I'm currently working on a problem where I am trying to model a stochastic process where a stochastic random variable is divided by a deterministic random variable following a normal distribution. Its been a while wince I had a look at stochastic calc so apologies in advance if my math is a bit rusty!
Consider a simple stochastic process with no drift that satisfies the following SDE
Y(t) = X(t)
dX(t) = $\sigma$dB(t) where B(t) is Brownian motion and B(t) ~ N($0$,t).
Solving such a process would give me Y(t) = Y$_0$ + $\sigma$B(t)
I'm currently trying to find out what the resultant stochastic process would be if Y(t) = $\frac{X(t)}{Z}$ where Z is a Normal distribution with mean $\mu_z$ and variance $\sigma_z^2$ i.e Z ~ N($\mu_z$,$\sigma_z^2$).
How would one apply Ito's lemma in such a case?