Chapter 2

Fundamentals Of Stochastic Analysis

Takashi Yasuoka

Abstract

This chapter brie y summarizes basic concepts of stochastic calculus, using intuitive examples. First, the fundamentals of probability spaces are intro- duced by working with a simple example of a stochastic process. Next, stochastic processes are introduced in connection with a natural ltration and a martingale. Then, we introduce a stochastic integral and Ito's formula, which is an important tool for solving stochastic differential equations. Finally, we address some funda- mental examples of stochastic differential equations, which simply model the price process of a nancial asset. Although these subjects are applied in practice to interest rate modeling, the de nitions are given for the one-dimensional case for the sake of simplicity. We complement this with some basic results for multi-dimensional cases in Section 2.7, at the end of this chapter.

Total Pages: 31-51 (21)

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