Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon
Résumé
When the unconditioned process is a diffusion living on the half-line x ∈] − ∞, 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 < +∞, the conditioning consists in imposing the probability distribution P∗(y, T ) to be surviving at time T at the position y ∈] − ∞, a[, as well as the probability distribution $\gamma ^∗$(Ta ) of the absorption time Ta ∈ [0, T ]. When the time horizon is infinite T = +∞, the conditioning consists in imposing the probability distribution $\gamma ^∗$(Ta ) of the absorption time Ta ∈ [0, +∞[, whose normalization [1 − S∗(∞)] determines the conditioned probability S∗(∞) ∈ [0, 1] of forever-survival. This case of infinite horizon T = +∞ 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 μ 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
Domaines
Physique [physics]
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