# Strong solution for geometric brownian motion with varying drift and volatility

I have an equation of the form:

$$dX_{t}=\mu(X_{t})X_{t}dt+\sigma(X_{t})X_tdZ_{t}$$

I know that if I wrote it as $dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dZ_{t}$, I would need strong assumptions on the functions $\mu(\cdot)$ and $\sigma(\cdot)$ to guarantee that a strong solutions exists (usually a Lipschitz thing).

But since I am putting more structure on my equation, by assuming that the drift and volatility can be written as $\mu(X_{t})X_{t}$ and $\sigma(X_{t})X_t$, I was wondering if there is any other theorem (with weaker conditions on $\mu(\cdot)$ and $\sigma(\cdot)$) that guarantees a strong solution.

Thanks!

## 1 Answer

A side remark on terminology.

It seems more accurate to refer to geometric Brownian motion as log Brownian motion, since the logarithm of a geometric Brownian motion is a Brownian motion. This would be analogous to the naming of lognormal random variables, whose log is normally distributed.

Back to the question at hand.

I don't see a significant advantage to that structure, in terms of regularity. In fact, if we approach this process as if it were a log Brownian motion, Ito's formula implies that $Y_t:=\log(X_t)$ satisfies the SDE: $$d Y_t = \left( \mu(\exp(Y_t)) - \frac{1}{2} \sigma(\exp(Y_t))^2 \right) dt + \sigma(\exp(Y_t)) dZ_t$$ Thus, even if the coefficients $\mu(\cdot)$ and $\sigma(\cdot)$ are globally Lipschitz continuous in the original variables, when viewed in logarithmic scaling, the coefficients of the SDE become locally Lipschitz continuous.

Note that in the original variables, this is also true; i.e., the drift and volatility are not necessarily globally Lipschitz continuous even when the coefficients $\mu(\cdot)$ and $\sigma(\cdot)$ are.

• It is not obvious to me that there some advantage either. I asked because I know that there are some results that relax the Lipschitz condition, but they require the volatility $\sigma(\cdot)$ to be larger than a constant $c>0$. This is clearly violated for the log brownian and those results are not of much use. Since these models (log brownian motions) are very popular in economics and finance, I think there may be some more general results on this case. – Pcw. Sep 6 '16 at 15:29
• That sounds like an ellipticity assumption, which is quite standard for one-dimensional SDEs; see: Zvonkin, A.K. (1974): A transformation of the phase space of a process that removes the drift. Math. USSR Sbornik, 93, No. 1, 129–149. To put Zvonkin's result in context, see Chapter 1 of: Cherny and Engelbert (2005): Singular Stochastic Differential Equations. Springer. The latter reference gives a thorough overview of the state of the art for these types of results, I think. – Nawaf Bou-Rabee Sep 6 '16 at 15:41
• That Cherny and Engelbert (2005) was very helpful, thanks. Although I didn't find any result on strong solutions, there is a lot about weak solutions and uniqueness in law. Do you happen to know such a reference that also deals with reflected processes? In fact, I am dealing with a geometric brownian that has a reflecting barrier. – Pcw. Sep 6 '16 at 17:44