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 the process that is conditioned on hitting the states $v_1,\dots ,v_n$ is obtained by a change of measure on the law of the unconditioned process. This results in an algorithm for obtaining weighted samples from the conditioned process. Our results are illustrated by numerical simulations.