Brownian Motion and Quadratic Variation

Medium·18 min read·Interactive lab

Stochastic CalculusBrownian MotionQuadratic Variation

Quick Quiz

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}

←Lab