Let $w_t$ denote a consumer's wealth at time $t$ and $c_t$, the amount she chooses to consume, so her savings exiting this time period are $w_t-c_t$. Given this savings decision, her savings $w_{t+1}$ at time $t+1$ are determined by a random process in which $w_{t+1}=\alpha w_t$ or $w_{t+1}=\beta w_t$, where $\alpha,\beta$ are postitive constants, each with probability $\dfrac{1}{2}$ and independent of past rates of return on her savings. At time $t$, when she is choosing $c_t$, she knows $w_t$ but she doesn’t know anything about future returns except the probabilistic law just given; she does know her past consumption decisions, of course, and past rates of return. Considering a log utility, i.e. $u(t)=ln(c_t)$ and assuming that the consumer is impatient, so that he discounts future utility by a factor $b$ each period, where $0 < b < 1$, then the consumer's decision problem can be written as follows: $$v(c_t)=\max_{c_t \geqslant 0}\sum_{t=0}^{T}b^{t}u(c_t)$$
If the consumer can not borrow, then what is the optimal consumption level for the consumer?
$\underline{Note:}$ The problem is based on David M. Kreps' microeconomic theory book, but it is adjusted to be a finite horizon problem. Kreps, in his book, solves this problem in a fuzzy way for $T=3$, that is not obvious to me. However, due to the fact that I know little of dynamic programming, since I am in the begining of examining this topic, I would appreciate it, if someone could provide a solution in the finite horizon. If the problem is not set in a rigorous way, I would be glad too to see somene making any appropriate change. I beleive that it is a classic problem in the field of economics.