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?


  • $\begingroup$ 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

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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