Transient one-dimensional diffusions conditioned to converge to a different limit point

TL;DR

This paper studies how one-dimensional diffusions modeling populations that tend to extinction can be conditioned to instead tend to infinity, resulting in a transformed process characterized by the scale function.

Contribution

It introduces a method to condition transient diffusions to change their asymptotic behavior, specifically transforming processes that go to extinction into those that tend to infinity, via Doob $h$-transforms.

Findings

The conditioned process is the Doob $h$-transform with $h=s$, the scale function.

Conditioning alters the asymptotic behavior of the diffusion.

Examples illustrate the effect of conditioning on specific diffusions.

Abstract

Let $(X_t)_{t\geq 0}$ be a regular one-dimensional diffusion that models a biological population. If one assumes that the population goes extinct in finite time it is natural to study the $Q$-process associated to $(X_t)_{t\geq 0}$. This is the process one gets by conditioning $(X_t)_{t\geq 0}$ to survive into the indefinite future. The motivation for this paper comes from looking at populations that are modeled by diffusions which do not go extinct in finite time but which go `extinct asymptotically' as $t\rightarrow \infty$. We look at transient one-dimensional diffusions $(X_t)_{t \geq 0}$ with state space $I=(\ell, \infty)$ such that $X_t\rightarrow \ell$ as $t\rightarrow \infty$, $\mathbb{P}^x$-almost surely for all $x\in I$. We `condition' $(X_t)_{t \geq 0}$ to go to $\infty$ as $t\rightarrow \infty$ and show that the resulting diffusion is the Doob $h$-transform of $(X_t)_{t\geq 0}$ with $h=s$ where $s$ is the scale function of $(X_t)_{t\geq 0}$. Finally, we explore what this conditioning does in two examples.