Markovian pure jump processes model a wide range of phenomena, including chemical reactions at the molecular level, dynamics of wireless communication networks and the spread of epidemic diseases in small populations. There exist algorithms like Gillespie's SSA or Anderson's Modified Next Reaction Method, that simulates a single trajectory with the exact distribution of the process, but this can be time consuming when many reactions take place during a short time interval. Gillespie's approximated tau-leap method, on the other hand, can be used to reduce computational time, but it may lead to non-physical values due to a positive one-step exit probability, and it also introduces a time discretization error. Here, we present a novel hybrid algorithm for simulating individual trajectories which adaptively switches between the SSA and the tau-leap method.  The switching strategy is based on a comparison of the expected inter-arrival time of the SSA and an adaptive time step derived from a Chernoff-type bound for the one-step exit probability.  Because this bound is non-asymptotic, we do not need to make any distributional approximation for the tau-leap increments. This hybrid method allows us (i) to control the global exit probability of any  simulated trajectory, and (ii) to obtain accurate and computable estimates of the expected value of any smooth observable of the process with minimal computational work. We present numerical examples that  illustrate the performance of the proposed method.

ISSN (print): 1540-3459 ISSN (online): 1540-3467