Markov bridges: SDE representation

Let X be a Markov process taking values in E with continuous paths and transition function (Ps,t). Given a measure μ on (E,E), a Markov bridge starting at (s,εx) and ending at (T∗,μ) for T∗<∞ has the law of the original process starting at x at time s and conditioned to have law μ at time T∗. We will consider two types of conditioning: (a) weak conditioning when μ is absolutely continuous with respect to Ps,t(x,⋅) and (b) strong conditioning when μ=εz for some z∈E. The main result of this paper is the representation of a Markov bridge as a solution to a stochastic differential equation (SDE) driven by a Brownian motion in a diffusion setting. Under mild conditions on the transition density of the underlying diffusion process we establish the existence and uniqueness of weak and strong solutions of this SDE.