Introduction to Stochastic Analysis: Integrals and Differential Equations

Chapter 4

Stochastic Integral with Respect to Brownian Motion

 

 

4.1. Preliminaries. Stochastic integral with respect to a step process

In the previous chapter, we discussed that in order to make sense of stochastic differential equations, it is important to define the integrals Y_sdB_s, where the integrand function Y is random, and the integrating function (also random) is a Brownian motion B.

Before defining the stochastic integral, recall the definition of the Stieltjes integral. Roughly speaking, the Stieltjes (or Riemann-Stieltjes) integral of a function f in the interval [a, b] with respect to a function g is the limit

where a = t₀ < t₁ < … < t_k = b is a partition of the interval [a, b], and is its intermediate partition, i.e.

To rigorously define the Stieltjes integral , consider any sequence Δⁿ = {a = = b}, n ∈ , of partitions of the interval [a, b] such that | Δⁿ | max_i Δ ...