Dual Stochastic Transformations of Solvable Diffusions

TL;DR

This paper introduces a dual transformation method to construct new families of solvable one-dimensional diffusions with closed-form transition densities, expanding the toolkit for analyzing stochastic processes.

Contribution

It develops a dual application of the diffusion canonical transformation, creating multi-parameter solvable diffusions with flexible drift and diffusion structures.

Findings

New classes of solvable diffusions with explicit transition densities

Complete boundary classification for the new diffusion families

Martingale characterization of the constructed diffusions

Abstract

We present new extensions to a method for constructing several families of solvable one-dimensional time-homogeneous diffusions whose transition densities are obtainable in analytically closed-form. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines smooth monotonic mappings and measure changes via Doob-h transforms. This gives rise to new multi-parameter solvable diffusions that are generally divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The theory is applicable to diffusions with either singular and/or non-singular endpoints. As part of the results in this paper, we also present a complete boundary classification and martingale characterization of the newly developed diffusion families.