Time Homogeneous Diffusions with a Given Marginal at a Deterministic Time

TL;DR

This paper proves the existence of a diffusion martingale with a prescribed marginal distribution at a fixed deterministic time, extending classical results to continuous and bounded state spaces.

Contribution

It establishes the existence of diffusion martingales with a given law at a fixed time, using fixed point theorems and discretization methods, without addressing uniqueness.

Findings

Existence of diffusion martingale with specified law at fixed time

Construction via discretization and fixed point theorem

Extension to bounded measurable state spaces

Abstract

In this article, it is proved that for any cumulative distribution function with compact support and a specified t > 0, there exists a diffusion martingale which has this law at time t. The article proves existence; no claims are made about uniqueness. After a discussion on strings and associated semigroups, the article gives a re-working of a standard approach to the problem of constructing an explicit discrete time martingale diffusion on a finite state space which, for a random geometrically distributed time that is independent of the diffusion, the law of the diffusion stopped at this random time has the prescribed law. This argument is developed, using a fixed point theorem, to determine conditions under which there is a discrete time martingale diffusion that has a prescribed law at an independent random time with negative binomial distribution. The step length for the time discretisation is then reduced and in the limit it is shown that for a finite state space, there exists a continuous time martingale diffusion such that $X_\tau$ has law $\mu$, where $\tau$ has a Gamma distribution. For fixed $t = E[\tau]$, the parameters of the Gamma distribution may be altered, reducing the coefficient of variation of $\tau$ to zero, to show that there is a martingale diffusion $X$ such that $X_t$ has law $\mu$. The argument is then extended to obtain the result for any state space that is a bounded measurable subset of $R$.