On Stochastic Differential Equations Driven By The Sub-fractional Brownian Motion

Recently, the long-range dependence property has become an important aspect of stochastic models in various scientific areas including hydrology, telecommunica-tion, turbulence, image processing and finance. The best known and most widely used process that exhibits the long-range dependence property is the fractional Brownian motion. As an extention of Brownian motion, in2004, Bojdeki et al. introduced and studied a rather special class of self-similar Gaussian processes which preserves many properties of the fractional Brownian motion (e.g. self-similarity, long-range depen-dence, Holder paths, neither a semimartingale nor a Markov process). This special process is called the sub-fractional Brownian motion. The biggest difference between the sub-fractional Brownian motion and the fractional Brownian motion is that the frac-tional Brownian motion has stationary increments whereas the former has no. Because the sub-fractional Brownian motion is neither a semimartingale nor a Markov process, many of the powerful techniques from stochastic analysis are not available. We study on stochastic differential equations driven by the sub-fractional Brownian motion by applying Malliavin calculus theory.In Chapter1, we introduce the backgrounds of investigations, the related prelimi-naries and list the main results obtained by this paper.In Chapter2, the sub-fractional Brownian motion is studied. At first, we obtain the L2-consistency and the strong consistency of the maximum likelihood estimators of the mean and variance of the sub-fractional Brownian motion with drift at discrete observation. By combining the Stein's method with Malliavin calculus, we get the central limit theorem and the Berry-Esseen bounds for these estimators. Then, based on a random walk approximation of the sub-fractional Brownian motion, we construct an approximated maximum likelihood estimator for the drift parameter in a simple linear model driven by the sub-fractional Brownian motion, and study the asymptotic behaviors of the estimator. The numerical simulations show that the estimator has superiority. Moreover, when0<H<1/6, we prove by means of Malliavin calculus that the convergence of weighted cubic variation of a sub-fractional Brownian motion SH holds in L2toward an explicit limit which only depends on SH.In Chapter3, we investigate the asymptotic properties of a least squares estimator for the parameter a of a sub-fractional bridge. Depending on the value of a, we prove that we may have strong consistency or not. When we have consistency, we obtain the rate of this convergence as well.In Chapter4, we begin to study the sub-fractional Ornstein-Uhlenbeck process, and obtain the Berry-Esseen bounds of the least squares estimator for the unknown parameter of the discretely observed sub-fractional Ornstein-Uhlenbeck process. Then, we investigate the asymptotic properties of the sequential maximum likelihood estimator of the drift parameter for the sub-fractional Ornstein-Uhlenbeck type process satisfying a stochastic differential equation driven by the sub-fractional Brownian motion.In Chapter5, we consider the theoretical reinsurance ruin problem. Assume that an insurance company has two types of independent claims, respectively modeled by a Markov additive process (large claims) and a sub-fractional Brownian motion (small claims), and chooses to reinsure both of them according to a quota share policy. This leads to study a bivariate risk process. We obtain asymptotic of the ruin probability of one of the risk processes when initial reserves tend to infinity along a direction by using large deviation theory.In the last Chapter, we investigate the problem of estimating the parameters for the sub-mixed fractional Brownian motion from discrete observations based on the maximum likelihood method. We obtain the asymptotic properties of these estimators. By combining the Stein's method with Malliavin calculus we prove the asymptotic normality for the corresponding estimators.