# CHAPTER 38

# STOCHASTIC DIFFERENTIAL EQUATIONS

A stochastic differential equation is a differential equation that involves stochastic processes. The solution of a stochastic differential equation is also a stochastic process. In this chapter, we present stochastic differential equations and their solutions.

# 38.1 Basic Concepts and Facts

**Definition 38.1** (Strong Solution of SDE). A stochastic process {*X*_{t} : *a* ≤ *t* ≤ *b*} is called a *strong solution* of the following stochastic differential equation

if it satisfies the following conditions:

*t, X*

_{t}) belongs to

_{ad}(Ω,

*L*

^{2}[

*a, b*]) so that for each is an Itô integral.

*f*(

*t, X*

_{t}) belong to

*L*

^{1}[

*a, b*].

*t*, the following equation

holds almost surely.

**Definition 38.2** (Weak Solution of SDE). If there exist a probability space with a filtration, a Brownian motion {_{t} : *t* ≥ 0}, and a process {_{t} : *a* ≤ *t* ≤ *b*} adapted to the filtration such that they satisfy the conditions given in ...

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