Converting Scalar Stratonovich SDEs

Why a Scalar SDE’s Drift Changes

A scalar stochastic differential equation can be written using either an Itô integral or a Stratonovich integral. The two forms describe the same process when their drift coefficients are adjusted appropriately. The adjustment is not an arbitrary convention: it comes from the quadratic covariation between the state-dependent diffusion coefficient and Brownian motion. In particular, the diffusion coefficient’s derivative enters the drift correction.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). An Itô integrand \(H\) on \([0,T]\) is progressively measurable and belongs to \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\); its integral is \(\int_0^t H_s\,dB_s\). The Stratonovich integral \(\int_0^t H_s\circ dB_s\) is defined, for continuous semimartingale integrands, by adding half the quadratic covariation to the Itô integral. When integrability is only local, these statements are understood after stopping on intervals where the required integrals are square-integrable.

For the conversion below, suppose \(X\) is a continuous semimartingale satisfying the displayed Stratonovich equation, \(b(X)\) is time-integrable, and the relevant Itô integrands are progressively measurable and in \(\mathcal{L}^2\) on each finite time interval under consideration. Assume \(\sigma\in C^2\), with its derivatives sufficiently locally bounded for Itô’s formula; localization permits the same calculation up to exit times from bounded regions. These assumptions justify the operations, but do not by themselves assert global existence of a solution.

Deriving the Drift Correction

The defining relation between the two integrals is \(\int_0^t H_s\circ dB_s=\int_0^t H_s\,dB_s+\tfrac12[H,B]_t\), where \([H,B]_t\) denotes quadratic covariation. To apply it, we need the covariation of \(H_t=\sigma(X_t)\) with \(B_t\). The finite-variation drift term has zero covariation with Brownian motion. The Stratonovich integral and its Itô part have the same covariation with \(B\), because their difference is a finite-variation process. Thus the equation gives \(d[X,B]_t=\sigma(X_t)\,dt\). Itô’s formula applied to \(\sigma(X_t)\) then gives \(d[\sigma(X),B]_t=\sigma'(X_t)\,d[X,B]_t\). Consequently the correction is \(\tfrac12\int_0^t\sigma'(X_s)\sigma(X_s)\,ds\).

Here is the coefficient conversion, with all terms displayed. The quadratic-variation multiplication rules are \(dB_t\,dB_t=dt\), \(dB_t\,dt=0\), and \(dt\,dt=0\). The correction uses the first rule; it is not obtained by applying the ordinary chain rule to Brownian paths.

Proof. Set \(H_t=\sigma(X_t)\). By the Stratonovich-to-Itô integral identity, the integral equation becomes \(X_t=X_0+\int_0^t b(X_s)\,ds+\int_0^t\sigma(X_s)\,dB_s+\tfrac12[\sigma(X),B]_t\). The drift integral has finite variation, so its covariation with \(B\) is zero. The correction term also has finite variation. Therefore the equation implies \(d[X,B]_t=\sigma(X_t)\,dt\). By Itô’s formula for the \(C^2\) function \(x\mapsto\sigma(x)\), its martingale part has diffusion coefficient \(\sigma'(X_t)\sigma(X_t)\); hence \(d[\sigma(X),B]_t=\sigma'(X_t)\sigma(X_t)\,dt\). Substituting this expression for the covariation into the integral equation yields the asserted Itô drift and leaves the diffusion coefficient unchanged. Conversely, starting from that Itô equation and using \(\int H\,dB=\int H\circ dB-\tfrac12[H,B]\), with \(H=\sigma(X)\) and \(d[\sigma(X),B]_t=\sigma'(X_t)\sigma(X_t)\,dt\), subtracts precisely the correction and recovers the Stratonovich equation. Thus both directions hold. \(\square\)

The theorem is an identity for a solution already under consideration: it does not prove that a solution exists or is unique. A strong solution is adapted to the given filtration and driven by its specified Brownian motion; a weak solution may use a newly constructed probability space and Brownian motion. Pathwise uniqueness means two solutions on the same space with the same Brownian motion and initial value are indistinguishable; uniqueness in law means their distributions agree. The conversion itself keeps the same process and Brownian motion and makes no claim about any of these existence or uniqueness properties.

Worked Conversions

Conditions, Interpretation, and a Common Error

The conversion concerns the drift only: the diffusion coefficient in the Itô equation is still \(\sigma(x)\). The derivative \(\sigma'(x)\) measures how the diffusion changes with the state, while the product \(\sigma(x)\sigma'(x)/2\) is the resulting noise-induced drift. For constant \(\sigma\), its derivative vanishes, so there is no correction. In time-dependent scalar equations \(dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\circ dB_t\), the analogous correction is \(\tfrac12\sigma(t,X_t)\partial_x\sigma(t,X_t)\,dt\); there is no time derivative in this term.

The regularity and integrability conditions matter. If \(\sigma\) is not differentiable, the displayed correction may not be defined, and the theorem cannot be applied in this form. If the coefficients are only locally regular, conversion is valid up to localization times, but it does not guarantee that a solution remains finite for all time. Under global Lipschitz and linear growth assumptions, standard existence and uniqueness results for Itô SDEs give a global strong solution; these conditions are sufficient, not necessary, and are not part of the conversion identity itself.

Common mistake: omitting the derivative. In the nonlinear example above, writing the correction as \(\tfrac12\sigma(x)^2=\tfrac12(1+x^2)^2\) is wrong. The correction is half the product of the diffusion and its derivative, not half the square of the diffusion. At \(x=1\), the incorrect expression gives \(\tfrac12(2)^2=2\), whereas the correct value is \(\tfrac12(2)(2)=2\); this point alone does not expose the error. At \(x=0\), however, the incorrect expression is \(\tfrac12\), while the correct correction is \(\tfrac12(1)(0)=0\). The resulting drift would be wrong at \(x=0\), demonstrating why the derivative must be included explicitly.

Another useful check is the constant-diffusion case. If \(\sigma(x)=c\), then \(\sigma'(x)=0\), and both notations have the same drift \(b(x)\). A proposed correction that remains nonzero for constant diffusion is inconsistent with the conversion formula. In every case, the conversion is between the same process on the same filtered probability space under \(\mathbb{P}\); it is not a change of measure.