Drift Corrections in Multidimensional Stratonovich SDEs

Why the Drift Changes in Several Dimensions

In a multidimensional stochastic differential equation, each Brownian component acts through its own diffusion vector field. The Stratonovich-to-Ito correction differentiates each such field in the direction in which that same Brownian component moves the state. The contributions are then added. This direction-by-direction structure matters: it is not generally correct to differentiate one diffusion column along another.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying an \(m\)-dimensional Brownian motion \(\mathbf{B}_t=(B_t^1,\ldots,B_t^m)\), whose components are independent standard Brownian motions relative to the filtration. Let \(X_t=(X_t^1,\ldots,X_t^n)\) be a continuous adapted state process. The drift is \(a=(a_1,\ldots,a_n)\), and the diffusion coefficients form an \(n\times m\) matrix \(\sigma=(\sigma_{ij})\).

Definition: The multidimensional Stratonovich SDE \(dX_t^i=a_i(t,X_t)\,dt+\sum_{j=1}^m\sigma_{ij}(t,X_t)\circ dB_t^j\) means, in integral form, \(X_t^i=X_0^i+\int_0^t a_i(s,X_s)\,ds+\sum_{j=1}^m\int_0^t\sigma_{ij}(s,X_s)\circ dB_s^j\). The Ito integral uses progressively measurable integrands in \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). For a continuous semimartingale integrand \(H\), the Stratonovich convention is \(\int_0^t H_s\circ dB_s^j=\int_0^t H_s\,dB_s^j+\tfrac12[H,B^j]_t\). All Ito integrals in the conversion below are assumed to have progressively measurable \(\mathcal{L}^2\) integrands on the time interval considered.

Here \([H,B^j]_t\) denotes quadratic covariation. The Brownian rules are \(dB_t^j\,dB_t^\ell=\mathbf{1}_{\{j=\ell\}}dt\), \(dB_t^j\,dt=0\), and \(dt\,dt=0\). In particular, the correction to the \(j\)-th Stratonovich integral depends on covariation with \(B^j\), not with every Brownian component.

The Multidimensional Conversion Formula

Theorem: Suppose \(X\) satisfies the Stratonovich SDE above, and the coefficients and solution have the integrability needed for the displayed integrals. Assume each \(\sigma_{ij}(t,x)\) is \(C^{1,2}\) in \((t,x)\), so the time-dependent Itô formula applies to \(\sigma_{ij}(t,X_t)\). Suppose also that the resulting correction terms are time-integrable. Then the same process satisfies the Ito SDE \(dX_t^i=\left(a_i(t,X_t)+\tfrac12\sum_{j=1}^m\sum_{k=1}^n\sigma_{kj}(t,X_t)\partial_{x_k}\sigma_{ij}(t,X_t)\right)dt+\sum_{j=1}^m\sigma_{ij}(t,X_t)\,dB_t^j\).

Proof. Fix indices \(i\) and \(j\). Apply the time-dependent Itô formula to the function \(g(t,x)=\sigma_{ij}(t,x)\) along \(X_t\). The formula explicitly reads

\begin{aligned} d\sigma_{ij}(t,X_t) ={}&\partial_t\sigma_{ij}(t,X_t)\,dt +\sum_{k=1}^n\partial_{x_k}\sigma_{ij}(t,X_t)\,dX_t^k\\ &+\frac12\sum_{k=1}^n\sum_{\ell=1}^n \partial_{x_kx_\ell}\sigma_{ij}(t,X_t)\,d[X^k,X^\ell]_t . \end{aligned}

In this formula, the time derivative and second spatial derivatives are present; the assumed \(C^{1,2}\) regularity justifies their use. The finite-variation terms have zero covariation with \(B^j\). The martingale part of \(dX_t^k\) is \(\sum_{\ell=1}^m\sigma_{k\ell}(t,X_t)dB_t^\ell\), whose covariation with \(B^j\) is \(\sigma_{kj}(t,X_t)dt\), since \(dB^\ell dB^j=\mathbf{1}_{\{\ell=j\}}dt\). Taking covariation of the displayed Itô formula with \(B^j\) therefore gives

[\sigma_{ij}(\,\cdot\,,X),B^j]_t =\int_0^t\sum_{k=1}^n \partial_{x_k}\sigma_{ij}(s,X_s)\sigma_{kj}(s,X_s)\,ds .

By the definition of the Stratonovich integral, each integral against \(B^j\) equals its Ito integral plus one half of this covariation. Sum the resulting identities over \(j=1,\ldots,m\), and add the ordinary drift integral. The coefficient of \(dt\) is exactly the drift in the theorem. This proves the conversion formula. \(\square\)

For a vector description, write the \(j\)-th column of \(\sigma\) as \(V_j=(\sigma_{1j},\ldots,\sigma_{nj})\). Its contribution to the correction has \(i\)-th component \(\sum_k V_j^k\partial_{x_k}V_j^i\). Thus each noise direction differentiates its own column. No derivative with respect to time appears in the correction: time variation contributes finite variation, whose covariation with Brownian motion is zero.

Worked Corrections

Worked Example: A Triangular Two-Noise System

Consider a two-dimensional state driven by two independent Brownian motions, with diffusion columns \(V_1(x_1,x_2)=(x_2,0)\) and \(V_2(x_1,x_2)=(x_1,x_2)\), and Stratonovich drift \(a=(0,1)\). Find the Ito drift and evaluate it at \(x=(2,3)\).

The matrix of diffusion coefficients is \(\sigma=\begin{pmatrix}x_2&x_1\\0&x_2\end{pmatrix}\). For the first column, the directional derivative of \(V_1\) along \(V_1\) is zero: its first component \(x_2\) has directional derivative \(x_2\partial_{x_1}x_2+0\partial_{x_2}x_2=0\), and its second component is constant.

For the second column, differentiate its first component \(x_1\) along \((x_1,x_2)\), obtaining \(x_1\partial_{x_1}x_1+x_2\partial_{x_2}x_1=x_1\). Differentiate its second component \(x_2\) along the same direction, obtaining \(x_1\partial_{x_1}x_2+x_2\partial_{x_2}x_2=x_2\). Hence the full correction is \(\tfrac12(0+(x_1,x_2))=(x_1/2,x_2/2)\). The Ito drift is therefore \((x_1/2,1+x_2/2)\), which at \((2,3)\) is \((1,5/2)\).

Check: In components, the correction for the first coordinate is \(\tfrac12(\sigma_{11}\partial_{x_1}\sigma_{11}+\sigma_{21}\partial_{x_2}\sigma_{11}+\sigma_{12}\partial_{x_1}\sigma_{12}+\sigma_{22}\partial_{x_2}\sigma_{12})=\tfrac12(0+0+x_1+0)=x_1/2\). For the second it is \(\tfrac12(0+0+0+x_2)=x_2/2\). At \((2,3)\), these are \(1\) and \(3/2\), so adding the original drift gives \((1,5/2)\) as above.

Worked Example: Three Brownian Motions and Two State Variables

Let the three diffusion columns be \(V_1=(x_1,0)\), \(V_2=(0,x_1)\), and \(V_3=(x_2,x_1)\). Compute the correction at \(x=(2,-1)\), and suppose the Stratonovich drift there is \((-1,2)\).

For \(V_1\), differentiating each component in direction \((x_1,0)\) gives \((x_1,0)\). For \(V_2\), its first component is zero, and the derivative of its second component \(x_1\) in direction \((0,x_1)\) is \(0\); its contribution is \((0,0)\). For \(V_3\), differentiating \(x_2\) in direction \((x_2,x_1)\) gives \(x_1\), while differentiating \(x_1\) in that direction gives \(x_2\). Its contribution is \((x_1,x_2)\). Summing and multiplying by \(1/2\), the correction is \((x_1,x_2/2)\).

At \((2,-1)\), the correction is \((2,-1/2)\). Adding the specified Stratonovich drift gives the Ito drift \((1,3/2)\).

Check: At the given point the columns are \((2,0)\), \((0,2)\), and \((-1,2)\). Their directional derivatives are respectively \((2,0)\), \((0,0)\), and \((2,-1)\). The sum is \((4,-1)\), and half of it is \((2,-1/2)\). Adding \((-1,2)\) again gives \((1,3/2)\).

Worked Example: Time-Dependent Diffusion Coefficients

For \(t\ge0\), let \(V_1(t,x)=(t x_1,x_1)\) and \(V_2(t,x)=(0,t x_2)\), with Stratonovich drift \(a=(1,-1)\). Find the correction at \(t=2\), \(x=(1,3)\).

For \(V_1\), the state derivatives of its components are \(\nabla_x(t x_1)=(t,0)\) and \(\nabla_x x_1=(1,0)\). Taking directional derivatives along \(V_1=(t x_1,x_1)\) gives \((t^2x_1,t x_1)\). For \(V_2\), the first component contributes zero. The state derivative of its second component \(t x_2\) is \((0,t)\), so its derivative along \((0,t x_2)\) is \(t^2x_2\). The total correction is \(\tfrac12(t^2x_1,t x_1+t^2x_2)\).

At \(t=2\), \(x=(1,3)\), the two directional derivatives are \((4,2)\) and \((0,12)\). The correction is \((2,7)\), and the Ito drift is \((3,6)\).

Check: Direct substitution into \(\tfrac12(t^2x_1,t x_1+t^2x_2)\) gives \(\tfrac12(4,2+12)=(2,7)\). The time derivatives of the coefficients, such as \(\partial_t(t x_1)=x_1\), do not enter this correction; in the Itô formula they multiply \(dt\), which has zero covariation with \(B^j\).

An Affine-Diffusion Corollary

Corollary: Suppose the \(j\)-th diffusion column is affine in the state, \(V_j(x)=A_jx+c_j\), where \(A_j\) is an \(n\times n\) matrix and \(c_j\in\mathbb{R}^n\). Then the Stratonovich-to-Ito correction is \(\tfrac12\sum_{j=1}^m A_j(A_jx+c_j)\).

Proof. The derivative matrix of \(V_j(x)=A_jx+c_j\) with respect to \(x\) is \(A_j\), because the derivative of the constant vector \(c_j\) is zero. The directional derivative of \(V_j\) along itself is therefore \(A_jV_j(x)=A_j(A_jx+c_j)\). Sum these contributions over the Brownian components and multiply by \(1/2\), as in the conversion theorem. \(\square\)

Common Mistakes and Why the Correction Matters

A frequent error is to mix the Brownian index and the state-coordinate index. For example, the correction is not generally formed by differentiating \(V_j\) in direction \(V_\ell\) and summing over all pairs \(j,\ell\). The covariation calculation selects the same Brownian component on both sides, so the terms are \(D V_j\,V_j\), one per \(j\).

In the three-noise example, the incorrect cross-direction expression would include terms such as \(D V_1\,V_3\), despite \(V_1\) being integrated against \(B^1\) and \(V_3\) against the independent \(B^3\). Those cross-covariations vanish because \(dB^1dB^3=0\). The correct sum gave correction \((x_1,x_2/2)\); adding unrequested cross terms would change the drift without any corresponding covariation to justify it.

The conversion is important whenever the same stochastic dynamics are expressed in the two integral conventions. The diffusion coefficient is unchanged, but the drift is not: the Ito drift is the Stratonovich drift plus the correction. In applications, confusing those drifts changes the generator and can therefore change expectations computed from the process. The conversion formula itself does not assert existence or uniqueness of a solution; it states that, under its regularity and integrability assumptions, a given solution has these equivalent Ito and Stratonovich descriptions.

Quantity	Role in the conversion
\(a_i(t,x)\)	Stratonovich drift before correction
\(\sigma_{ij}(t,x)\)	Coefficient of Brownian component \(B^j\) in state coordinate \(i\)
\(\sum_k\sigma_{kj}\partial_{x_k}\sigma_{ij}\)	Coordinate \(i\) of the directional derivative of column \(j\) along itself
\(\tfrac12\sum_{j,k}\sigma_{kj}\partial_{x_k}\sigma_{ij}\)	Drift added to obtain the Ito form

Check Your Understanding

For each question, keep the Brownian-component index distinct from the state-coordinate index.

Write the correction formula for a three-dimensional state driven by two Brownian motions, using component indices.
If every diffusion column is constant in the state, what is the Stratonovich-to-Ito drift correction? Explain from the derivative formula.
For \(V_1(x_1,x_2)=(x_1,0)\) and \(V_2(x_1,x_2)=(0,x_2)\), compute the correction.
Explain why \(\partial_t\sigma_{ij}\) does not appear in the correction, even when the diffusion coefficients depend on time.
For affine columns \(V_j(x)=A_jx+c_j\), state the correction and identify where the constant vector \(c_j\) enters.