How to solve Mean-reverting diffusion

Asked Mar 07 '22 at 20:18

Active Mar 07 '22 at 20:18

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I'd like to learn how to solve the following mean-reverting process equation: \begin{equation} dS_{t}=a(b-S_{t})dt + S_{t}\sigma dw \end{equation}

I have solved the equation without the presence of $S_{t}$ before $\sigma$. My solution is like the following: \begin{equation} dS_{t}=a(b-S_{t})dt + \sigma dw \end{equation} \begin{equation} d(e^{at}(b-S_{t}))=ae^{at}(b-S_{t})dt -ae^{at} dS_{t} \end{equation} and replace $dS_{t}$ and solve the problem. However, when $S_{t}$ exists as the multiplier before $\sigma$ in the problem, I could not come up with a practical function to solve it analytically. What is the effective way to solve the problem? In other words, what would be the practical function, instead of $e^{at}(b-S_{t})$, for differentiation?

asked Mar 07 '22 at 20:18

Farhad Tahmasebi

I don't know of any nice closed form solution $S_t$ when the diffusion coefficient is $\sigma S_t$. My guess would be there is no "nice" solution for this. – oliverjones Mar 08 '22 at 00:57
This answer reproduces an explicit solution from Wikipedia for the case $b=0.$ There might be more. – Kurt G. Mar 08 '22 at 08:13
I though the only way is to solve it using the Kolmogorov forward pde. No other solution has crossed my mind yet! – Farhad Tahmasebi Mar 08 '22 at 15:17
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@FarhadTahmasebi , in fact, for $b\not=0$ there is also a solution. – Kurt G. Mar 09 '22 at 11:41

How to solve Mean-reverting diffusion

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