First consider a standard GBM process of the form,
$$\frac{dS_t}{S_t} = \mu \, dt+ \sigma \, dW_t$$
but instead of the normal $W_t \sim N(0,1)$ , instead we have that, $$W_t \sim \operatorname{EMG}^-(0,1,\lambda)$$
where $\operatorname{EMG}^-$ is the Exponentially Modified Gaussian$^{[1]}$ distribution corresponding to the random variable $W=X-Y$, where $X\sim N(\cdot)$ and $Y \sim \operatorname{EXP}(\lambda)$. Basically what this distribution gives is a standard normal variable with a slight left skew due to the subtraction of the exponential.
Now that we have the problem set up, two things jump out:
- $W_t$ is not a Wiener process (because it does not satisfy having Gaussian increments) so this is not a GBM.
- Because $\frac{dS_t}{S_t}$ is not a GBM we cannot do ito calculus with it (at least as far as I understand it).
If this was a standard GBM you could do the standard thing of saying, look $\frac{dS_t}{S_t}=\ln(S_t)=f(S_t)$ and then compute the ito derivative of $f(S_t)$ as $$d(f(S_t))=\frac{\partial f}{\partial S}\, dS_t+\frac{1}{2} \frac{\partial^2 f}{\partial S^2} \,dS_t^2$$
So at long last my question is, how do you handle a GBM-like process when the shocks are not iid standard normal? How do you solve for this process $S_t$ and find the first moments of it?
[1]: For more information on the $\operatorname{EMG}^+$ distribution, see here: https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution
EDIT: Doing some more reading since I have posted this leads me to believe that $W_t$ is a semi-martingale and there are methods of dealing with semi-martingales in Ito calculus. A would expect a solution would probably start there. I am currently trying that way to tackle it myself but if anyone else comes up with an answer in the meantime it would be very welcome.
