Risk sharing with deep neural networks

Abstract

We consider the problem of optimally sharing a financial position among agents with potentially different reference risk measures. The problem is equivalent to computing the infimal convolution of the risk metrics and finding the so-called optimal allocations. We propose a neural network-based framework to solve the problem and we prove the convergence of the approximated inf-convolution, as well as the approximated optimal allocations, to the corresponding theoretical values. We support our findings with several numerical experiments.

Keywords:

• Risk sharing
• Deep neural networks
• Risk allocation
• Inf-convolution
• Universal approximation theorem

JEL Classification:

• C45
• C63
• G32

Acknowledgments

The authors thank two anonymous referees for precious comments, and F.-B. Liebrich for addressing them to the reference (Shapiro 2013) and for pointing out the delicate point of the standardness requirements on the underlying probability space.

Disclosure statement

The authors report there are no competing interests to declare.

Notes

1 Increasing is understood in the non-strict sense.

1 As a consequence of translation invariance, we can assume, without loss of generality, that $\mathbb{Q}$ gives positive mass to every neighborhood of 0. Then, given two optimal allocations ${\phi }_1,{\phi }_2\in {\mathcal{A}}_{\text{Lip}}^0$, proposition 2.3 implies that ${\phi }_1-{\phi }_2=c$ $\mathbb{Q}$-a.s., for some $c\in \mathbb{R}$. However, the definition of ${\mathcal{A}}_{\text{Lip}}^0$ necessarily implies c = 0.

3 We thank an anonymous referee for pointing out this fact.

4 These examples were all suggested by an anonymous referee whom we thank.