Some Distributional Properties of Linear Stochastic Differential Equations

TL;DR

This paper establishes conditions for the existence and properties of density functions and support for the solutions of general linear stochastic differential equations, enhancing understanding of their distributional characteristics.

Contribution

It provides a complete characterization of when the transition distribution of a linear SDE has a density and details its differentiability and support properties.

Findings

Necessary and sufficient conditions for density existence

Characterization of the support of the distribution

Analysis of the differentiability of the density and quantile functions

Abstract

In this paper, we prove a sufficient and necessary condition for the transition probability distribution of a general, time-inhomogeneous linear SDE to possess a density function and study the differentiability of the density function and the transition quantile function of the SDE. Moreover, we completely characterize the support of the marginal distribution of this SDE.