# Log-normal mean reversion SDE

I study the Tataru-Fisher 2003 LSV model (implemented in the Bbg terminal for FX exotics pricing), the volatility has the following dynamics : $$dV_t = \kappa (1 - V_t) dt + \xi V_t dB_t$$ In the paper it is said that this dynamics is log normal :

We have chosen a lognormal process for the volatility process... the volatility process Vt cannot reach zero

and they use the log-normality of this process for describing the whole study of the model. I've tried to show this log-normality by applying Ito's formula to $$ln(V_t)$$ : $$d(ln(V_t)) = \big(\kappa(\frac{1}{V_t} - 1) - \frac{\xi^2}{2}\big)dt + \xi dB_t$$ So i don't get apparent normal law for $$ln(V_t)$$ process.

Moreover i try to test normality of the process by doing an Euler Scheme diffusion and basic adequation statistics tests don't reject normality assumption. I read in some papers here or here that this process is called log-normal mean reversion. In the first source they caracterize the log-normality diffusion as something where $$d \langle V \rangle _t = V_t^2 \xi^2 dt$$

Do you know how to prove log-normality of such process ? Or at least strict positivity.

• Log normality only comes from the fact that the process dVt depends on Vt. Commented Jul 16 at 12:18
• How can we show that ? I can't see direct link between what you call "dVt depends on Vt" and law of the process Vt. How do you state rigorously this last sentence btw ? Commented Jul 16 at 12:57
• Hi: What you wrote at the top for the dynamics is a version of geometric brownian motion. That's why you get the log normaility. See this for details. www-users.cse.umn.edu/~dodso013/docs/GBM-primer.pdf Commented Jul 16 at 16:41
• Sorry i don't understand what you call "a version of gbm" ? For me the definition of a GBM is the solution to the SDE $dX_t = X_t(\mu dt + \sigma dW_t)$. But here for $dV_t$ the drift isn't factorizable by $V_t$. Commented Jul 17 at 8:35
• Hi Justin: are you talking about the $\kappa (1 - V_t) dt$ term ? and comparing it to the $X_t \mu dt$ term ? It's late and I'm having trouble following. I could have made a mistake somewhere ? Commented Jul 17 at 9:19