Paper 2, Section II, 27 J27 \mathrm{~J}

Stochastic Financial Models
Part II, 2017

(a) What is a Brownian motion?

(b) Let (Bt,t0)\left(B_{t}, t \geqslant 0\right) be a Brownian motion. Show that the process B~t:=1cBc2t\tilde{B}_{t}:=\frac{1}{c} B_{c^{2} t}, cR\{0}c \in \mathbb{R} \backslash\{0\}, is also a Brownian motion.

(c) Let Z:=supt0BtZ:=\sup _{t \geqslant 0} B_{t}. Show that cZ=(d)Zc Z \stackrel{(d)}{=} Z for all c>0c>0 (i.e. cZc Z and ZZ have the same laws). Conclude that Z{0,+}Z \in\{0,+\infty\} a.s.

(d) Show that P[Z=+]=1\mathbb{P}[Z=+\infty]=1.