Parameter Estimation in Several Classes of Non-Markovian Random Processes Defined by Stochastic Differential Equations.

[摘要] This work is concerned with parameter estimation of solutions of stochastic evolutionequations driven by Gaussian processes. Two different classes of problems areconsidered. We study certain stochastic differential equations of the formdXt = f(Xt; Yt; t; theta1)dt + g(Xt; Yt; t; theta2)dYtwhere (Yt) is a given Gaussian process with known covariance kernel, and f andg are some known drift and volatility functions which depend on unknown parametersof interest (theta1, theta2). For both problems considered, the resulting process Xtis generally non-Markovian, which makes the problems interesting from a mathematicalviewpoint and useful in many applications where the Markov assumption isimpractical.We first consider the case of a non-semimartingale driving the dynamics of whereis a Gaussian random field with covariance structure of the formfor a general Volterra kernel K. Volterra processes are one of the most recentadditions to the field of continuous Gaussian processes and represent generalizationsof the popular fractional Brownian motion (fBm). For this problem we derive estimatesof the drift parameter theta1, as well as derive several asymptotic properties ofour estimate.Next we study a monotone increasing integral functional of a standard Brownianmotion, which can be formally regarded as a solution to the degenerate stochasticdifferential equation with g = 0. This choice is motivated by physical properties ofmany degradation processes which have continuous and monotone increasing randomtrajectories. In many applications one is interested in estimating the time to failureof various devices thus, given some failure threshold, D > 0 , it is natural to studythe ;;time to failure;; random variable TD defined byTD := inf t > 0 : Xt = DWe first estimate theta1 based on observing several paths of the process X and thennumerically estimate the entire distribution of TD. We establish several estimatesof theta1, based on different data observation assumptions, as well as derive consistencyresults for these estimators. Additionally, we provide a consistent estimator of themean of TD.

[发布机构] University of Michigan

[效力级别] Statistics and Numeric Data

[关键词] Inference for Stochastic Differential Equations;Statistics and Numeric Data;Science;Statistics