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.
Highlights
Classical optimal control problems in insurance mathematics include finding risk measures like the probability of ruin or the expected discounted future dividend payments
In this paper we propose a novel deep neural network algorithm for semilinear elliptic partial differential equation (PDE) associated to infinite time horizon control problems in high dimensions
In this subsection we recall the connection between semilinear elliptic PDEs and backward stochastic differential equations (BSDEs) with random terminal time
Summary
Classical optimal control problems in insurance mathematics include finding risk measures like the probability of ruin or the expected discounted future dividend payments. These are problems over a potentially infinite time horizon, ending at an unbounded random terminal time—the time of ruin of the insurance company. In recent models which take multiple economic factors into account, the problems are high dimensional For computing these risk measures, optimal control problems need to be solved numerically. In this paper we propose a novel deep neural network algorithm for semilinear (degenerate) elliptic PDEs associated to infinite time horizon control problems in high dimensions
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