Stochastic Processes Cheat Sheet

Brownian motion, Itô's lemma, martingales and random walks.

Brownian motion

W_{t} - W_{s} \sim N (0, t - s)

Cov (W_{s}, W_{t}) = min (s, t)

df = f_{t} d t + f_{x} d X + \frac{1}{2} f_{xx} (d X)^{2}

S_{t} = S_{0} exp ((μ - \frac{1}{2} σ^{2}) t + σ W_{t})

P (reach N from k) = \frac{k}{N}

E [T] = k (N - k)