Fundamentals of Stochastic Analysis

Abstract

This chapter briefly summarizes basic concepts of stochastic calculus, using intuitive examples. First, the fundamentals of probability spaces are introduced by working with a simple example of a stochastic process. Next, stochastic processes are introduced in connection with a natural filtration 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 fundamental examples of stochastic differential equations, which simply model the price process of a financial asset.

Although these subjects are applied in practice to interest rate modeling, the definitions 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.

Keywords: Abstract Bayes’ rule, Augmented filtration, Brownian motion, Conditional expectation, Distribution function, Euler approximation, Equivalent measure, Exponential martingale, Filtered probability space, Girsanov theorem, Ito’s formula, Market measure, Martingale, Measure change, Natural filtration, Probability space, Quadratic covariation, Random walk, Real-world measure, Risk-neutral measure, Radon–Nykodim derivative, Random variable, Sample space, Stochastic differential equation, Stochastic integral, Stochastic process