I have the following problem:
Consider the stochastic differential equation $$d X_t=\mu(X_t)dt+dB_t,$$ where $B_t$ is a standard Brownian motion, and $X_0=x\in\Re$ a.s.. Assume that the drift $\mu(\cdot)$ is $$\mu(x)=\begin{cases}\mu_1, &\text{if } x>\bar x\\\mu_2, &\text{if }x\le \bar x\end{cases}$$ where $\mu_1>\mu_2>0$ and $\bar x\in\Re$. Since $\mu(\cdot)$ is bounded and Borel, we know that there exists a unique strong solution $\{X_t\}_{t\ge 0}$ to the SDE above. But since it fails the Lipschitz condition ($\mu$ is even discontinuous), the standard results cannot apply.
I am curious about:
(i) Is $\{X_t\}_{t\ge 0}$ a continuous semimartingale?
(ii) Is $\{X_t\}_{t\ge 0}$ a diffusion process?
It seems that I cannot prove (i) and (ii) rigorously. Can anyone give me some advices or references? Many thanks!
Yes. The strong/weak existence implies that the solution is a semimartingale. If you check the Definition 2.1 of Karatzas 1991, it implies that $\mu(X_t)$ is an adapted process. Also $\int^t_0 d B_t$ is trivially a (local) martingale. So, $X_t$ is a sum of a local martingale and an adapted process, which is a semimartingale.
It depends on what do you mean by diffusion process. In the widest sense a diffusion process is said to be a continuous-time Markov process (see the book by Karatzas 1991).