Browsing by Author "Dippon, Jürgen (PD Dr. )"

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    Stochastic differential equations driven by Gaussian processes with dependent increments and related market models with memory
    (2006) Schiemert, Daniel; Dippon, Jürgen (PD Dr. )
    In the last ten years fractional Brownian motion BHt received a lot of attention (e.g. [Be],[GrNo], [HuOk],[HuOkSa], [So] [DeUs], [Oh] and [DuHuPa]). This process has dependent increments, which make it interesting for many applications such as finance (e.g. [HuOk], [Be]) and network simulations (e.g. [No]). However, BHt has a covariance function E(BHt BHs ),which depends only on the Hurst parameter H 2 (0, 1). It follows for example that one cannot model a process with short range dependency with BHt . As a generalization a class of centered Gaussian processes with dependent increments Bvt is defined. If Bvt is used as noise process, it is often important to have an integral driven by this process.Several authors defined the stochastic integral driven by fractional Brownian motion RR XsdBHs . In order to use this integral to explain stochastic differential equations it is desirably that the stochastic integral driven by fractional Brownian motion has expectation value 0. If the integral is defined by use of the Wick product ([Be], [HuOk]) the expectation value of the stochastic integral driven by BHt is zero. Thus this definition is interesting for applications. In order to use the Wick product it was helpful to use white noise distribution theory, because the Wick product is not closed in L2( ). This theory ([HiKuPoSt], [Ku], [PoSt])offers also the possibility to derivate fractional Brownian motion in the Hida distribution sense. Further, it has a lot of advantages in the treatment of the Wick product, e.g. the Wick calculus is closed in the space of the Hida distributions [Be], [La]. So one can define stochastic differential equations by integrating the Wick product of the integrand with the derivative of the fractional Brownian motion and solve the bilinear equation (e.g. [HuOk] ). This approach is formulated here for stochastic differential equations driven by Bvt. In the first chapter the auxiliary results of white noise theory are summarized. It begins in Section 2.1 with the construction of the Schwartz space and its dual, which is later used to define the Hida test and distribution space. In Section 2.2 the class of Gaussian processes Bvt is defined and several properties are shown. The Section 2.3 introduces the Hida test and Hida distribution space. Their construction is using the chaos decomposition theorem and the definition of the Schwartz space. As mentioned before one can derivate certain stochastic processes in the Hida distribution space. This is based on the derivative of a deterministic function in the sense of tempered distributions. Further tools to examine the convergence in the Hida distribution space are part of the following section. There the S-transform and the Wick product are defined. The S-transform is a mapping, which allows to examine stochastic processes in a deterministic manner, as we will see in Chapter 3. At the end of Chapter 2, 4 1 Introduction there are some characterization and convergence theorems referring to elements of the Hida distribution space. In Chapter 3 the derivative of the Gaussian process Bvt will be declared by a derivative in the sense of tempered distributions. In Section 3.1 we define the stochastic integral RR Xs dBvs . Further, conditions are presented under which the integral can be approximated by Riemann sums. Several typical applications of the stochastic integrals are developped like partial integration or solving the bilinear stochastic differential equation driven by Bvt . An existence and uniqueness theorem for solutions of stochastic differential equations is formulated a general one in Section 3.4. In Section 3.2 a stochastic integral driven by a in the Hida sense continuously differentiable stochastic process is discussed. As an example the related Ornstein-Uhlenbeck process is treated. In Section 3.3 we derive a chain formula and variants thereof for stochastic distribution processes, which concides with Itˆo’s rule in the case of Brownian motion. By the S-transform and the Wick product it is possible to obtain the chain rule directly be the derivative of a deterministic function, which makes applications practical and proofs easy. As already mentioned, Section 3.4 presents several theorems on the existence and uniqueness of solutions of stochastic differential equations. The results of this dissertation are already published in the preprints [DiSc] and [DiSc2].