Calculating Quadratic Variation With a Drift Term

For

X_t=A_t+\sigma W_t

with continuous finite-variation drift

A

and constant

\sigma

, the quadratic variation is

[X]_t=\sigma^2t

. The drift contributes zero squared variation and zero covariation with the Brownian part. For

X_t=A_t+\int_0^t b_s\,dW_s

, the corresponding formula is

[X]_t=\int_0^t b_s^2\,ds

under the usual stochastic-integral assumptions.

Bound the drift and mixed terms

For a partition $\pi$ of $[0,T]$ , expand

\sum_i(\Delta X_i)^2=\sum_i(\Delta A_i)^2+ 2\sigma\sum_i\Delta A_i\Delta W_i+\sigma^2\sum_i(\Delta W_i)^2.

Because $A$ is continuous with finite variation,

\sum_i(\Delta A_i)^2 \le \max_i|\Delta A_i|\,\operatorname{TV}_{[0,T]}(A) \longrightarrow0.

The cross sum is bounded in absolute value by

\left(\sum_i(\Delta A_i)^2\right)^{1/2} \left(\sum_i(\Delta W_i)^2\right)^{1/2}

by Cauchy–Schwarz. The first factor tends to zero and the second converges in probability to

\sqrt T

, so the cross sum vanishes in probability. The last sum converges in probability to

\sigma^2T

. This argument also works for continuous random finite-variation drift, because the first bound holds path by path. The drift and the Brownian motion need not be independent.

For the mixed sum, Cauchy–Schwarz bounds its absolute value by the square root of the squared-drift sum times the square root of the squared-Brownian sum. The first factor tends to zero because $A$ is continuous with finite variation. The second stays bounded in probability because it converges to $\sqrt T$ . This is enough to show the mixed term vanishes even if the increments have different signs on every interval. A large net drift over the year can therefore coexist with small quadratic variation.

For an original exercise, take $X_t=3t+0.4W_t$ . At $T=2$ , the deterministic move is $6$ , while $[X]_2=0.4^2\cdot2=0.32$ . The covariation with the underlying Brownian motion is $[X,W]_2=0.4\cdot2=0.8$ . You can calculate it directly by multiplying each increment of $X$ by its corresponding Brownian increment and taking the limit. Quadratic variation squares the Brownian loading and covariation takes the product of the two loadings.

Now suppose volatility is

0.4

in the first year and

0.2

in the second. The same time horizon then has quadratic variation

0.4^2+0.2^2=0.20

. The average annual variance is

0.10

, and the corresponding annualized volatility is

\sqrt{0.10}\approx0.316

. Averaging the two volatility levels first gives

0.30

, which answers a different question.

The same calculation applies to the log price in a geometric Brownian motion model. A change in the expected return modifies its drift, while the quadratic variation of the log price remains

\sigma^2 T

for constant volatility. This distinction makes finely sampled log returns useful for estimating variance under the model.

Calculating Quadratic Variation With a Drift Term

Bound the drift and mixed terms

Explore a Brownian path at different resolutions

References and next steps