SIG-BSDE for Dynamic Risk Measures

TL;DR

This paper explores dynamic risk measures derived from backward stochastic differential equations, introduces a numerical algorithm with convergence proof, and applies deep learning techniques to complex financial problems.

Contribution

It develops a novel numerical scheme for BSDEs, proves its convergence, and integrates deep learning for solving ambiguous interest rate issues.

Findings

The backward Euler-Maruyama scheme effectively solves BSDEs.

The convergence theorem guarantees the numerical method's reliability.

Deep learning enhances solutions for ambiguous interest rate problems.

Abstract

In this paper, we consider dynamic risk measures induced by backward stochastic differential equations (BSDEs). We discuss different examples that come up in the literature, including the entropic risk measure and the risk measure arising from the ambiguous interest rate problem. We develop a numerical algorithm for solving a BSDE using the backward Euler-Maruyama scheme and the universal approximation theorem for the signature of a path. We prove the convergence theorem and use the algorithm to solve some examples of dynamic risk measures induced by BSDEs. At last a deep learning approach is included for solving the ambiguous interest rate problem as well.