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At the beginning of each year, you can buy assets for $1$ unit of money that are worth $a$ unit of money at the end of the year or stocks that are worth a random amount $v\ge 0$. If you always invest a fixed proportion $p$ of your wealth in buying assets, then your wealth at the end of the year $n+1$ is $w_{n+1}=\left(xp+(1-p)v_n\right)w_n$, where $v_1,v_2,\dots, v_n$ are i.i.d with distribution $v$ which has $\mathbb E\left[v^{\pm 2}\right]<\infty$.

Could anyone tell me how to show that

(1) $\frac{1}{n}\log(w_n)\to b(p)$ a.s for some $b(p)$.

(2) Show that $b''(p)\le 0$ for some $0<p<1$, hence $c(p)$ is concave.

(3) Suppose $\mathbb P(v=1)=\mathbb P(v=4)=\frac{1}{2}$, then find the optimal $p$ as a function of $x$.

I am not sure, how to proceed and what tools from probability theory I need to use to solve this problem. Thanks for any help.

https://math.stackexchange.com/questions/4036817/asset-buying-wealth-calculation-almost-sure-convergence

I have asked there too, let me know from which website I should delete the question.

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