# FxPaul

Math in finance or vice versa

## Process definition

In this article we deduce the closed-from solution of the modified version of Ornstein-Uhlenbeck process:
$dS = \theta(\mu - S) dt + \sigma S dW_t$
where $\theta$ – mean reversion parameter, $\mu$ – mean and $\sigma$ – volatility.

## Integrating factor approach

There exists a general approach to non-linear stochastic differential equations of the form:
$dX_t = f(t,X_t) dt + c(t) X_t dW_t \, , \, X_0 = x$
where $f$ and $g$ are given continuous and deterministic functions.

The method consists of:

1. Define the integrating factor:
$F_t = F_t(\omega) = \exp\left( - \int_0^t {c(s) dW_s} + \frac{1}{2}\int_0^t {c^2(s) dW_s} \right)$
2. So the original equation could be written as $d\left(F_t X_t\right) = F_t f(t,X_t) dt$
3. Now define
$Y_t(\omega) = F_t(\omega) X_t(\omega)$
so that $X_t = F^{-1}_t Y_t$
4. And it yields the deterministic differential equation for each $\omega\in\Omega$
$\frac{dY_t(\omega)}{dt} = F_t(\omega) f\left(t, F^{-1}_t(\omega) Y_t(\omega)\right); \, Y_0 = x$

We can therefore solve it with $\omega$ as a parameter to find $Y_t(\omega)$ and then obtain $X_t(\omega)$

Written by fxpaul

May 27, 2011 at 08:39