I am investigating a simple model related to the random walk (Just that you work with the product here).
What you do is setting $S_0=1$ and $S_{k}=u\cdot S_{k-1}$ with some probability $p$ and $S_{k}=d\cdot S_{k-1}$ with some probability $q=1-p$. Those choices are of course independent. Therefore $S_n=\prod _{k=1}^n X_k$ with $X_k \in \{u,d\}$
Now the question is $\operatorname{Cov}(S_n,S_l)=?$
Also I would like to know a way how to test some formula I found for this with mathematica if that exists. For example I generated two tables $A,B$ that simulate $S_n$ for $n=10$ (with $100.000$ values each). Now $\text{Variance}(A)$ gives me (almost) the same result in mathematica as $\text{Variance}(B)$ or $\text{Covariance}(A,A)$. But even plugging in $\text{Covariance}(A,B)$ I start getting really tiny values, is that correct?
Now I will provide some own thoughts I did about this problem:
Certainly you have $P(S_n=u^k d^{n-k})=\binom{n}{k} p^k (1-p)^{n-k}$.
With this you can easily see that
$E(S_n)=\sum _{k=0}^n u^k d^{n-k} P(S_n=u^k d^{n-k}) = (d q+p u)^n$
and simillary
$E(S_n^2)=(d^2 q+p u^2)^n$
This also provides the variance, but the covariance seems very tough (at least for me).