# asset buying, wealth calculation, almost sure convergence

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 $$0, hence $$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.