# how to extend lognormal model so that $\sigma$ is correlated to $\mu$?

Consider a log-normal model, $dx / x = \mu dt + \sigma dW$, where $W(t)$ is a Wiener process.

Let's say $\mu$ and $\sigma$ change with time, slowly, so we note them by $\mu(t)$ and $\sigma(t)$.

Consider $dx / x$, where the drift rate is $\mu$, and the volatility is $\sigma \sqrt{dt}$. Here, $\mu(t)$ and $\sigma(t)$ is not correlated.

Now, if in some cases the data shows a strong correlation, such as when $\mu(t)$ goes up, $\sigma(t)$ would also go up -- the 2 are almost in a linear relationship, something like $\sigma(t) = \sigma_0 + k \mu(t)$ -- how could I set a model for that?

Of course, I could just put it as $$dx/x = \mu(t) dt + (\sigma_0 + k \mu(t)) dW$$

But I wonder, is there some already established model/methods for such situation? for example for general stochastic model there are HJM, for mean-reverse there are Hull-White, for stock price there is the log-normal.

Is there some model already researched or even better, used in industry, that extend lognormal model $dx / x = \mu dt + \sigma dW$, so that $\mu(t)$ and $\sigma(t)$ would be correlated?

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What do you mean saying mu(t) is correlated to sigma(t)? They are determenistic functions, there can only be case when sigma = f(mu(t)), something you have already mentioned. –  Rustam Aug 16 at 9:25
sigma = f(mu(t)), is what i observed in saving account, i'm wondering if there are also such case in, say, stock, interest rate, and if so, are there already some established models? –  athos Aug 16 at 13:48
As @Rustam notes, "correlation" of deterministic functions in the sense you describe is a special case of allowing $\mu$ and $\sigma$ to have a term structure of arbitrary shape. Since the latter is easy to treat, no one bothers with restricted forms of it.
Now, there quite a few people who deal with models that let $\sigma$ change with $S$. I am thinking in particular of local volatility models, which have an explicit surface $\sigma(S,t)$ to match vanilla option markets. These are used on exotics desks (and the models sometimes have jump terms also).