Multidimensional Milstein and Commutative Noise

From Scalar Milstein to Several Noise Channels

For a system driven by one Brownian motion, the Milstein correction uses the squared Brownian increment minus the time step. With several independent Brownian motions, there are also cross terms: the change in one diffusion direction can interact with a different noise direction. In general those terms involve iterated Ito integrals, which cannot be recovered from the Brownian increments alone. When the diffusion vector fields commute, however, the cross terms combine into products of increments. This is the useful case in which a multidimensional Milstein step avoids simulation of Lévy areas.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})\) carrying an \(m\)-dimensional Brownian motion \(\mathbf{B}_t=(B_t^1,\ldots,B_t^m)\), whose components are independent standard Brownian motions. Consider an \(n\)-dimensional state \(X_t\) satisfying \(dX_t=\mu(X_t)\,dt+\sum_{j=1}^m \sigma_j(X_t)\,dB_t^j\), where \(\mu:\mathbb{R}^n\to\mathbb{R}^n\) is the drift and each \(\sigma_j:\mathbb{R}^n\to\mathbb{R}^n\) is a diffusion vector field. Assume the coefficients are globally Lipschitz and satisfy linear growth, so the SDE has a unique strong solution on finite time intervals. For the derivative expressions below, assume also that the \(\sigma_j\) are continuously differentiable. Stronger smoothness and boundedness assumptions are used when establishing global strong-order error bounds.

On a grid \(t_n=nh\), write \(\Delta B_n^j=B_{t_{n+1}}^j-B_{t_n}^j\). For a vector field \(v\), \(Dv(x)\) denotes its derivative matrix at \(x\), and \(Dv(x)u\) is its directional derivative in direction \(u\). Define the coefficient vector \(C_{jk}(x)=D\sigma_k(x)\sigma_j(x)\). The indexing here means that the inner increment is from Brownian component \(j\), and the outer Ito integral is with respect to component \(k\).

The drift term contributes at scale \(h\), the diffusion terms at scale \(\sqrt h\), and the iterated-integral correction at scale \(h\). The correction accounts for how the diffusion coefficients change as the state moves during the step. It is the multi-noise counterpart of the scalar Milstein correction.

Commutativity and the Increment-Only Formula

The equivalence in the definition follows directly from the definition of \(C_{jk}\): \(C_{jk}=D\sigma_k\,\sigma_j\), while \(C_{kj}=D\sigma_j\,\sigma_k\). This condition says that differentiating one diffusion field in the direction of another gives the same result after swapping the fields. It is sometimes expressed by saying that the Lie bracket of each pair of diffusion fields is zero.

Proof. Apply the Ito product rule to the two Brownian increments from \(t_n\) to \(s\). Since \(dB_s^j\,dB_s^k=\delta_{jk}\,ds\), integration over the step gives \(\Delta B_n^j\Delta B_n^k=I_{jk}^{(n)}+I_{kj}^{(n)}+\delta_{jk}h\). Now sum this identity after multiplying by the symmetric vector coefficient \(C_{jk}(Y_n)\). Renaming the indices \(j\) and \(k\) in the second iterated-integral sum gives \(\sum_{j,k}C_{jk}I_{kj}^{(n)}=\sum_{j,k}C_{kj}I_{jk}^{(n)}=\sum_{j,k}C_{jk}I_{jk}^{(n)}\), where the last equality is commutativity. Therefore \(\sum_{j,k}C_{jk}(\Delta B_n^j\Delta B_n^k-\delta_{jk}h)=2\sum_{j,k}C_{jk}I_{jk}^{(n)}\). Dividing by \(2\) proves the formula. \(\square\)

The subtraction of \(h\) occurs only in diagonal terms. It is the Ito correction: for one Brownian component, the product identity reads \((\Delta B^j)^2=2I_{jj}^{(n)}+h\). Cross products have no deterministic \(h\) term because independent Brownian components have zero covariation. This is consistent with the differential rules \(dB_t^j\,dB_t^k=\delta_{jk}dt\), \(dB_t^j\,dt=0\), and \(dt\,dt=0\).

Why Noncommuting Fields Leave an Extra Term

If \(C_{jk}\ne C_{kj}\), the product identity still determines the sum \(I_{jk}^{(n)}+I_{kj}^{(n)}\), but not the two integrals separately. The difference between them is an antisymmetric iterated-integral contribution, commonly called a Lévy area. A general noncommutative Milstein scheme therefore needs additional information beyond the Brownian increments. The next step in the course treats those area terms; here the key distinction is whether the coefficients make them cancel.

A Scalar State with Several Commuting Noises

Commutativity is not restricted to systems with as many state coordinates as Brownian components. In one state dimension, each diffusion field is a scalar function and the condition becomes \(\sigma_j'(x)\sigma_k(x)=\sigma_k'(x)\sigma_j(x)\). The condition may hold even when the noise channels have different sizes. In multiple dimensions, it is a genuine restriction on how the vector fields interact.

Practical Interpretation and Common Pitfall

The increment-only scheme is useful because a simulation already generates \(\Delta B_n^j\) for each time step. Under commutativity, no separate random variables for the iterated integrals are needed. The correction still matters: replacing Milstein by Euler–Maruyama drops the full \(C_{jk}\)-weighted term, even in the commuting case. The correction is not an adjustment to the drift; it is the discretized contribution from changes in the diffusion fields along the step.

Common mistake: applying the symmetric formula without checking commutativity. For the noncommuting example, a tempting but invalid substitution would replace \(I_{12}^{(n)}\) by \(\tfrac12\Delta B_n^1\Delta B_n^2\). The product identity only gives \(I_{12}^{(n)}+I_{21}^{(n)}=\Delta B_n^1\Delta B_n^2\); it does not imply either integral equals half that product. Since the coefficients of these two integrals differ, that substitution changes the correction. First test \(D\sigma_k\,\sigma_j=D\sigma_j\,\sigma_k\) for every pair of noise channels. Only then use the increment-only expression.

Commutativity is a sufficient structural condition for this simplification. It is not a claim that every noncommuting system must have a nonzero area contribution in every step: particular coefficients or cancellations may remove terms in a special case. The reliable general test is the vector-field condition, not an assumption based only on the number of state coordinates or noise channels.