Stochastic differential equations

Abstract

Stochastic differential equations serve as the foundation for many sections of applied sciences, such as mechanics, statistical physics, diffusion theory, cosmology, financial mathematics, economics, etc. The number of works devoted to various issues related to specific equations considered in individual areas of science listed above is very large. In this study, we consider only the general theory of stochastic differential equations, which is based on the approach initiated by K. Itô. In addition, we discuss simple analytical and numerical methods for solving such equations and, finally, we present an application in finance, namely, the Black and Scholes option price formula. In Chapter 1 we introduce basic notations and facts from the theory of stochastic processes, needed for the concept of Itô integrals in Chapter 2. In Section 1.2 we present the concept and some properties of Brownian motion which is one of the fundamental processes in mathematics and physics, as well as in natural science in general. In Chapter 2 we develop the Itô stochastic calculus, which has important applications in mathematical finance and stochastic differential equations. The theory of stochastic integration with respect to Brownian motion is developed in Section 2.1. In Section 2.2 we present the chain rule for stochastic calculus, commonly known, as the Itô formula. Chapter 3 returns to our main theme of stochastic differential equations. In this chapter, we present the stochastic differential equations, driven by Brownian motion, and the notions of strong and weak solutions. Section 3.1 is devoted to the theorem on the existence and uniqueness of a solution to a stochastic differential equation. In Section 3.2 we introduce the concept of a weak solution and the method for constructing such solutions by the Girsanov theorem. In Section 3.3 we give examples of stochastic differential equations and some analytical methods for solving them. In Section 3.4 we discuss two most popular numerical methods for solving (or simulating from) stochastic differential equations: the Euler-Maruyama method and the Milstein method. In Chapter 4 we introduce the necessary concepts from finance, and, finally, we present a proof of the Black-Scholes formula, which gives a theoretical estimate of the price of European-style options. Chapter 5 highlights areas for further research and perspectives.