Stochastische Differentialgleichungen [chapter]

In this diploma thesis, a short introduction into the theory of stochastic differential equations is given. Informally spoken, a stochastic differential equation is a way to mathematically describe processes whose time evolution is not only deterministic, but also influenced by random events. Hereby, the random events are assumed to be normally (Gaussian) distributed. In the following, the contents of every chapter will be described shortly. Chapter 1 comprises a short motivation why randomness
more » ... should be included in the theory of ordinary differential equations. With the help of a first example of a stochastic differential equation, the requirements on the differential equation which are needed in this context are illustrated. The necessary theoretical background for this work is presented in chapter 2. First, the most important theorems and definitions from probability and measure theory are recapitulated. Then, some important properties of the expectation of a random variable are deduced, which will be an important tool in the following chapters and sections. Furthermore, several types of convergence are defined and an important limit theorem is stated. In section 1.4, the basic principles of stochastic processes are introduced and these stochastic processes, which fulfil certain additional properties as e.g. continuity, are presented. The Brownian Motion is the most important stochastic process for the theory of differential equations. The next sections describe how with the help of the Brownian Motion, one defines stochastic integrals which have important properties, here in particular the martingale property. At the end of this first sub-chapter, the white noise which represents the random influence in stochastic differential equations is presented and identified with the Brownian Motion. The next subsection tackles the meanwhile unavoidable definition of stochastic integrals. After having defined it for step functions, a larger class of functions is included in this definition. In section 2.3 the most important properties for the use of the stochastic integral are outlined. Next, its definition is extended once more to a slightly larger class of functions and the Itô formula is deduced, which is an important tool in the theory of stochastic differential equations. In the last chapter, stochastic differential equations are introduced theoretically and with the help of two examples. Section 3.1 tackles the most important definitions and the examples are explained in the context of ordinary differential equations. Later, randomness will be introduced in these examples, and the resulting changes will be presented. The most important theorem of existence and uniqueness for the stoastic initial value problem III is proven in detail. It imposes certain requirements on the coefficients and the initial condition. The examples from the first section are revisited, now including randomness. In the following, dependence on the initial condition, continuity and boundedness of the solution is proven. Since most times, it is not possible to analytically solve a stochastic differential equation, the concept of numerical approximation of solutions is presented. In particular, the Euler approximation is introduced which gives a numerical solution for discrete times. The approximation error is estimated both analyically and numerically with estaimation of the confidence interval. IV Zusammenfassung Mit dieser Diplomarbeit wird eine kurze Einführung in die Theorie der stochastischen Differentialgleichung gegeben. Eine stochastische Differentialgleichung ist, salopp gesagt, eine Möglichkeit um mathematische Prozesse zu beschreiben, deren zeitliche Entwicklung nicht nur deterministisch ist, sondern auch durch Zufälligkeiten beeinflusst wird. Hier werden die zufälligen Ereignisse als normalverteilt angenommen.
doi:10.1007/3-540-33414-9_26 fatcat:t3fad7jiyfez5jh2ewm64mvphy