Let {Sₙ}n≥0 be a symmetric simple random walk under the measure P. that is, in the notation of Example 2.3.7, P = 1/2. Show tha. {Sₙ²}n≥0 is a P-sub martingale and that
{Sₙ² - n)n≥0 is a P-martingale. Let T = inf{n : Sₙ ∉ (-a, a)), where a ∉ N. Use the Optional Stopping Theorem (applied to a suitable sequence of bounded stopping times) to show that E[T] = a².