cp's OEIS Frontend

A monotonic lattice path is one which starts at (0,0) consists entirely of steps of length 1, moving only rightwards (x,y) -> (x+1,y) or upwards (x,y) -> (x,y+1).

a(n+1) <= 2*a(n).

Consider a stock with an initial price of x. At every time step it either gains or loses 50% of its value with equal probability (x -> 0.5*x or x-> 1.5*x). An investor buys this stock and sells it if its current price exceeds its initial price and holds onto it otherwise. What is the probability the stock is not sold after n time steps?

The answer to this is a(n) / 2^n. Limit_{n->oo} a(n) / 2^n = c ~ 0.2863153965300.

Notice that the expected value of the stock remains constant after each time step ((0.5*x + 1.5*x)/2 = x), but the expected log of the stock price is constantly decreasing, at a rate of (1/2) * log(3/4), each time step.

Using the central limit theorem, there is a 100% likelihood that the stock price falls below any arbitrarily small positive value, p > 0 in the long run.

Since there is a probability 1-c that the investment yields a profit, a probability c that the stock is never sold, and the stock maintains a constant expected value, the expected profit if sold is cx/(1-c) ~ 0.401179169535*x.