Suppose an investor wants to invest a certain amount of his assets, worth $ dollars, into a particular stock. After enough statistical analysis of the history of the stock, the investor believes that a good model for how the stock will behave for the next few days or weeks is that, for every second, the probability that the stock fluctuations yield an increase of one in the value he currently has invested is u and the probability that it decreases by one is d (where u + d = 1). The investor sets up an automatic trader software that will automatically sell all of his stock as soon as the value he has invested in this stock reaches a limit value L.

Our goal is to compute the probability that the stock achieves the limit valuation. To that end, let V_t be a random variable giving the valuation of the investor's stocks after t seconds have passed. We can define the time it takes for the investor either to go broke or to have his stocks sold by the autotrader as T = min {t : V_t = 0 or V_t = L}. It turns out that T is finite; this means that it is never the case that the stock will fluctuate between 0 and L of valuation forever (namely, the event that it fluctuates forever with never going broke or reaching the limit value has 0 probability; i.e., P(T < ) = 1). Now we want to compute the following: p_$ = P(V_t = L | V₀ = $), namely, the probability that, given that the investor starts with $ value in assets invested in this stock, they end up with the limiting value.

a) What are the values of p₀ and p_L?

b) Write a recurrence for p_$.

c) Solve the recurrence (for simplicity, assume that u d). Your final solution should give p_$ depending only on u (or d), $, and L.

(Hint: a linear homogenous equation)