Brownian Driver Rotation and Conversion Invariance

Why the Brownian Driver Basis Matters

A multidimensional diffusion can be represented using different sets of independent Brownian drivers. For example, two drivers can be replaced by two orthogonal linear combinations of those drivers without changing their joint standard Brownian law. The diffusion coefficients change under this replacement, even though the random fluctuations of the state have not changed.

This raises an important question when converting between Itô and Stratonovich forms: does changing the driver basis also change the conversion drift? For a constant orthogonal rotation, the answer is no. The correction is built from all diffusion directions together, and the orthogonal mixing terms cancel when they are summed.

The key restriction is that the rotation is constant. The results below concern ordinary finite-dimensional SDEs with differentiable diffusion coefficients; they do not imply invariance under arbitrary state-dependent changes of noise representation.

Conversion Formula and Rotation

Definition: Let \(X_t\in\mathbb{R}^d\) be driven by an \(m\)-dimensional standard Brownian motion \(W_t\). Write the Stratonovich equation as \(dX_t=a(X_t)\,dt+\sum_{j=1}^m V_j(X_t)\circ dW_{j,t}\), where each diffusion column \(V_j:\mathbb{R}^d\to\mathbb{R}^d\) is continuously differentiable. Its equivalent Itô drift is \(b(x)=a(x)+\frac12\sum_{j=1}^m DV_j(x)V_j(x)\). Here \(DV_j(x)\) is the Jacobian matrix of \(V_j\), and \(DV_j(x)V_j(x)\) is the directional derivative of \(V_j\) along itself.

In coordinates, the \(i\)th component of the correction is \(\frac12\sum_{j=1}^m\sum_{k=1}^d V_{kj}(x)\,\partial V_{ij}(x)/\partial x_k\), where \(V_{ij}\) is component \(i\) of column \(V_j\). This is the standard multidimensional Itô–Stratonovich conversion. It applies locally for continuously differentiable coefficients; usual local Lipschitz and growth conditions can additionally be imposed when a global solution is required.

Let \(Q\) be a constant \(m\times m\) orthogonal matrix, so \(Q^TQ=QQ^T=I_m\). Define \(\widetilde W_t=Q^TW_t\). Then \(W_t=Q\widetilde W_t\), and the new diffusion columns are \(\widetilde V_\ell=\sum_{j=1}^m Q_{j\ell}V_j\), for \(\ell=1,\ldots,m\).

Theorem: Under a constant orthogonal rotation of the Brownian drivers, the multidimensional conversion correction is unchanged: \(\sum_{\ell=1}^m D\widetilde V_\ell(x)\widetilde V_\ell(x)=\sum_{j=1}^m DV_j(x)V_j(x)\). Consequently, the Itô and Stratonovich drifts associated with the rotated representation are the same as those associated with the original representation.

Why the Correction Is Invariant

First, \(\widetilde W\) is again a standard \(m\)-dimensional Brownian motion. Its increments are obtained by applying the fixed linear map \(Q^T\) to increments of \(W\), so they remain independent across disjoint time intervals and have mean zero. Since Brownian increments are Gaussian, \(\operatorname{Cov}(\widetilde W_t-\widetilde W_s)=(t-s)Q^TI_mQ=(t-s)I_m\) for \(s<t\). Thus the rotated components are independent standard Brownian motions.

The SDE keeps the same state process under this change of drivers. Indeed, substituting \(dW=Q\,d\widetilde W\) into the noise term gives the columns \(\widetilde V_\ell\) above. For Stratonovich integrals, linearity under a constant matrix gives the same substitution; for Itô integrals, the same constant-matrix linearity applies.

It remains to establish the correction identity. Since \(Q\) is constant, differentiation acts only on the vector fields: \(D\widetilde V_\ell=\sum_{k=1}^m Q_{k\ell}DV_k\). Multiplying by \(\widetilde V_\ell\) and summing gives

\begin{aligned} \sum_{\ell=1}^m D\widetilde V_\ell\,\widetilde V_\ell &=\sum_{\ell=1}^m \left(\sum_{k=1}^m Q_{k\ell}DV_k\right) \left(\sum_{j=1}^m Q_{j\ell}V_j\right)\\ &=\sum_{k=1}^m\sum_{j=1}^m \left(\sum_{\ell=1}^m Q_{k\ell}Q_{j\ell}\right)DV_kV_j\\ &=\sum_{k=1}^m\sum_{j=1}^m \delta_{kj}DV_kV_j\\ &=\sum_{j=1}^m DV_jV_j. \end{aligned}

The third equality follows from \(QQ^T=I_m\), which says that \(\sum_\ell Q_{k\ell}Q_{j\ell}=\delta_{kj}\). The terms involving two different original columns therefore vanish in the sum, while each original directional-derivative term remains once. Multiplying both sides by \(1/2\) proves equality of the conversion corrections. \(\square\)

Worked Calculations

Worked Example: One State With Two Drivers

Consider the scalar Stratonovich equation \(dX_t=a(X_t)\,dt+X_t\circ dW_{1,t}+2X_t\circ dW_{2,t}\). Its diffusion columns are \(V_1(x)=x\) and \(V_2(x)=2x\). In one state dimension, each Jacobian is an ordinary derivative, so

\frac12\left(V_1'(x)V_1(x)+V_2'(x)V_2(x)\right) =\frac12(x+4x)=\frac52x.

Rotate the two drivers by \(45\) degrees: \(Q=\frac{1}{\sqrt2}\begin{pmatrix}1&-1\\1&1\end{pmatrix}\). The rotated columns are \(\widetilde V_1=3x/\sqrt2\) and \(\widetilde V_2=x/\sqrt2\). Their correction is

\frac12\left(\frac{3}{\sqrt2}\frac{3x}{\sqrt2} +\frac{1}{\sqrt2}\frac{x}{\sqrt2}\right) =\frac12\left(\frac92x+\frac12x\right) =\frac52x.

The individual columns have changed, but the sum of their derivative contributions has not. The Itô drift is \(a(x)+5x/2\) in either representation.

Worked Example: Two Compartments With Cross-Coupled Noise

Let \(x_1\) and \(x_2\) denote two compartment states. Consider Stratonovich diffusion columns \(V_1(x_1,x_2)=(0.2x_1,\,0.1x_2)^T\) and \(V_2(x_1,x_2)=(0.3x_2,\,0.4x_1)^T\). The noise in each driver affects both states, so the second column also creates cross-compartment coupling.

The Jacobians and directional derivatives are

DV_1= \begin{pmatrix}0.2&0\\0&0.1\end{pmatrix}, \qquad DV_2= \begin{pmatrix}0&0.3\\0.4&0\end{pmatrix},

DV_1V_1= \begin{pmatrix}0.04x_1\\0.01x_2\end{pmatrix}, \qquad DV_2V_2= \begin{pmatrix}0.12x_1\\0.12x_2\end{pmatrix}.

Thus the conversion correction is \(\frac12(DV_1V_1+DV_2V_2)=(0.08x_1,\,0.065x_2)^T\). The cross-coupled column contributes \(0.12x_1\) to the first component and \(0.12x_2\) to the second; omitting those terms would give an incorrect conversion.

For the \(45\)-degree rotation in the preceding example, \(\widetilde V_1=(V_1+V_2)/\sqrt2\) and \(\widetilde V_2=(-V_1+V_2)/\sqrt2\). To check the correction explicitly, write \(A=DV_1V_1\), \(B=DV_1V_2\), \(C=DV_2V_1\), and \(D=DV_2V_2\). The two rotated directional derivatives sum to \(\frac12(A+B+C+D)+\frac12(A-B-C+D)=A+D\). Hence the rotated correction is again \(\frac12(A+D)=(0.08x_1,\,0.065x_2)^T\). This calculation verifies invariance for both compartment states, including the cross-coupled noise.

Worked Example: A Redundant Driver That Contributes No Correction

Suppose a two-state model has three Brownian drivers with columns \(V_1=(0.1x_1,0)^T\), \(V_2=(0,0.2x_2)^T\), and \(V_3=(0,0)^T\). The third driver is redundant. The correction is

\frac12\sum_{j=1}^3 DV_jV_j =\frac12 \begin{pmatrix}0.01x_1\\0.04x_2\end{pmatrix} = \begin{pmatrix}0.005x_1\\0.02x_2\end{pmatrix}.

Any constant \(3\times3\) orthogonal rotation mixes all three columns, so the rotated representation may have three nonzero columns. Nevertheless, the theorem still applies: the sum of all three rotated directional derivatives is exactly the sum above. A zero original column does not make the invariance argument fail; the proof uses orthogonality and does not require the diffusion matrix to have full rank.

Diagnostics, Alternatives, and Limits

When checking a rotated model, use the following sequence:

Check the rotation.
Confirm that \(Q\) is square and \(Q^TQ=QQ^T=I_m\). A general scaling or shear does not preserve independent standard Brownian drivers.

Transform the columns.
With \(\widetilde W=Q^TW\), use \(\widetilde V_\ell=\sum_j Q_{j\ell}V_j\). The state dimension indexes rows; the driver dimension indexes columns.

Differentiate the transformed fields.
For a constant rotation, \(D\widetilde V_\ell=\sum_jQ_{j\ell}DV_j\). If extra derivatives of \(Q\) appear, the transformation being used is not the constant rotation treated here.

Sum over every driver.
Keep the full sum \(\sum_\ell D\widetilde V_\ell\widetilde V_\ell\). Checking only one rotated column can suggest a false change in the correction.

A useful independent check is the instantaneous covariance matrix: \(\widetilde V\widetilde V^T=VQQ^TV^T=VV^T\). This confirms that the instantaneous state-noise covariance is unchanged. However, covariance equality by itself is not a substitute for the correction calculation when comparing arbitrary diffusion representations: the theorem above specifically proves the identity for a constant orthogonal rotation of the given columns.

The main alternative is to keep the original driver basis and apply the conversion formula directly. This is often clearer when drivers have distinct scientific interpretations. A rotated basis can be mathematically convenient, but its columns may no longer correspond to separate biological sources of variability. Both representations describe the same diffusion when related by the stated rotation; neither automatically gives a more mechanistic account.

Common mistake: rotating only the Brownian motion. If \(W=Q\widetilde W\), the diffusion columns must also change to \(VQ\). Keeping the old columns while replacing the drivers generally changes the SDE.
Common mistake: assuming any mixing preserves the correction. Orthogonality and constancy are essential to the proof. A state-dependent rotation introduces derivatives of the rotation into the transformed vector fields, so this result does not apply as written.
Common mistake: changing the conversion drift after rotation. Individual column contributions can change, but their complete sum does not. Recomputing and retaining only one contribution is not a valid conversion.

In practical PK models, this invariance means that a constant re-expression of independent process-noise drivers does not by itself alter the Itô drift corresponding to a specified Stratonovich drift. It does not select which interpretation is appropriate, establish that a noise model is biologically plausible, or justify a state-dependent change of drivers.

Check Your Understanding

Answer each question, then compare with the solutions.

For scalar state \(X\) with columns \(V_1(x)=3x\) and \(V_2(x)=4x\), calculate the conversion correction.
Why does \(\widetilde W=Q^TW\) remain standard Brownian motion when \(Q\) is constant and orthogonal?
In a two-state model, suppose one diffusion column is identically zero. Does a constant orthogonal rotation invalidate conversion invariance? Explain.
What is the error in rotating the Brownian drivers but leaving the diffusion matrix unchanged?

Solutions. (1) The correction is \(\frac12(3\cdot3x+4\cdot4x)=\frac12(9x+16x)=25x/2\). (2) The transformed increments are Gaussian, retain independent increments, and have covariance \((t-s)Q^TQ=(t-s)I_m\); hence their components are independent standard Brownian motions. (3) No. The proof uses the full column sum and orthogonality, not full rank; the zero column may become part of nonzero rotated columns, while the total correction remains unchanged. (4) The stochastic integral has changed its driver basis but not its coefficients. To represent the same noise term one must replace the column matrix \(V\) by \(VQ\) when \(W=Q\widetilde W\).