What is the canonical way of simulating discrete time stochastic dynamical systems in Mathematica using the new functionality of Random processes?
To take a concrete example, lets consider the optimal gambling problem. A gambler comes to a casino with an initial fortune $x_1$ and let $X_n$ denote his fortune at time $n$. At each time he bets a fraction $\alpha x_n$. At each time he wins with probability $p$ and loses with probability $(1-p)$. Thus, the dynamics of the system can be written as:
$$ X_{n+1} = \begin{cases} (1+\alpha) X_n, & \text{with probability } p \\ (1-\alpha) X_n, & \text{with probability } 1-p \\ \end{cases} $$
I want to simulate this system for 20 time steps with $x_1 = 12$, $p=0.8$, and then plot 50 sample paths and the mean value of $X_{20}$.