Continuous-time perpetuities and time reversal of diffusions

TL;DR

This paper develops methods to estimate the distribution of continuous-time perpetuities and their underlying factors in ergodic Markov models, combining PDEs and time reversal techniques for efficient simulation.

Contribution

It introduces a PDE-based approach and a time reversal method to characterize the joint distribution, enabling efficient Monte Carlo estimation.

Findings

PDE approach ensures existence of perpetuity density under certain conditions.

Time reversal method allows sampling from the joint distribution via a single reversed process.

Efficient Monte Carlo estimation of perpetuities in ergodic Markov models.

Abstract

We consider the problem of estimating the joint distribution of a continuous-time perpetuity and the underlying factors which govern the cash flow rate, in an ergodic Markov model. Two approaches are used to obtain the distribution. The first identifies a partial differential equation for the conditional cumulative distribution function of the perpetuity given the initial factor value, which under certain conditions ensures the existence of a density for the perpetuity. The second (and more general) approach, identifies the joint law as the stationary distribution of an ergodic multi-dimensional diffusion using techniques of time reversal. This later approach allows for efficient use of Monte-Carlo simulation when estimating the distribution, as the distribution is obtained by sampling a single path of the reversed process.