Stochastic Differential Equation

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1. Definition
2. Ito process and Ito diffusion

1. Definition

A stochastic differential equation [SDE] is a differential equation where some components are considered to be stochastic. Therefore the solution to a SDE is actually a random function.

Consider an ordinary differential equation [ODE]

$\fs4 \frac{d X(t)}{dt }= f(t, X(t))$

The solution of this ODE can be expressed as

$\fs4 X(t) = X(0) + \int_0^t f(s,X(s))ds$

Now the solution to the SDE

$\fs4 dX(t) = \mu(t, X(t))dt + \sigma(t,X(t)) dB(t)$

(where $\fs4 B(t)$ is a Brownian motion) is the function

$\fs4 X(t) = X(0) + \int_0^t \mu(s,X(s))ds + \int_0^t \sigma(s, X(s)) dB(s)$

The first integral in the above expression is a normal Lebesgue integral, while the second one is an Ito integral. Apparently the solution to a SDE will be a random function, which can be though of a collection of random variables; that is to say the solution is a stochastic process. Such a process is called an Ito process

2. Ito process and Ito diffusion

The solution to a SDE of the above form can be though of as an Ito diffusion in the following way: if it describes the position of a particle, which is $\fs4 X(t)$ at time $\fs4 t$ , then its position after time $\fs4 \triangle$ will be normally distributed with mean $\fs4 \mu(t,X(t))\triangle$ and variance $\fs4 \sigma^2(t,X(t))\triangle$ .

Since the motion only depends on the path through its latest position, the process is Markov .

In general though Ito processes do not need to be Markov. One can construct the solution to

$\fs4 X(t) = X(0) + \int_0^t \mu(s, \omega)ds + \int_0^t \sigma(s, \omega) dB(s, \omega)$

Of course for the solution to make sense we want the functions $\fs4 \mu$ and $\fs4 \sigma$ to be $\fs4 (\mathscr{B} \times \mathscr{F}_t)$ -adapted. That is to say, we should be able to ascertain their values at time $\fs4 t$ by the history of the particle up to time $\fs4 t$ ; the behaviour of the particle must not be influenced by its future.

An example of an Ito process which would not be an Ito diffusion would be the integral

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Stochastic Differential Equation

1. Definition

2. Ito process and Ito diffusion