## 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

## Modified Ornstein-Uhlenbeck process solution

Let’s apply the described approach to the process. Thus we’ve got in notation of the method:

Integrating factor transforms to:

and ODE for it is:

and initial conditions are:

Thus the solution is:

And recovering solution:

where

**UPD:** Fixed signs in 2 last equations.

Thanks for sharing the solution. I have been looking everywhere for this :) I have a few questions though.

In the expression for S(t), shouldn’t the second term be (1-exp(-alpha t)) instead of (1+exp(-alpha t)). Just feel weird thats its so different from the non-stochastic version.

It seems that the variance either converge to infinity or zero. I wonder if there is there any lognormal process that has bounded variance? (like the OU process)

Thanks!

Boon Teik OoiJuly 14, 2013 at 22:02

You are right! Thanks for the proof reading of the solution. I’ve just made an update.

I think that the variance converge to something like OU process with as mean and as mean-reversion parameter.

fxpaulJuly 16, 2013 at 08:43

Excellent stuff! I was looking for this problem.

quant3December 24, 2013 at 16:14

Hi,

I was also looking for the solution of this problem, but i am afraid what you derive for S(t) is not correct. Nevertheless the integrating factor technique is interesting, but i am not sure it works this way in this case.

As far as i know in the literature it is known as Generalized OU process, or sometimes Nelson’s diffusion process, or even continuous GARCH(1,1) process.

Anyhow, i think the correct solution is more complicated, it should depend on the full path of the Brownian motion up to time t, not just on the value W(t). I think Eq. (6) in this paper (https://www.tu-braunschweig.de/Medien-DB/stochastik/lindner5.pdf) is the correct solution. Also, see Eq. (9) and the text below that.

Cheers,

andras

AndrasDecember 26, 2013 at 21:33

Hi Andras,

Sorry for very long reply. I’ve got too much work over this Christmas.

There is an error in the derivation of .

It should be .

I will fix it over this weekend.

Thanks a lot!

fxpaulFebruary 8, 2014 at 22:02

As Boon Teik Ooi pointed out, the variance of the solution converges to either 0 or infinity as time -> infinity (simply because the solution is a non-random term + exponential BM * exp(-theta*t)). However, the solution is incorrect – the variance is in fact finite, just as it is for the plain OU. You can find the solution in Shreve, p.300.

AlexFebruary 11, 2016 at 11:33