Each day, a very volatile stock rises 70% or drops 50% in price, with equal probabilities and different days independent. Suppose a hedge fund manager always invests a fraction $\alpha$ of her current fortune into the stock each day. Let $Y_n$ be her fortune after $n$ days, starting from an initial fortune of $Y_0 = 100$, find the function $g(\alpha)$ such that $(\log Y_n)/n \rightarrow g(\alpha)$ with probability 1 as $n\rightarrow \infty$, and prove that $g(\alpha)$ is maximized when $\alpha=2/7$.
Here is what I tried: We have $Y_1 = (1-\alpha)Y_0+\alpha Y_0\cdot 1.7^{U_1}\cdot 0.5^{1-U_1} = Y_0(1+\alpha (1.7^{U_1}\cdot 0.5^{1-U_1}-1))$ where $U_1$ is the Bernoulli variable.
Hence we have $Y_n = Y_0\prod_{k=1}^n(1+\alpha (1.7^{U_k}\cdot 0.5^{1-U_k}-1))$, where $U_k$ is the indicator variable of the event that the stock goes up on the k-th day. But this expression is too complicated for me to do infer anything on $(\log Y_n)/n$.
Any help would be helpful, thanks in advance.