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vVIENER-HOPF FACTORIZATION FOR TIME-INHOMOGENEOUS MARKOV CHAINS AND STATISTICAL INFERENCE FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
Title
vVIENER-HOPF FACTORIZATION FOR TIME-INHOMOGENEOUS MARKOV CHAINS AND STATISTICAL INFERENCE FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
Creator
Huang, Yicong
Date
2018, 2018-05
Description
The thesis consists of two major parts, and it contributes to two topics in stochastic analysis – Wiener-Hopf factorization (WHf) for Markov...
Show moreThe thesis consists of two major parts, and it contributes to two topics in stochastic analysis – Wiener-Hopf factorization (WHf) for Markov chains and statistical inference for Stochastic Partial Differential Equations (SPDEs). The first part deals with Wiener-Hopf factorization for finite state time inhomogeneous Markov chains. To the best of our knowledge, this study is the first attempt to investigate the WHf for the time-inhomogeneous Markov chains. In this work we only deal with a special class of time-inhomogeneous Markovian generators, namely piece-wise constant, which allows to derive the corresponding WHf by using an appropriately tailored randomization technique. Besides the mathematical importance of the WHf methodology, there is also an important computational aspect: it allows for efficient computation of important functionals of Markov chains. In this work, we also provide an efficient algorithm to compute the quantities in the Wiener-Hopf factorization for the time-inhomogeneous Markov chains. Finally, we provide a comparison (based on numerical simulations) between our algorithm and the brute-force Monte Carlo simulations. The second part is dedicated to statistical inference for Stochastic Partial Differential Equations (SPDEs). First, we study the problem of estimating the drift/viscosity coefficient for a large class of linear, parabolic SPDEs driven by an additive space-time noise. We propose a new class of estimators, called trajectory fitting estimators (TFEs). The estimators are constructed by fitting the observed trajectory with an artificial one, and can be viewed as an analog to the classical least squares estimators from the time-series analysis. As in the existing literature on statistical inference for SPDEs, we take a spectral approach, and assume that we observe the first N Fourier modes of the solution, and we study the consistency and the asymptotic normality of the TFE, as N ₀₀. Next we consider a parameter estimation problem for one dimensional stochastic heat equation, when data is sampled discretely in time or spatial component. We establish some general results on derivation of consistent and asymptotically normal estimates based on computation of the p-variations of stochastic processes and their smooth perturbations. We apply these results to the considered SPDEs, by using some convenient representations of the solutions. For some equations such representations were ready available, while for others classes of SPDEs we derived the needed representations along with their statistical asymptotic properties. We prove that the real valued parameter next to the Laplacian, and the positive parameter in front of the noise can be consistently estimated by observing the solution at a fixed time and on a discrete spatial grid, or at a fixed space point and at discrete time instances of a finite interval, assuming that the mesh size goes to zero.
Ph.D. in Applied Mathematics, May 2018
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