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.



Stochastic process with discontinuous drift
You are absolutely on the right track, and in fact, your change of measure is well-defined. In order to fully leverage this change of measure, one must *eliminate* the time integrals that appear in it. This is nicely done in **Brownian Motion and Stochastic Calculus** by Ioannis Karatzas and Steven E. Shreve (1991); see in particular page 439 of Chapter 6 entitled *Paul Levy's Theory of Brownian Local Time*. 


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


