 Dec 2020 |

## Keywords

- backward stochastic differential equations
- semilinear elliptic partial differential equations
- stochastic optimal control
- unbounded random terminal time
- machine learning
- deep neural networks