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I'm trying to figure out how to solve the following SDE,

$$ dZ_t = -\kappa(Z_t-\mu)dt + Z_tdW_t $$

It looks really similar to the OU process but applying the integrating factor approach which solves the OU process doesn't seem to quite work out.

Can someone kindly give some pointers?

Thanks!

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  • $\begingroup$ Have a look at this question: math.stackexchange.com/q/592995 ... or just search for "linear SDE", then you will find several questions which are very similar to your question. $\endgroup$ – saz Jun 7 '15 at 14:55
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I suppose we are in the $1$-dimensional case. In order to solve $$ \begin{cases} dZ=−κ(Z−μ)dt+ZdW \\ Z(0) = Z_0 \end{cases} $$ we look for a solution of the form $$ Z(t) = e^{W(t)}y(t), $$ where $y(t)$ is non-stochastic (notice $y(0) = Z_0$). By the Itô product rule and Itô's formula we get $$ dZ = e^{W} dy + d(e^W) \, y \qquad \qquad \quad \qquad \\ = e^W dy + y \, e^W dW + \frac{1}{2} y \, e^W dt. $$ Substituting it into the original SDE, and taking into account the transformation, we have $$ e^W dy + y \, e^W dW + \frac{1}{2} y \, e^W dt = -\kappa (e^W y -\mu) dt + e^W y \, dW, $$ that is $$ dy = -\left( \kappa+ \frac{1}{2} \right) y \, dt + \kappa \mu e^{-W} dt, $$ i.e., a random ODE that has solution $$ y(t) = e^{-(\kappa + \frac{1}{2})t} Z_0 + \kappa \mu \int_0^t e^{-(\kappa + \frac{1}{2})(t-s)} e^{-W(s)} ds. $$ And then you can find $Z(t)$ just substituting y(t) it into the aforementioned equation.

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