Vector Fields and Noise-Induced Drift

Diffusion Coefficients as Vector Fields

In a multidimensional stochastic differential equation, each independent Brownian noise pushes the state in a particular direction, and that direction can vary with the state. The diffusion coefficient for one noise is therefore naturally viewed as a vector field. When the direction changes from point to point, the Stratonovich-to-Itô conversion adds a drift: each field is differentiated in its own direction. This additional term is called the noise-induced drift.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard \(m\)-dimensional Brownian motion \(\mathbf{B}_t=(B_t^1,\ldots,B_t^m)\) with independent components. Let the state \(X_t\) take values in \(\mathbb{R}^d\), and let \(V_k:\mathbb{R}^d\to\mathbb{R}^d\) be the diffusion vector field corresponding to \(B^k\). The notation \(DV_k(x)\) denotes the \(d\times d\) derivative matrix of \(V_k\) at \(x\).

For clarity, the \(i\)-th coordinate of the correction from one field is \((DV_k(x)V_k(x))_i=\sum_{j=1}^d \partial_j V_k^i(x)V_k^j(x)\). It is the directional derivative of the \(i\)-th component of \(V_k\) along \(V_k\) itself. With independent Brownian components, there is one such term for each field, and the terms are summed. There are no cross terms between distinct fields in this conversion.

All stochastic integrals below are Itô integrals in the class \(\mathcal{L}^2\): their integrands are progressively measurable and satisfy \(\mathbb{E}\int_0^T |H_s|^2\,ds<\infty\) for each finite \(T\). Stratonovich integrals are understood through the Itô integral plus one half of the relevant covariation. For locally bounded coefficients, the same conversion can be applied up to exit times from bounded regions; the displayed global examples have linear, globally Lipschitz coefficients. The standard existence-and-uniqueness result for globally Lipschitz coefficients of at most linear growth gives their solutions on finite time intervals.

Why the Directional Derivative Appears

Proof. By the definition of the Stratonovich integral, its \(i\)-th coordinate for field \(k\) is \(\int_0^t V_k^i(X_s)\,dB_s^k+\tfrac12[V_k^i(X),B^k]_t\). We compute the covariation. Write the Itô diffusion coefficient of \(X\) in coordinate \(j\) against \(B^\ell\) as \(V_\ell^j(X_s)\); a finite-variation drift has zero covariation with Brownian motion. Apply Itô's formula to \(g(x)=V_k^i(x)\). Its first derivatives are \(\partial_jg=\partial_jV_k^i\), and its second derivatives are \(\partial_{j r}g=\partial_{j r}V_k^i\). The formula includes the correction explicitly:

The second-derivative term and the finite-variation part of \(dX_t^j\) have zero covariation with \(B^k\). Since \(d[B^\ell,B^k]_t\) is \(dt\) when \(\ell=k\) and zero otherwise, the remaining term gives \(d[V_k^i(X),B^k]_t=\sum_j\partial_jV_k^i(X_t)V_k^j(X_t)\,dt\). Consequently the correction in coordinate \(i\) is half the sum over \(k\) of these directional derivatives. This is exactly \(\tfrac12\sum_k DV_k(X_t)V_k(X_t)\), proving the asserted drift. \(\square\)

This argument uses the Itô multiplication rules \(dB_t^k\,dB_t^\ell=\mathbf{1}_{\{k=\ell\}}dt\), \(dB_t^k\,dt=0\), and \(dt\,dt=0\). In particular, the correction is not found by differentiating the drift \(b\), nor by differentiating one field along a different independent noise field.

Worked Two-Dimensional Systems

The Generator View

The same correction can be recognized by comparing the differential operators associated with the two descriptions. For a twice continuously differentiable test function \(f:\mathbb{R}^d\to\mathbb{R}\), define \(V_k f(x)=\nabla f(x)\cdot V_k(x)\). In Stratonovich form the generator is \(b\cdot\nabla+\tfrac12\sum_k V_k(V_k f)\), where the field acts as a directional derivative. The corresponding Itô generator uses the corrected drift.

Proof. For each \(k\), differentiate \(V_k f=\sum_i V_k^i\partial_i f\) along \(V_k\): \(V_k(V_k f)=\sum_{j,i}V_k^j(\partial_jV_k^i)\partial_i f+\sum_{j,i}V_k^jV_k^i\partial_{ji}f\). The first sum is \((DV_kV_k)\cdot\nabla f\), and the second is \(V_k^\mathsf{T}(D^2f)V_k\). Summing these identities, multiplying by one half, and adding \(b\cdot\nabla f\) gives the displayed equality. \(\square\)

This equality explains why the correction matters beyond rewriting an equation: it ensures that the Stratonovich and Itô descriptions give the same infinitesimal action on smooth test functions. It also provides a useful independent check of a conversion: the first-order drift term must contain exactly the directional derivatives of the diffusion fields.

A Common Mistake

A frequent error is to differentiate each coordinate of a field with respect to the coordinate bearing the same index, rather than calculate the full directional derivative. For \(V(x,y)=(y,1)\), that shortcut might incorrectly report \(DV\,V=(0,0)\), and hence no drift. But the derivative matrix is \(\begin{pmatrix}0&1\\0&0\end{pmatrix}\), and multiplying it by \((y,1)\) gives \((1,0)\). The correct correction is therefore \((1/2,0)\). The worked solution \(X_t=B_t^2/2\) verifies the correction: its Itô differential contains \(\tfrac12dt\), which the incorrect conversion would omit.

The distinction is between differentiating a field and differentiating it in the direction of that field. In coordinates, every partial derivative that contributes must be multiplied by the corresponding component of the same diffusion field. When there are several independent Brownian drivers, calculate this product field by field and then sum.