# Is the solution to an SDE a continuous semimartingale/diffusion process?

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) 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!

• Why the strong solution is unique, $\mu(x)$ doesn't satisfy the Lipschits' condition, if $\mu_1\ne\mu_2$. But if $X_0\ne\overline{x}$, $\tau=\inf\{t>0: X_t=\overline{x}\}$, then on $[0,\tau)$, $X$ may be a diffusion process. Apr 9, 2017 at 2:20
• @JGWang The Lipschitz condition is only a sufficient condition to guarantee existence and uniqueness of a strong solution. There are many different ways to prove the conclusion claimed above, and one possible way is to use the Zvonkin's transformation. See, for example, mathnet.ru/php/…
– OnoL
Apr 9, 2017 at 2:37
• I think this might contain what you are looking for Apr 10, 2017 at 15:04
• @Vincent.W. Thanks a lot. But it seems that the literature does not establish the properties of the unique strong solution (path continuity, semimartingale, etc...).
– OnoL
Apr 11, 2017 at 17:25
• Yes, $X$ is a $\sigma(B)$-semimartingale. You can find this result in Kallenberg, Foundations of Modern Probability. May 15, 2017 at 18:21

1. 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.