UC Merced

In this dissertation, we consider the problem of inferring unknown parameters of stochastic differential equations (SDE) from time-series observations. In particular, we develop and test numerical methods to perform frequentist and Bayesian inference for SDE. A key challenge in developing practical inference algorithm is the computation of the likelihood. To compute the likelihood, we propose a novel, fast method that tracks the probability density of the SDE. Our method does not rely on sampling; instead, it evolves the density in time using repeated quadrature on the Chapman-Kolmogorov equation of the Markov chain that results from a time discretization of the SDE. We name our method density tracking by quadrature (DTQ). Our method enables accurate, parallelizable computation of the likelihood when the data is collected with large inter-observation time or when the data consists of one or more time series. In this dissertation, we focus on a particular case of the DTQ method that arises from applying the Euler-Maruyama method in time and the trapezoidal quadrature rule in space. Under some regularity condition for the drift and the diffusion terms of SDE, we theoretically prove that the density computed by the DTQ method converges in $L^1$ to the exact density with a first-order convergence rate in temporal step size. Numerical tests show that the empirical performance of the DTQ method complies with the theoretical convergence results.

To perform inference using maximum likelihood approach, we develop methods to compute the gradient of the likelihood. We propose a direct method to compute the gradient from the DTQ likelihood and use this direct method to perform parametric inference of the SDE. We also propose a more efficient adjoint-based method to compute the gradient information with a computational cost (in time) that does not scale with the dimension of the unknown parameter vector. Therefore, we use this adjoint-based method to perform nonparametric inference of SDE. Using the DTQ method to compute the likelihood, we also develop a Markov Chain Monte Carlo (MCMC) algorithm using a Metropolis scheme to perform Bayesian inference. We apply this Bayesian inference method for coupled SDE. In this work, we derive a coupled, nonlinear SDE to model the chaser’s pursuit of the runner in a basketball game. We perform Bayesian inference using NBA tracking data to show the appropriateness of the model for basketball fast break situations.