On the time a diffusion process spends along a line

TL;DR

This paper analyzes the distribution of the time a diffusion process spends along a line, deriving explicit formulas and revealing new exponential distribution families, with implications for statistical estimation and process theory.

Contribution

It provides a detailed characterization of the local time along a line for diffusion processes, including explicit distribution formulas and new exponential family constructions.

Findings

The local time is either infinite or a mixture of exponential and zero.

Explicit formulas relate distribution parameters to process coefficients.

Introduces new multivariate exponential distribution families.

Abstract

For an arbitrary diffusion process $X$ with time-homogeneous drift and variance parameters $\mu(x)$ and $\sigma^2(x)$, let $V_\varepsilon$ be $1/\varepsilon$ times the total time $X(t)$ spends in the strip $[a+bt-(1/2)\varepsilon,a+bt+(1/2)\varepsilon]$.The limit $V$ as $\varepsilon\rightarrow0$ is the full halfline version of the local time of $X(t)-a-bt$ at zero, and can be thought of as the time $X$ spends along the straight line $x=a+bt$. We prove that $V$ is either infinite with probability 1 or distributed as a mixture of an exponential and a unit point mass at zero, and we give formulae for the parameters of this distribution in terms of $\mu(\cdot)$, $\sigma(\cdot)$, $a$, $b$, and the starting point $X(0)$. The special case ofa Brownian motion is studied in more detail, leading in particular to a full process $V(b)$ with continuous sample paths and exponentially distributed marginals. This construction leads to new families of bivariate and multivariate exponential distributions. Truncated versions of such `total relative time' variables are also studied. A relation is pointed out to a second order asymptotics problem in statistical estimation theory, recently investigated in Hjort and Fenstad (1992a, 1992b).