Exercise: Quadratic Variation by Simulation

Prerequisites: Brownian Motion

Problem

The lesson states that the quadratic variation of Brownian motion on $[0, T]$ is deterministic:

\sum_{i=0}^{n-1}(W_{t_{i+1}} - W_{t_i})^2 \xrightarrow{L^2} T \quad \text{as } \max_i(t_{i+1} - t_i) \to 0

This exercise asks you to verify the statement numerically and quantify the rate of convergence.

Simulate a single Brownian path on $[0, 1]$ using a uniform grid of $n$ points. For $n = 10, 100, 1000, 10000$ , compute the sample quadratic variation $Q_n = \sum_{i=0}^{n-1}(W_{t_{i+1}} - W_{t_i})^2$ . Report each value.
Compare to the first-order variation $V_n = \sum_{i=0}^{n-1}|W_{t_{i+1}} - W_{t_i}|$ at the same grid sizes. Describe how $V_n$ scales with $n$ , and contrast with $Q_n$ .
For the $n = 1000$ case, repeat the simulation $N = 1000$ times and plot (or summarise statistics of) the empirical distribution of $Q_n$ . Is the limit exactly $T = 1$ , or does it fluctuate? Report the sample mean and standard deviation, and compare the standard deviation to the theoretical $\sqrt{2T/n}$ prediction.
Conceptually: explain in one paragraph why $V_n \to \infty$ is consistent with $Q_n \to T$ .

Use rng.standard_normal(n) to generate the i.i.d. Gaussian increments. For part 3, the theoretical variance of

Q_n

under a uniform grid is

\operatorname{Var}(Q_n) = 2T^2/n

Jump to the solution when you're ready.