## Archive for **May 27th, 2011**

## Closed-form solution of modified Ornstein-Uhlenbeck process

## Process definition

In this article we deduce the closed-from solution of the modified version of Ornstein-Uhlenbeck process:

where – mean reversion parameter, – mean and – volatility.

## Integrating factor approach

There exists a general approach to non-linear stochastic differential equations of the form:

where and are given continuous and deterministic functions.

The method consists of:

- Define the integrating factor:

- So the original equation could be written as
- Now define

so that - And it yields the deterministic differential equation for each

We can therefore solve it with as a parameter to find and then obtain