Use mixture normal to approximate lognormal shocks in dyanre, will Kalman filter be applicable? Please give me some guidance, thank you very much!

Dear Johannes,
Thank you very much for your previous help and guidance.
In my DSGE model, I am trying to make one of the shocks have heteroskedasticity.
for example, return=sigma*eps
eps is the shock and is iid distributed, sigma is a state variable and is time-varying.
after loglinearisation, log(eps) is no longer normal but becomes log normal, so Kalman filter is not applicable.
My solution is: I approximate the log(eps) using finite mixture of normally distributed shocks, then will Kalman filter be able to apply so as to estimate the model?

Thank you again and look forward to hearing from you.
Best wishes,

That sounds very much like what Kim/Shephard/Chib (1998) suggest to do. Theoretically, that should be feasible, but you need to be careful when the linearization is applied.