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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!

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  • $\begingroup$ 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. $\endgroup$
    – JGWang
    Apr 9, 2017 at 2:20
  • $\begingroup$ @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/… $\endgroup$
    – OnoL
    Apr 9, 2017 at 2:37
  • $\begingroup$ I think this might contain what you are looking for $\endgroup$
    – Vincent.W.
    Apr 10, 2017 at 15:04
  • $\begingroup$ @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...). $\endgroup$
    – OnoL
    Apr 11, 2017 at 17:25
  • $\begingroup$ Yes, $X$ is a $\sigma(B)$-semimartingale. You can find this result in Kallenberg, Foundations of Modern Probability. $\endgroup$
    – 0xbadf00d
    May 15, 2017 at 18:21

1 Answer 1

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  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.

  2. 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).

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