Parameter Estimation in Several Classes of Non-Markovian Random Processes Defined by Stochastic Differential Equations.
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Abstract
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(Xt; Yt; t; theta1)dt + g(Xt; Yt; t; theta2)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
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 theta1, 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 t > 0 : Xt = D
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.Ph.D.StatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/77849/1/bcreiner_1.pd