I have difficulty modeling the following problem with a random walk:
Consider that at time $t=0$ Mike has $M$ dollars. He receives a dollar at each of the times $R_1, R_1+ R_2, R_1+R_2+R_3,\cdots$, where $R_i$s are independent random variables according to probability density function (pdf) $p_1(x)$, and spends a dollar at times $S_1, S_1+ S_2, S_1+S_2+S_3,\cdots$, where $S_i$s are independent random variables according to pdf $p_2(x)$. If he has to spend a dollar but he doesn't have any money, he borrows that.
Question: What is the probability that he doesn't borrow money, from the beginning until the time that he spends the $k^{th}$ dollar?
If I model this by a random walk, then I will have to find the survival probability which may be already in the literature. Let's try model this problem by a random walk: Mike starts at point $x=M$, when he receives a dollar he goes to $x+1$, and when he spends a dollar he goes to $x-1$. Next, we have to find out every time that Mike moves from $x$, what is the probability of moving to right ($x+1$) and what is the probability of moving to left ($x-1$). But, the probability that Mike goes to left or right depends on $x$ and is not fixed. For random walks I have seen these probabilities are fixed.What should I do?
I am wondering if you have faced similar problem and know how I should deal with it. I appreciate any helps regarding that including useful references. Of course, a full answer is the best!
Thank you