The uniqueness of solution for stochastic differential equation involved with sign function.

Question

When I read a paper about Levy distribution thoerem (http://www.maphysto.dk/publications/MPS-RR/1998/22.pdf). In the first page, the author mentioned the following:

There is a unique strong solution of this SDE:

$d X^{\lambda}_{t} = -\lambda sign(X^{\lambda}_{t}) dt + d B_{t}$ with $ X^{\lambda}_{0}=0$

$B_{t}$ is the standard Brownian motion. I am confused that how the author directly concludes the uniqueness and existence of the solution. Since $sign(x)$ is not a Lipschitz function with respect to $x$.

A similar question related to the Levy process case is as follows.

$d X^{\lambda}_{t} = -\lambda sign(X^{\lambda}_{t}) dt + d B_{t} +dN_{t} \quad X^{\lambda}_{0}=0 \quad (2)$.

Here $B_{t}$ is the standard Brownian motion. $N_{t}$ is a pure jump Levy process with generating triplet $(0, 0, \nu)$. $\nu$ is the Levy measure.

How to describe the existence and uniqueness of the above equation?? I did not know any results about this. Only I guess when $\nu$ is symmetric, there may be a solution. I am new to the theory of SDE. Any reference are very appreciated.

Answer 1

Well it seems that the correct reference for this result is in the book by Revuz and Yor "Continuous Martingales and Brownian Motion" at the Chapter IX - Stochastic Differential Equation, Section 2 Existence and Uniqueness in the Case of Lipschitz Coefficients precisely treated as an Exercise (2.11) about "Zvonkin's Method".

The result is far from trivial.

The original reference on Zvonkin's method in R & Y 's book seems to be the following paper :

Zvonkin, A.K. "A transformation of the phase space of a process that removes the drift"

Math. USSR Sbornik 2 (1974) 129-149.

I don't know if it is available in english.

Best regards

Answer 2

The result is also mentioned (and proved) in

N. Ikeda, S. Watanabe: Stochastic Differential Equations and Diffusion Processes, Section IV.2 and IV.3.

In fact, one can show that the SDE

$$dX_t = b(X_t) \, dt+ dB_t, \qquad X_0 = x$$

has a (strong) unique solution for any Borel-measurable bounded function $b$.

Answer 3

Thanks for answers. It's very helpful. It mentioned that the existence and uniqueness is true if $b(ｘ)$ is bounded measurable function.

I have an idea (not a proof) for this question. $X_{t}$ is the solution of $dX_{t}=b(X_{t})+dB_{t}$ may be equivalent to the fact that there is a new probability measure $\bar{P}$ such that $$law(X_{t}|\bar{ P})=law(B|P)$$ (1). One of the sufficient condition for latter claim can be given as $$ E(\exp(\int_{0}^{\infty} b(s,\omega))<\infty$$. (2)

If this is true, then that $b$ is bounded is one of the cases.