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 $x \in ]-\infty,a[$ 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.