HardStochastic Processes

Random Walk — Expected Steps

Jane StreetTwo Sigma

A symmetric random walk starts at 3 and stops when it hits 0 or 10 (each step ±1, equally likely). What is the expected number of steps until it stops?

Approach

Expected absorption time satisfies $E_k = 1 + \tfrac12 E_{k-1} + \tfrac12 E_{k+1}$ .

The solution is $k(N-k)$ .