# Is this geometric Brownian Motion?

The SDE for GBM is usually specified as:

$$dX(t) = X(t)[\mu dt + \sigma dW(t)]$$

If we model diffusion as stochastic, is the following still GBM?

$$dX(t) = X(t)[\mu dt + \sigma_t dW(t)]$$

$\sigma_t$ is stochastic and is driven to an independent Brownian motion.

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$\sigma_t$ deterministic ? – mike Dec 12 '12 at 10:42
$\sigma(t)$ is stochastic and has a separate Wiener process inside of it. – Jase Dec 12 '12 at 11:34
Then you won't get geometric brownian motion. An example is $\sigma_t = \frac 1 {\sqrt{X_t}}$ giving you a well known square root diffusion. (If $\sigma_t$ is deterministic you get a process whose log in has gaussian independent increments which are not stationary) – mike Dec 12 '12 at 13:16