Consider a market with $d = 1$ risky asset with prices $(S_n)_{n \geq 0}$ and interest rate $r$. Suppose an investor has initial wealth $X_0 > 0$, consumes $C_n$, and holds $\theta_n$ shares over the time interval $(n - 1, n]$, where $C_n$ and $\theta_n$ are $\mathcal{F}_{n-1}$-measurable. The investor’s goal is to maximize $\mathbb{E}[\sum_{k = 1}^\infty \beta^{k - 1} U(C_k)]$, where $U(x) = \sqrt{x}$ is the investor’s utility function and $0 < \beta < 1$ is the investor’s rate of discounting. Assume $S_n = S_{n-1}\xi_n$ where $(\xi_n)_{n \geq 1}$ are independent copies of the positive random variable $\xi$, and assume that $0 \leq C_n \leq X_{n - 1}$ for all time $n$.
Let $\alpha = \max_t{\mathbb{E}[U(1 + r + t[\xi - (1 + r)])]}$ and let $t^*$ be the maximizer. Assuming that $\alpha \beta < 1$, show that the optimized wealth equation is
$$X_n = X_0 \alpha^{2n}\beta^{2n} \prod_{k = 1}^n (1 + r + t^{*}[\xi_k - (1 + r)])$$
where the investor consumes $C_n = (1 - \alpha^2 \beta^2)X_{n - 1}$ and holds $\theta_n = \alpha^2 \beta^2 t^* X_{n - 1}/S_{n - 1}$ number of shares.
I'm not really sure where to start with this. I believe the wealth dynamics given the consumption and investment policy can be modeled with $$X_n = (1 + r)(X_{n - 1} - C_n) + \theta_n(S_n - S_{n - 1}(1 + r)).$$ If we substitute $C_n, \theta_n$ into the equation, we get
\begin{align*} X_n &= X_{n - 1}[(1 + r) - (1 + r)(1 - \alpha^2 \beta^2 t^{*}) + \alpha^2 \beta^2 t^{*}(\xi_n - (1 + r))] \\ &= X_{n - 1}\alpha^2\beta^2(1 + r + t^{*}[\xi_n - (1 + r)]), \end{align*}
so it's clear how the wealth equation is the one provided, but how does one get those expressions for $C_n, \theta_n$, especially without more information on the distribution of $\xi$?