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.

**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:

(a) The stochastic process σ(*t, X*_{t}) belongs to _{ad}(Ω, *L*^{2}[*a, b*]) so that for each is an Itô integral.

(b) Almost all sample paths of *f*(*t, X*_{t}) belong to *L*^{1}[*a, b*].

(c) For each *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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