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Euler approximation for stochastic Volterra equations and central limit theorem to a system of stochastic heat equations
Saikia, Bhargobjyoti
Saikia, Bhargobjyoti
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Abstract
In the first chapter we start by introducing stochastic Volterra equations and state some well-known results. In the course of the chapter, we establish the convergence in distribution of the normalized error in the Euler approximation scheme, for a particular type of stochastic Volterra equations driven by a standard Brownian motion, with a kernel of the form (t-s)^a, where a in (-1/2, 1/2). We use the asymptotic version of the Knights theorem along with some sophisticated analysis of the equation, the solution satisfies, to study the convergence of the normalized error. In chapter 2, we consider a system of stochastic heat equations, where each coordinate is a one dimensional stochastic heat equation driven by a multiplicative space time white noise. We show that the spatial integral of this solution from -R to R converges in Wasserstein distance to a multivariate normal distribution as R tends to infinity, after re-normalization. We also attempt to get a corresponding functional central limit theorem which states that the spatial integral average converges in distribution to a d-dimensional centered Gaussian martingale. We use Malliavin Calculus and Stein's method as the basic tools in the analysis. The first chapter of this thesis corresponds to a manuscript of the author and David Nualart. And the second chapter is an extension of the paper Huang et al. (2022)
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Date
2022-08-31
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University of Kansas
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Keywords
Mathematics, Applied mathematics, Central limit theorem, Euler approximation, Malliavin Calculus, Stochastic heat equation, Stochastic Volterra equation