I am given a discrete-time process ${X_i}$ where $X_{i+1} = uX_i$ with probability $p$ and $X_{i+1} = dX_i$ with probability $1-p$. $X_0$ is equal to $x_0$ as well. Stated another way, $X_{i+1} = X_iu^{I_i}d^{1-I_i}$ where $I_1, I_2,...$ are iid Bernoulli RVs. Now for $n\geq1$ I need to compute $$E[X_n \mid X_{n-1}]$$ and $$Var(X_n \mid X_{n-1})$$
This seems very simple to accomplish, but I am having trouble. So far, the direction I am taking is the following: $$E[X_n \mid X_{n-1}] = E[X_{n-1}u^{I_{n-1}}d^{1-I_{n-1}} \mid X_{n-2}u^{I_{n-2}}d^{1-I_{n-2}}]$$
Following this, I am not sure if it's possible to bring in probabilities, and use Bayes theorem, or if I am even on the right track here. All help would be appreciated.
$E(X_n|X_{n-1})=E(X_{n-1}u^{I_{n-1}}d^{1-I_{n-1}}|X_{n-1})=X_{n-1}E(u^{I_{n-1}}d^{1-I_{n-1}}|X_{n-1})$ because $X_{n-1}$ can be pulled out of the expectation. Now assuming that the $I_j$'s are independent of the $X_n$'s (without which the question cannot be answered!) we get $E(X_n|X_{n-1})=X_{n-1} (pu+(1-p)d)$. You can use a similar method to calculate $E(X_n^{2}|X_{n-1})$ and this gives you the variance also.