A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics

TL;DR

This paper introduces a deep neural network algorithm designed to solve high-dimensional semilinear elliptic PDEs, which are crucial in insurance mathematics for risk assessment, leveraging the connection to backward stochastic differential equations.

Contribution

The paper presents a novel deep learning approach for high-dimensional elliptic PDEs in insurance mathematics, utilizing the link to backward stochastic differential equations with random terminal time.

Findings

Effective in high dimensions

Applicable to risk measure computations

Bridges PDEs and stochastic processes

Abstract

In insurance mathematics optimal control problems over an infinite time horizon arise when computing risk measures. Their solutions correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In this paper we propose a deep neural network algorithm for solving such partial differential equations in high dimensions. The algorithm is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with random terminal time.