Importance sampling for a simple Markovian intensity model using subsolutions

TL;DR

This paper develops efficient importance sampling algorithms for rare-event probability estimation in Markovian jump processes, particularly in credit risk modeling, using subsolutions to handle complex state-dependent Hamilton-Jacobi equations.

Contribution

It introduces a novel approach using subsolutions to design asymptotically optimal importance sampling algorithms for complex Markovian jump processes.

Findings

Algorithms outperform standard Monte Carlo in numerical tests.

Theoretical performance bounds are established.

Asymptotic optimality is demonstrated for specific examples.

Abstract

This paper considers importance sampling for estimation of rare-event probabilities in a specific collection of Markovian jump processes used for e.g. modelling of credit risk. Previous attempts at designing importance sampling algorithms have resulted in poor performance and the main contribution of the paper is the design of efficient importance sampling algorithms using subsolutions. The dynamics of the jump processes causes the corresponding Hamilton-Jacobi equations to have an intricate state-dependence, which makes the design of efficient algorithms difficult. We provide theoretical results that quantify the performance of importance sampling algorithms in general and construct asymptotically optimal algorithms for some examples. The computational gain compared to standard Monte Carlo is illustrated by numerical examples.