Parameter estimation in several classes of non-Markovian random processes defined by stochastic differential equations

This work is concerned with parameter estimation of solutions of stochastic evolution equations driven by Gaussian processes. Two different classes of problems are considered. We study certain stochastic differential equations of the form dXt=f&parl0;Xt,Yt,t,q 1&parr0;dt+g&parl0;Xt,Yt,t,q 2&parr0;dYt where (Yt) is a given Gaussian process with known covariance kernel, and f and g are some known "drift" and "volatility" functions which depend on unknown parameters of interest (theta1, theta2). For both problems considered, the resulting process Xt is generally non-Markovian, which makes the problems interesting from a mathematical viewpoint and useful in many applications where the Markov assumption is impractical.;We first consider the case of a non-semimartingale driving the dynamics of where is a Gaussian random field with covariance structure of the form 0t 0sK&parl0;t,u&parr0;K&parl0;s,v&parr0;dvdu for a general Volterra kernel K. Volterra processes are one of the most recent additions to the field of continuous Gaussian processes and represent generalizations of the popular fractional Brownian motion (fBm). For this problem we derive estimates of the drift parameter theta 1, as well as derive several asymptotic properties of our estimate.;Next we study a monotone increasing integral functional of a standard Brownian motion, which can be formally regarded as a solution to the degenerate stochastic differential equation with g ≡ 0. This choice is motivated by physical properties of many degradation processes which have continuous and monotone increasing random trajectories. In many applications one is interested in estimating the time to failure of various devices thus, given some failure threshold, D > 0, it is natural to study the "time to failure" random variable TD defined by TD:=inf&cubl0;t>0:Xt =D&cubr0;;We first estimate theta1 based on observing several paths of the process X and then numerically estimate the entire distribution of TD. We establish several estimates of theta1, based on different data observation assumptions, as well as derive consistency results for these estimators. Additionally, we provide a consistent estimator of the mean of TD.