Guided simulation of conditioned chemical reaction networks

Let $X$ be a chemical reaction process, modeled as a multi-dimensional continuous-time jump process. Assume that at given times $0<t_1 < \cdots < t_n$, linear combinations $v_i = L_i X(t_i),\, i = 1,\dots, n$ are observed for given matrices $L_i$. We show how to sample the process conditioned on hitting the states $v_1,\dots, v_n$ by a change of measure on the law of the unconditioned process. We derive sufficient conditions for this change of measure and complement our results using numerical illustrations.