Discretizations of Langevin diffusions provide a powerful method for sampling and Bayesian inference. However, such discretizations require evaluation the gradient potential function. In several real-world scenarios, obtaining evaluations might either be computationally expensive, or simply impossible. this work, we propose analyze stochastic zeroth-order algorithms discretizing overdamped unde...

Hybrid switching diffusions, also known as switching diffusions or regimeswitching systems, have drawn a signifi cant amount of attention in recent decades in a variety of applications due to their ability to model complex systems with uncertainty. Control and systems engineers have long been familiar with the fundamental idea of using a continuous stochastic differential equation to describe a...

The fractional Fokker–Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the nonlocal property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time frac...

We consider a certain class of non-symmetric Markov chains and obtain heat kernel bounds and parabolic Harnack inequalities. Using the heat kernel estimates, we establish a sufficient condition for the family of Markov chains to converge to non-symmetric diffusions. As an application, we approximate non-symmetric diffusions in divergence form with bounded coefficients by non-symmetric Markov ch...

This paper introduces quasi-maximum likelihood estimator for discretely observed diffusions when a closed-form transition density is unavailable. Higher order Wagner-Platen strong approximation is used to derive the first two conditional moments and a normal density function is used in estimation. Simulation study shows that the proposed estimator has high numerical precision and good numerical...

Abstract. In this paper we present the first results on substructuring methods for nonlocal operators, specifically, an instance of the nonlocal p-Laplace operator. We present a nonlocal variational formulation of this operator, proving a nonlocal Poincaré inequality and upper bound to establish a spectral equivalence. We then introduce a nonlocal two-domain variational formulation utilizing no...

ON SWITCHING DIFFUSIONS: THEFEYNMAN-KAC FORMULA ANDNEAR-OPTIMAL CONTROLSbyNICHOLAS BARANAugust 2014Advisor: Dr. George YinMajor: MathematicsDegree: Doctor of Philosophy We consider diffusions in two different contexts. First, we consider the so-calledFeynman-Kac formula(s) for switching diffusions. These formulas provide stochasticrepresentations for ...

In this paper we look at the properties of limits of a sequence of real valued inhomogeneous diffusions. When convergence is only in the sense of finite-dimensional distributions then the limit does not have to be a diffusion. However, we show that as long as the drift terms satisfy a Lipschitz condition and the limit is continuous in probability, then it will lie in a class of processes that w...

We introduce two models of biological aggregation, based on randomly moving particles with individual stochasticity depending on the perceived average population density in their neighborhood. In the first-order model the location of each individual is subject to a density-dependent random walk, while in the second-order model the density-dependent random walk acts on the velocity variable, tog...

This paper is devoted to numerical solutions for a class of jump-diffusions with regime switching. After briefly reviewing the notion of jump-diffusions with regime switching, finite-difference procedures are constructed. Under simple conditions, it is proved that the algorithm converges to the desired limit by means of a martingale problem formulation. Numerical experiments are carried out to ...