Exact Variance-Reduced Simulation of Lattice Continuous-Time Markov Chains with Applications in Reaction Networks

Abstract

We propose an algorithm to reduce the variance of Monte Carlo simulation for the class of countable-state, continuous-time Markov chains, or lattice CTMCs. This broad class of systems includes all processes that can be represented using a random-time-change representation, in particular reaction networks. Numerical studies demonstrate order-of-magnitude reduction in MSE for Monte Carlo mean estimates using our approach for both linear and nonlinear systems. The algorithm works by simulating pairs of negatively correlated, identically distributed sample trajectories of the stochastic process and using them to produce variance-reduced, unbiased Monte Carlo estimates, effectively generalizing the method of antithetic variates into the domain of stochastic processes. We define a method to simulate anticorrelated, unit-rate Poisson process paths. We then show how these antithetic Poisson process pairs can be used as the input for random time-change representations of any lattice CTMC, in order to produce anticorrelated trajectories of the desired process. We present three numerical parameter studies. The first examines the algorithm’s performance for the unit-rate Poisson process, and the next two demonstrate the effectiveness of the algorithm in simulating reaction network systems: a gene expression system with affine rate functions and an aerosol particle coagulation system with nonlinear rates. We also prove exact, analytical expressions for the time-resolved and integrated covariance between our antithetic Poisson processes for one technique.