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

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 γ^{*}(T_{a}) of the absorption time T_{a}∈[0,T]. When the time horizon is infinite T=+∞, the conditioning consists in imposing the probability distribution γ^{*}(T_{a}) of the absorption time T_{a}∈[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.