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Introduction to Stochastic Analysis: Integrals and Differential Equations by Vigirdas Mackevičius

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Chapter 6

Stochastic Differential Equations

 

 

Now, we can already make sense of the integral equation

[6.1]images

derived in Chapter 3 (most often, I = [0,T] or [0, )). Though partially for historical reasons, but rather for practical convenience, it is usually written in the formal differential form

[6.2]images

Both equations are called stochastic differential equations, though here the word integral would fit better. A formal definition is as follows.

DEFINITION 6.1.– A continuous random process Xt, tI, is a solution of the stochastic differential equation [6.2] in an interval I if for all t I, it satisfies equation [6.1] with probability one.

EXAMPLE 6.2.- Consider the stochastic differential equation

images

This equation describes, for example, the dynamics of stock price in financial mathematics, or the development of some populations in biology. Let us check that the random process Xt = x0 exp{(µ−σ2 /2)t+σBt}, t image 0, is its solution. Applying the Itô rule (Theorem 5.2) to the function F(t, x) = x exp {(µ − σ2/2)t + σx},we have:

In particular, for µ = 0 and σ = 1, we get that the random process ...

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