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

Area of Science:

• Computational Science
• Stochastic Processes
• Chemical Kinetics

Background:

• Monte Carlo simulations are widely used for analyzing complex systems.
• Variance reduction is crucial for improving the efficiency and accuracy of these simulations.
• Lattice countable-state, continuous-time Markov chains (CTMCs) are common models for systems like reaction networks.

Purpose of the Study:

• To develop a novel algorithm for variance reduction in Monte Carlo simulations of lattice CTMCs.
• To generalize the method of antithetic variates to stochastic processes.
• To demonstrate significant mean estimate accuracy improvements.

Main Methods:

• Simulating pairs of negatively correlated, identically distributed sample trajectories.
• Developing a method to simulate anticorrelated, unit-rate Poisson process paths.
• Utilizing antithetic Poisson process pairs as input for random time-change representations of lattice CTMCs.

Main Results:

• Achieved order-of-magnitude reduction in Mean Squared Error (MSE) for Monte Carlo mean estimates.
• Demonstrated effectiveness for both linear and nonlinear systems, including reaction networks.
• Provided analytical expressions for covariance between antithetic Poisson processes.

Conclusions:

• The proposed algorithm effectively reduces variance in Monte Carlo simulations for lattice CTMCs.
• The method offers a powerful tool for analyzing complex stochastic systems, such as gene expression and aerosol coagulation.
• This work generalizes antithetic variates to a broader class of stochastic processes.