Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon
Cécile Monthus, Alain Mazzolo
Abstract
When the unconditioned process is a diffusion living on the half-line $x\ensuremath{\in}]\ensuremath{-}\ensuremath{\infty},a[$ 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 $T<+\ensuremath{\infty}$, the conditioning consists in imposing the probability distribution ${P}^{*}(y,T)$ to be surviving at time $T$ at the position $y\ensuremath{\in}]\ensuremath{-}\ensuremath{\infty},a[$, as well as the probability distribution ${\ensuremath{\gamma}}^{*}({T}_{a})$ of the absorption time ${T}_{a}\ensuremath{\in}[0,T]$. When the time horizon is infinite $T=+\ensuremath{\infty}$, the conditioning consists in imposing the probability distribution ${\ensuremath{\gamma}}^{*}({T}_{a})$ of the absorption time ${T}_{a}\ensuremath{\in}[0,+\ensuremath{\infty}[$, whose normalization $[1\ensuremath{-}{S}^{*}(\ensuremath{\infty})]$ determines the conditioned probability ${S}^{*}(\ensuremath{\infty})\ensuremath{\in}[0,1]$ of forever-survival. This case of infinite horizon $T=+\ensuremath{\infty}$ 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 $\ensuremath{\mu}$ 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.