Brownian Scaling and Quadratic Variation

Scaling Time and Amplitude

Brownian motion has fluctuations of size proportional to the square root of elapsed time. Consequently, speeding up its clock changes its quadratic variation even when its amplitude is unchanged. The aim here is to calculate that change directly from partition sums and then check it using Brownian scaling and increment variances.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\) with respect to \((\mathcal{F}_t)\). Fix a finite horizon \(T>0\), a real amplitude \(a\), and a time-scaling parameter \(c\ge0\). Define \(X_t=aB_{ct}\) for \(0\le t\le T\). When \(c=0\), this is the constant-zero process because \(B_0=0\); when \(a=0\), it is constant for any \(c\). We include both cases explicitly.

The value of \(c\) changes the Brownian time increment from \(t_i-t_{i-1}\) to \(ct_i-ct_{i-1}=c\Delta t_i\). The amplitude \(a\) multiplies each path increment by \(a\), so it multiplies each squared increment by \(a^2\). Both effects must be included.

Quadratic Variation from Partition Sums

Proof. If \(c=0\) or \(a=0\), every increment of \(X\) is zero, and the stated mean, variance, and limit are all zero. Suppose now \(c>0\) and \(a\ne0\). Brownian increments over the disjoint intervals \([ct_{i-1},ct_i]\) are independent and have distributions \(N(0,c\Delta t_i)\). Therefore \(\Delta X_i\) are independent centered normal random variables with variance \(a^2c\Delta t_i\). A centered normal random variable with variance \(v\) has second moment \(v\) and fourth moment \(3v^2\). It follows that \(\mathbb{E}[(\Delta X_i)^2]=a^2c\Delta t_i\) and \(\operatorname{Var}((\Delta X_i)^2)=3(a^2c\Delta t_i)^2-(a^2c\Delta t_i)^2=2a^4c^2(\Delta t_i)^2\). Independence of the increments makes their squares independent, so expectations and variances add. Hence \(\mathbb{E}[Q_\Pi(X)]=a^2c\sum_i\Delta t_i=a^2cT\), and \(\operatorname{Var}(Q_\Pi(X))=2a^4c^2\sum_i(\Delta t_i)^2\). Since \(\Delta t_i\le|\Pi|\), we have \(\sum_i(\Delta t_i)^2\le|\Pi|\sum_i\Delta t_i=T|\Pi|\). Thus \(\mathbb{E}[(Q_\Pi(X)-a^2cT)^2]=\operatorname{Var}(Q_\Pi(X))\le2a^4c^2T|\Pi|\to0\). This is convergence in \(L^2(\mathbb{P})\), and therefore also convergence in probability. \(\square\)

The theorem obtains the limit from the increments at the original time points. Equivalently, the corresponding Brownian partition points are \(ct_0,\ldots,ct_n\), a partition of \([0,cT]\) with mesh \(c|\Pi|\) when \(c>0\). The scaled process therefore accumulates the same Brownian quadratic variation over the longer interval, multiplied by \(a^2\).

Brownian Space-Time Scaling

For \(c>0\), define \(\widetilde B_t=B_{ct}/\sqrt{c}\) and the time-changed filtration \(\mathcal{G}_t=\mathcal{F}_{ct}\). Brownian scaling says that \(\widetilde B_t\) is itself a standard Brownian motion with respect to \((\mathcal{G}_t)\). This is a statement about the distribution of the rescaled process, not a pathwise identity between \(B_{ct}\) and \(\sqrt{c}B_t\).

Proof. Continuity follows from continuity of \(B\), and \(\widetilde B_0=0\). For \(s<t\), the increment equals \((B_{ct}-B_{cs})/\sqrt{c}\). The numerator is independent of \(\mathcal{F}_{cs}=\mathcal{G}_s\) and is normally distributed with mean zero and variance \(ct-cs=c(t-s)\). Dividing by \(\sqrt{c}\) gives a centered normal variable with variance \(t-s\), independent of \(\mathcal{G}_s\). Increments over disjoint intervals remain independent because the corresponding Brownian increments are independent. These properties, together with continuity and the initial value, are the defining properties of standard Brownian motion. \(\square\)

Since \(B_{ct}=\sqrt{c}\,\widetilde B_t\), the process \(X_t=aB_{ct}\) has the same law as \((a\sqrt{c})B_t\). The latter process has quadratic variation \((a\sqrt{c})^2T=a^2cT\), agreeing with the direct partition calculation. Equality in law does not assert that \(B_{ct}=\sqrt{c}B_t\) for the same sample path; it asserts that the two processes have the same finite-dimensional distributions.

Increment Variances as an Independent Check

For \(s<t\), direct use of the Brownian increment law gives

This variance check identifies the scale of a typical squared increment: its expectation is \(a^2c\Delta t_i\). Summing these expectations over a partition gives \(a^2cT\). The full partition calculation also quantifies the residual randomness: its variance is \(2a^4c^2\sum_i(\Delta t_i)^2\), which vanishes as the mesh shrinks. Increment variances alone give the expected sum; independence and the Gaussian fourth moment justify the convergence statement.

Common Mistake: Squaring the Time Factor

A frequent error is to treat a time change by \(c\) as though it multiplied each Brownian increment by \(c\). Brownian scaling instead multiplies the typical increment size by \(\sqrt{c}\), since the increment variance is multiplied by \(c\). The coefficient in quadratic variation is therefore \(a^2c\), not \(a^2c^2\).

For example, with \(a=2\), \(c=3\), and \(T=1\), the incorrect calculation \(a^2c^2T=2^2\cdot3^2\cdot1=36\) squares the time factor. The correct increment variance over an interval of length \(\Delta t\) is \(a^2c\Delta t=4\cdot3\Delta t=12\Delta t\). Summing these expected squared increments over \([0,1]\) gives \(12\), and the partition theorem confirms the quadratic variation \(a^2cT=4\cdot3\cdot1=12\).

In differential notation, the result is often summarized as \(dX_t\,dX_t=a^2c\,dt\). This is shorthand for the partition-sum limit, not ordinary multiplication of infinitesimal quantities. When \(a=0\) or \(c=0\), the process is constant and this quadratic variation is zero, in agreement with the formula.