Closed-form solution of modified Ornstein-Uhlenbeck process
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
- 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:
UPD: Fixed signs in 2 last equations.