Complex Stochastic Systems and the Effect of Discretization

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Ubiquitous presence of randomness in behaviors of various physical and biological systems is universally acknowledged. For example, cellular processes, fluctuations in stock prices, weather patterns, movement of microscopic particles exhibit different types of random behaviors. Furthermore, randomness is a vital ingredient in most modern algorithms that are popular in data-science. Hence, it is of fundamental importance that various types of randomness are properly characterized for detailed understanding of system properties. Mathematical models incorporating such randomness are typically expressed in terms of stochastic equations, which often have complex dynamics. An important component of mathematical studies of these equations includes computer simulations of their temporal evolution, which are necessary for understanding their behavioral patterns over time. Accurate statistical estimation of some key parameters of these stochastic equations from observed data is also crucial, as it leads to the "most appropriate mathematical model" underlying these data points. This in turn leads to better predictive power of such models. The research supported by this award will undertake proper mathematical analyses of various numerical schemes that are used for these purposes. More specifically, research in this direction will involve precise calculations of probabilities of getting accurate results from these schemes and will describe conditions under which these probabilities are very high. There is a critical need for such theoretical results, since they will inevitably lead to design of faster and more efficient algorithms, which in turn will be beneficial to society. The project will involve undergraduate and graduate students and will help them to gain valuable analytical and computational skills. The results of the project will be published in well-known scientific journals and will also be presented at domestic and international conferences. The research will focus on the limiting behaviors of complex stochastic systems discretized by properly scaled step sizes. Discretization is at the heart of various numerical schemes, but its effect on the desired convergence properties is not always clearly understood. For example, it is well known that approximating stationary distribution of an ergodic stochastic differential equation (SDE) by time averages of sample paths obtained from an Euler-Maruyama type discretization scheme (using fixed step-size) could be problematic. In particular, such an estimator will have a bias, which may or may not be quantifiable. These are infinite-time horizon problems, and there is a deficit of proper asymptotic results in this direction even for regular Ito-diffusions. Exploring proper scaling techniques to get desired convergence along with convergence rates for such discretization-based schemes is the central goal of this project. Stochastic models that will be considered covers switching jump-diffusion models, multiscale systems and systems of interacting SDEs with possible jumps. The last class is particularly important for understanding particle-based methods for approximating nonlinear integro-differential equations including Boltzmann-type equations. These equations are connected to appropriate systems of interacting SDEs with jumps through McKean-Vlasov type limits. The stochastic equations of interest will also include Langevin SDEs, which model dynamics of molecular systems in presence of particle interaction potential, damping and random forces, and which also play a pivotal role in certain Markov Chain Monte Carlo algorithms. The research puts special emphasis on large deviation asymptotics of error probabilities, which, importantly, because of presence of both upper and lower bounds, give the optimal exponential decay rate. This project is jointly funded by Probability program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.