Measuring Quadratic Variation from Simulated Paths

What a Simulated Quadratic-Variation Sum Measures

A Brownian path is simulated at finitely many time points, so its quadratic variation is not observed as a continuous-time limit. Instead, one computes a squared-increment sum on a grid and compares it with the elapsed time. Refining the grid gives a sequence of statistics whose expected value stays fixed while their variability decreases. This distinction matters: a single finite-grid sum can differ noticeably from elapsed time without contradicting Brownian quadratic variation.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). Its increments on disjoint intervals are independent, and \(B_{t_i}-B_{t_{i-1}}\) has distribution \(N(0,t_i-t_{i-1})\). An Ito integral, if used, is defined for a progressively measurable integrand \(H_s\) in \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). No stochastic integral is needed for the finite-grid calculation below.

The target \(T\) is elapsed time, not a value that each squared increment must equal. The rule \(dB_t\,dB_t=dt\) summarizes the limiting behavior of these sums as the mesh tends to zero; it is not an assertion that \((\Delta B_i)^2=\Delta t_i\) on every interval. In particular, the variance of a finite-grid statistic quantifies the fluctuations that remain in a simulation.

Finite-Grid Mean, Variance, and Convergence

Proof. Each increment is independent of the other increments and has distribution \(N(0,\Delta t_i)\). If \(Z\sim N(0,v)\), then \(\mathbb{E}[Z^2]=v\) and \(\mathbb{E}[Z^4]=3v^2\). Therefore \(\mathbb{E}[(\Delta B_i)^2]=\Delta t_i\), and

Linearity of expectation gives \(\mathbb{E}[Q_\Pi]=\sum_i\Delta t_i=T\). Independence of the increments implies independence of their squares, so the variance of their sum is the sum of their variances. This gives the stated variance. Since the mean is \(T\),

where \(|\Pi|=\max_i\Delta t_i\) and \((\Delta t_i)^2\le|\Pi|\Delta t_i\). The right side tends to zero when the mesh tends to zero, which is precisely convergence in \(L^2(\mathbb{P})\). \(\square\)

For equal intervals, \(\operatorname{Var}(Q_n)=2T^2/n\). Consequently the standard deviation is \(T\sqrt{2/n}\): doubling the number of grid intervals decreases the standard deviation by a factor of \(1/\sqrt{2}\), not by a factor of \(1/2\). This is a statement about the distribution across simulated paths. It does not guarantee that the statistics from one particular sequence of simulated paths decrease monotonically toward \(T\).

Computing a Nested Simulated Path Record

For an actual simulation, generate Brownian increments on a grid and add them cumulatively to obtain the path values. A nested grid retains the old time points when new ones are inserted. The increments on the finer grid then split the coarser increments: the two finer increments within an interval must sum to that interval's original increment. The following exact decimal record illustrates the calculation. Treat the displayed values as the simulated increments being analyzed; the arithmetic is exact for those displayed values.

A useful implementation check is to verify the nesting identity before comparing levels. If an old increment is \(\delta\) and its two children are \(\delta_1,\delta_2\), confirm \(\delta_1+\delta_2=\delta\). Then recompute the statistic directly from the increments at each level. Do not infer the finer statistic by merely copying the coarser one: in general, \(\delta_1^2+\delta_2^2\) differs from \((\delta_1+\delta_2)^2\).

Averaging Across Simulated Paths

Grid refinement and replication address different sources of variation. Refinement decreases the variance of \(Q_n\) for an individual path. Averaging across independent simulated paths decreases the variance of the estimated mean for a fixed grid. To see the second effect, let \(Q_n^{(1)},\ldots,Q_n^{(M)}\) be the statistics from \(M\) independent Brownian paths on the same grid, and let \(\overline Q_{n,M}=M^{-1}\sum_{j=1}^M Q_n^{(j)}\).

Proof. Each \(Q_n^{(j)}\) has expectation \(T\) and variance \(2T^2/n\) by the theorem. Therefore linearity gives \(\mathbb{E}[\overline Q_{n,M}]=M^{-1}\sum_{j=1}^M T=T\). The path simulations are independent, so their statistics are independent and the variance of their sum is the sum of their variances. Thus

This calculation includes \(M=1\), when it reduces to the single-path variance. \(\square\)

Common Pitfalls and Interpretation

Do not treat the target as a pathwise identity. A mistaken calculation may replace every \((\Delta B_i)^2\) by \(\Delta t_i\), concluding that \(Q_\Pi=T\) on every grid. The correction is that each increment is random: its squared value has mean \(\Delta t_i\), but it is not fixed at that value. For instance, on a single interval of length \(1/4\), an increment of \(0.5\) has square \(0.25\), while an increment of \(0.25\) has square \(0.0625\). Both are possible values of an increment with distribution \(N(0,1/4)\); neither changes the expected square \(1/4\).

Do not confuse convergence in mean square with one-run behavior. The theorem's \(L^2\) conclusion says that the expected squared error tends to zero as the mesh shrinks. It does not imply that the values from one finite run must approach the target monotonically. Compare several independently simulated paths, or report the grid statistic alongside its theoretical standard deviation. If the average over paths is being used, state both the number of paths and the number of time intervals: its variance depends on their product \(nM\).

For equal grids on \([0,T]\), the identity \(\operatorname{Var}(Q_n)=2T^2/n\) also gives a diagnostic for code. If simulated values across many independent paths have a mean far from \(T\), check the increment scaling: increments over a step \(T/n\) should have variance \(T/n\), not variance \(T\). If the mean is plausible but the spread is too large, check whether the grid has the intended number of intervals and whether increments are being squared before summation.