Random WALK, BROWNIAN MOTION and SDEs Continuation…

First Passage Time (Escape Time) Let 𝑎,𝑏 be positive real numbers, and consider the first time the random walk or the brownian motion starting at 0 reaches the boundary of the interval [−𝑏,𝑎]. Proposition 1: The probability that the escape happens at 𝑎 (rather than −𝑏) is exactly 𝑏 𝑎+𝑏

First Passage Time (Escape Time) Proposition 2: The expected length of the escape time from [−𝑏,𝑎] is 𝑎𝑏. Proposition 3: The Arcsine Law of Brownian motion holds that for 0≤ 𝑡 1 ≤ 𝑡 2 , the probability that a path does not cross zero in the time interval [𝑡 1 , 𝑡 2 ] is 2 𝜋 arcsin 𝑡 1 𝑡 2

First Passage Time (Escape Time) Group exercise: Provide numerical examples for propositions 1, 2 and 3 in MS Excel.

“Brownian motion is a model of random behavior, proposed by Robert Brown (biologist) in 1827. His initial interest was to understand the erratic movement of pollen particles floating on the surface of water, buffeted by nearby molecules.”