Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

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

(38.1)

if it satisfies the following conditions:

(a) The stochastic process σ(t, X_t) belongs to _ad(Ω, L²[a, b]) so that for each is an Itô integral.

(b) Almost all sample paths of f(t, X_t) belong to L¹[a, b].

(c) For each t, the following equation

(38.2)

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 ...