This thesis consists of two major parts, and contributes to two areas of research in stochastic analysis: (i) Wiener-Hopf factorization (WHf)... Show moreThis thesis consists of two major parts, and contributes to two areas of research in stochastic analysis: (i) Wiener-Hopf factorization (WHf) for Markov Chains, (ii) statistical inference for Stochastic Partial Differential Equations (SPDEs).WHf for Markov chains is a methodology concerned with computation of expectation of some types of functionals of the underlying Markov chain. Most results in WHf for Markov chains are done in the framework of time-homogeneous Markov chains. The major contribution of this thesis in the area of WHf for Markov chains are:
• We extend the classical theory to the framework of time-inhomogeneous Markov chains.
• In particular, we establish the existence and uniqueness of solutions for a new class of operator Riccati equations.
• We connect the solution of the Riccati equation to some expectations of interest related to a time-inhomogeneous Markov chain.
Statistical inference for SPDEs regards estimating parameters of a SPDE based on available and relevant observations of the underlying phenomenon that is modeled by the given SPDE. We summarize the contribution of this thesis in the area statistical inference for SPDEs as follows:
• We conduct the statistical inference for a diagonalizable SPDE driven by a multiplicative noise of special structure, using spectral approach. We show that the corresponding statistical model fits the classical uniform asymptotic normality (UAN) paradigm.
• We prove a Bernstein-Von Mises type result that strengthens the existing results in the literature.
• We prove the asymptotic consistency, asymptotic normality and asymptotic efficiency of two Bayesian type estimators. Show less
