Markov bridges: SDE representation

Let $X$ be a Markov process taking values in $\mathbf{E}$ with continuous paths and transition function $(P_{s,t})$. Given a measure $μ$ on $(\mathbf{E}, \mathscr{E})$, a Markov bridge starting at $(s,\varepsilon_x)$ and ending at $(T^*,μ)$ for $T^* <\infty$ 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) {\em weak conditioning} when $μ$ is absolutely continuous with respect to $P_{s,t}(x,\cdot)$ and b) {\em strong conditioning} when $μ=\varepsilon_z$ for some $z \in \mathbf{E}$. 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.

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