Brownian Motion and Related Processes 411 For 0 < s < t, do (M(s), M(t)) have the same joint distribution as (|B(s)|, |B(t)|)? 8.2.4 Use the reflection principle to obtain Pr{M(t) z, B(t) x} = Pr{B(t) 2z - x} = 1 - 2z - x t for 0 < x < m. (M(t) is the maximum defined in (8.19).) Differentiate with respect to x, and then with respect to z, to obtain the joint density function for M(t) and B(t): f M(t),B(t) (z, x) = 2z - x 2 2z - x . t t t 8.2.5 Show that the joint density function for M(t) and Y(t) = M(t) - B(t) is given by f M(t),Y(t) (z, y) = z + y 2 z + y . t t t