OFFSET
0,3
COMMENTS
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.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..3323
EXAMPLE
n, a(n), list of paths
n = 0: 1 {(0,0)}
n = 1: 1 {(0,0) -> (1,0)}
n = 2: 2 {(0,0) -> (1,0) -> (2,0), (0,0) -> (1,0) -> (1,1)}
n = 3: 3 {(0,0) -> (1,0) -> (2,0) -> (3,0), (0,0) -> (1,0) -> (2,0) -> (2,1) (0,0) -> (1,0) -> (1,1) -> (2,1)}
CROSSREFS
KEYWORD
nonn
AUTHOR
C. Krishna, Jul 18 2022
STATUS
approved