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How do I solve this stochastic differential equation?

  • Thread starter gangsta316
  • Start date
  • #1
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Homework Statement


I am trying to solve this
\begin{align}
d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t
\end{align}

where $b$ is a constant.

Note that I have the answer here and can provide it if necessary. But I want to know how one would come up with it.

Homework Equations




The Attempt at a Solution


I do not know many general methods for finding analytical expressions for the solutions of SDEs. The main method I know is this. Apply Ito's formula to $h(X_t)$ and where $d X_t$ appears, of course use the SDE given. Then you hope that you can choose h such that both the integrands are equal to a constant, or don't depend on $X_t$. Then you'd have an exact expression for $h(X_t) - h(X_0)$ and you would be able to proceed from there. That method did not seem to work here. Another method that I tried is to hope that we can get $X_t = f(W_t)$. We apply Ito's formula to $f(W_t)$ and match the integrands to those in $dX_t$, replace $X_t$ everywhere with $f(W_t)$. But I can't seem to match both integrands. I matched the $dW_t$ term to get $f(W_t) = sin(b W_t)$. But this will not match with the $dt$ integrand. So both of the methods that I know did not work. How can I solve it? Many thanks.
 

Answers and Replies

  • #2
18,178
7,768
Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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