# Stochastic process with discontinuous drift

While studying a portfolio optimization problem, I came across the process $$dX(t) = X(t)\,\Big(\,\big(\mu - \alpha\,1_{\{X(t)\,\geq\,C\}}\big)\,dt + \sigma\,dW(t) \Big)$$ which has a discountinuous drift: once the process crosses a pre-defined and fixed threshold $C>0$, it obtains an additional negativ drift of $-\alpha$.

I would like to understand if the SDE above has a solution and/or if it is possible to obtain the distribution/moments of the process at a given point in time.

Has anybody seen something similar or point me to relevant results in the literature? Any comments, ideas and suggestions are highly appreciated. Thank you!

EDIT 1

We can assume $\mu>0$ and $\alpha>0$.

EDIT 2

I would like to propose a solution approach using Girsanov's theorem: $$\begin{eqnarray*} dX(t) &=& X(t)\,\Big(\,\big(\mu - \alpha\,1_{\{X(t)\,\geq\,C\}}\big)\,dt + \sigma\,dW(t) \Big) \\ &=& X(t)\,\Big(\,\mu\,dt + \sigma\,\Big(dW(t) - \frac{\alpha}{\sigma}\,1_{\{X(t)\,\geq\,C\}}\,dt\Big) \Big) \\ &=& X(t)\,\Big(\,\mu\,dt + \sigma\,d\tilde{W}(t) \Big) \\ \end{eqnarray*}$$ with Brownian motion $\tilde{W}(t)=W(t)+\int_{0}^{t}\gamma(s)\,ds$, associated measure $$\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}} \bigg|_{\mathcal{F}_{t}} = e^{-\int_{0}^{t}\gamma(s)\,dW(s) - \frac{1}{2}\int_{0}^{t}\gamma(s)^{2}ds}$$ and $\gamma(t):=-\frac{\alpha}{\sigma}1_{\{X(t)\,\geq\,C\}}$. The SDE under $\tilde{\mathbb{P}}$ has a (strong) solution, which would imply a (strong) solution for the original problem under $\mathbb{P}$.

The pressing question is of course if this change of measure is well-defined. Novikov's condition clearly holds, but I might have missed a measurability/integrability criterion.

• I guess $\mu>0$, but is $\alpha$ < or >$\mu$ ? – Jean Duchon Jul 7 '16 at 8:00
• @JeanDuchon Yes, $\mu>0$. $\alpha$ does not stand in a natural realtion to $\mu$, but if either of the cases $\alpha\leq\mu$ or $\alpha\geq\mu$ is easier to solve, we can simply assume this to be true. – Mark Jul 12 '16 at 7:03

Minor point: I think your derivation would be more transparent if you apply Ito's formula to $\log X(t)$ and then use Girsanov.