Brownian Bridge™

1. Which of the following is NOT one of the four defining properties of standard Brownian motion?

W_0 = 0

almost surelyStationary independent increments with

W_t - W_s \sim \mathcal{N}(0, t-s)

Almost surely continuous pathsAlmost surely differentiable paths

2. Let $Q_n = \sum_{i=1}^n (W_{t_i} - W_{t_{i-1}})^2$ over an equal-mesh partition of $[0,t]$ with mesh $h = t/n$ . What is $\mathrm{Var}(Q_n)$ ?

2t^2

2t^2/n

t^2/n

2h

3. Why can the Lebesgue–Stieltjes integral $\int_0^T f_t \, dW_t$ not be defined pathwise for Brownian motion?

Brownian motion is not measurableBrownian motion has infinite quadratic variationBrownian motion has infinite total variationBrownian motion is not a martingale

4. For a continuously differentiable function $f: [0,T] \to \mathbb{R}$ , the quadratic variation $[f]_T = 0$ .

truefalse

5. Lévy's characterisation theorem states that a continuous local martingale $M$ with $M_0 = 0$ is a standard Brownian motion if and only if:

\mathbb{E}[M_t^2] = t

for all

t

[M]_t = t

for all

t \geq 0

M_t \sim \mathcal{N}(0, t)

for each fixed

t

M

has independent increments

6. Which of the following processes is a martingale?

W_t^2

W_t^2 - t

W_t^2 + t

e^{W_t}

Quiz: Brownian Motion and Quadratic Variation

Quick Quiz