Positivity and Log-Normality in a Mean-Reverting Volatility SDE

Summary

The document examines a mean-reverting volatility process with drift toward a fixed level and diffusion proportional to the current volatility. The author applies Itô’s formula to the logarithm of the process and observes that the resulting drift depends on the reciprocal of volatility, so the logarithm does not appear to follow a normal process. This raises a question about the common description of the process as log-normal mean reversion.

The discussion also asks whether proportional diffusion, expressed through quadratic variation scaling with the square of the process, is enough to establish log-normality or strict positivity. It includes an attempted Euler-scheme simulation and normality checks, but no derivation or resolution. The document therefore identifies a distinction between multiplicative noise and a genuinely log-normal process without proving positivity; its simulation evidence is limited and does not settle the mathematical question.

Key ideas

• The process combines mean-reverting drift with diffusion proportional to its current level.
• Applying Itô’s formula gives the log process a state-dependent drift involving the reciprocal of volatility.
• Multiplicative diffusion alone does not make the log process have an immediately apparent normal law.
• Quadratic variation proportional to the squared process is raised as a possible characterization, but the document does not establish it as sufficient.
• The simulation and normality tests described do not resolve strict positivity or the exact distribution.

Tags

# Log-normal mean reversion SDE


# 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.