Probability Density Functions for Random Walk

If X_t is a random walk that starts from 0 at t=0, X_t​~N(μt,σ²t). Its probability density is given by

p(x, t) = \frac{1}{\sqrt{2\pi\sigma^2 t}}e^{\frac{-(x-\mu t)^2}{2\sigma^2t}}

For a pure Brownian motion (which is a continuous RW), μ = 0 and σ = 1.

p(z, t) = \frac{1}{\sqrt{2\pi t}}e^{\frac{-z^2}{2t}}

This PDF satisfies the diffusion equation.

For a random walk that starts elsewhere, its probability density is given by

p(x_T, T; x_0, t_0) = \frac{1}{\sqrt{2\pi\sigma^2(T-t_0)}}e^{\frac{-[(x_T - x_0)-\mu(T-t_0)]^2}{2\sigma^2(T-t_0)}}

P(x_T, T; x₀, t₀) = probability of being at x_T at time T, given that we started at x₀ at time t₀.