Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon

TL;DR

This paper develops methods to construct diffusion processes conditioned on survival or absorption probabilities over finite or infinite horizons, with explicit applications to Brownian motion with drift and links to large deviations and control theory.

Contribution

It introduces a unified framework for conditioned diffusion processes on a half-line with absorbing boundaries, including explicit solutions for Brownian motion with drift.

Findings

Constructed conditioned processes for finite and infinite horizons.

Derived explicit conditioned trajectories for Brownian motion with drift.

Linked the framework to large deviations and stochastic control theory.

Abstract

When the unconditioned process is a diffusion living on the half-line $x \in ]-\infty,a[$ in the presence of an absorbing boundary condition at position $x=a$, we construct various conditioned processes corresponding to finite or infinite horizon. When the time horizon is finite $T<+\infty$, the conditioning consists in imposing the probability $P^*(y,T ) $ to be surviving at time $T$ and at the position $y \in ]-\infty,a[$, as well as the probability $\gamma^*(T_a ) $ to have been absorbed at the previous time $T_a \in [0,T]$. When the time horizon is infinite $T=+\infty$, the conditioning consists in imposing the probability $\gamma^*(T_a ) $ to have been absorbed at the time $T_a \in [0,+\infty[$, whose normalization $[1- S^*(\infty )]$ determines the conditioned probability $S^*(\infty ) \in [0,1]$ of forever-survival. This case of infinite horizon $T=+\infty$ can be thus reformulated as the conditioning of diffusion processes with respect to their first-passage-time properties at position $a$. This general framework is applied to the explicit case where the unconditioned process is the Brownian motion with uniform drift $\mu$ in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, we describe the links with the dynamical large deviations at Level 2.5 and the stochastic control theory.