[Paper Review] Approximation of quasi-stationary distributions for 1-dimensional killed diffusions with unbounded drifts

Denis Villemonais|ArXiv.org|May 22, 2009

Stochastic processes and statistical mechanics24 references3 citations

TL;DR

This paper presents a simulation-based method to approximate quasi-stationary distributions (QSDs) for one-dimensional Itô diffusions with unbounded drifts killed at the origin. By first restricting the process to bounded intervals with bounded drifts and then applying a Fleming-Viot-type interacting particle system, the method enables numerical approximation of the Yaglom limit, validated through simulations on the logistic Feller and Wright-Fisher diffusions.

ABSTRACT

The long time behavior of an absorbed Markov process is well described by the limiting distribution of the process conditioned to not be killed when it is observed. Our aim is to give an approximation's method of this limit, when the process is a 1-dimensional Itô diffusion whose drift is allowed to explode at the boundary. In a first step, we show how to restrict the study to the case of a diffusion with values in a bounded interval and whose drift is bounded. In a second step, we show an approximation method of the limiting conditional distribution of such diffusions, based on a Fleming-Viot type interacting particle system. We end the paper with two numerical applications : to the logistic Feller diffusion and to the Wright-Fisher diffusion with values in $]0,1[$ conditioned to be killed at 0.

Motivation & Objective

• To develop a numerically feasible approximation method for quasi-stationary distributions (QSDs) in one-dimensional diffusions with unbounded drifts killed at the boundary.
• To address the challenge that standard interacting particle systems may not be well-defined when drifts are unbounded.
• To establish conditions under which the QSDs of bounded approximations converge to the true QSD of the original unbounded drift diffusion.
• To provide a practical, simulatable method for computing QSDs in cases where spectral theory methods fail to yield explicit values.

Proposed method

• The method begins by approximating the original unbounded drift diffusion on (0, ∞) with a sequence of diffusions on (ε, 1/ε) with bounded drifts, ensuring well-definedness.
• For each ε > 0, the QSD νε of the bounded diffusion is approximated using a Fleming-Viot-type interacting particle system with N particles.
• Each particle evolves as a diffusion on (ε, 1/ε) until absorption; upon absorption, it jumps to a uniformly random surviving particle's position.
• The empirical measure of the particle system converges in law to νε as N → ∞, and νε converges to the true QSD ν as ε → 0.
• Theoretical convergence is established under conditions (H1) and (H3) on the drift, ensuring tightness and weak convergence of νε.
• Numerical approximations are computed via time-averaging of empirical measures over long time horizons, leveraging the ergodic theorem.

Experimental results

Research questions

• RQ1 Can quasi-stationary distributions be approximated for one-dimensional diffusions with unbounded drifts killed at the origin?
• RQ2 Under what conditions does the sequence of QSDs νε on bounded intervals converge to the true QSD of the unbounded diffusion?
• RQ3 Is a Fleming-Viot-type interacting particle system well-defined and effective for approximating QSDs when drifts are unbounded?
• RQ4 How can the Yaglom limit be numerically estimated in cases where spectral methods fail to yield explicit expressions?
• RQ5 What is the impact of drift parameters on the shape and support of the quasi-stationary distribution?

Key findings

• The QSD of the logistic Feller diffusion with unbounded drift is successfully approximated using the proposed particle system, showing that increasing the quadratic drift coefficient c shifts the QSD support closer to 0.
• For the Wright-Fisher diffusion, the particle system approximation closely matches the known analytical QSD density 2−2x, confirming the method’s accuracy.
• The empirical measure of the N-particle system converges to νε as N → ∞, and νε → ν as ε → 0, establishing the method’s theoretical validity.
• The method remains effective even when the drift q(x) explodes at the boundary, as long as conditions (H1) and (H3) are satisfied.
• Numerical simulations with ε = 0.001 and N = 1000 yield stable and accurate approximations of the limiting QSD, demonstrating practical feasibility.
• The convergence of the empirical measure to the true QSD is validated through time-averaged simulations, confirming the ergodic behavior of the particle system.