Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 27J (Stochastic Financial Models)

A standard Brownian motion is a real process $(B_t{)}_{t\ge 0}$ with $B_0=0$ almost surely, continuous paths, independent increments, and $B_t-B_s\sim N(0,t-s)$ for $0\le s<t$. If a filtration is specified, require adaptedness and independence of each increment from the past sigma-algebra. Thus the increments are centered Gaussian with variance equal to elapsed time.

The scaled process starts at zero and has continuous paths. Disjoint time intervals map to disjoint intervals under $t\mapsto c^{2}t$, preserving independence of increments, andThis holds for either sign of $c\ne 0$. Hence $\boxed{{\tilde{B}}_t=c^{-1}{B}_{c^{2}t}\text{ is standard Brownian motion}}$.

For $c>0$, Brownian scaling and the bijection $t\mapsto c^{2}t$ of $[0,\infty )$ give $Z\stackrel{d}{=}{c}^{-1}Z$, or $\boxed{cZ\stackrel{d}{=}Z}$. The supremum is a measurable extended random variable because continuity allows the supremum over rational times, and $Z\ge 0$.

For $z>0$, the function $F(z)=\mathbb{P}(Z\le z)$ must satisfy $F(z)=F(z/c)$ for every $c>0$, so it is constant on $(0,\infty )$. Letting $z\downarrow 0$ and $z\uparrow \infty$ shows $\mathbb{P}(Z=0)=\mathbb{P}(Z<\infty )$, hence $\boxed{Z\in \{0,+\infty \}\text{ almost surely}}$. This scaling argument alone does not decide the mass at zero; part (d) does.

For $a>0$, the Brownian reflection principle gives $\mathbb{P}({\sup }_{0\le s\le T}{B}_s\ge a)=2\mathbb{P}(B_T\ge a)$. Reflection of the path after its first hit of $a$ preserves its law and pairs paths ending below $a$ with paths ending above $a$, proving this identity. ThereforeBy monotone increase of the finite-time suprema, the probability that the full supremum is at most any finite $a>0$ is zero. Taking a countable union over integer $a$ gives $\boxed{\mathbb{P}(Z=+\infty )=1}$.