Simulation of forward-reverse stochastic representations for conditional diffusions

TL;DR

This paper develops stochastic representations for the distributions of conditioned multidimensional diffusions, enabling efficient Monte Carlo estimators that overcome the curse of dimensionality, with applications to stochastic volatility models.

Contribution

It introduces new stochastic representations for conditioned diffusions and analyzes their convergence, extending previous methods to more general conditioning scenarios.

Findings

Monte Carlo estimators achieve root-N accuracy.

Estimators are free from curse of dimensionality.

Numerical example demonstrates application to stochastic volatility models.

Abstract

In this paper we derive stochastic representations for the finite dimensional distributions of a multidimensional diffusion on a fixed time interval, conditioned on the terminal state. The conditioning can be with respect to a fixed point or more generally with respect to some subset. The representations rely on a reverse process connected with the given (forward) diffusion as introduced in Milstein, Schoenmakers and Spokoiny [Bernoulli 10 (2004) 281-312] in the context of a forward-reverse transition density estimator. The corresponding Monte Carlo estimators have essentially root-$N$ accuracy, and hence they do not suffer from the curse of dimensionality. We provide a detailed convergence analysis and give a numerical example involving the realized variance in a stochastic volatility asset model conditioned on a fixed terminal value of the asset.