How to Test a Stratonovich Conversion
A conversion formula is easy to misremember because the correction is a drift term, not a change to the diffusion coefficient. Three details determine it: the sign comes from the definition of the Stratonovich integral, the factor is one half, and the derivative acts on the diffusion vector field. With several noise components, covariations also matter. Rather than relying on memory, we can test a proposed conversion against an exact Brownian integral identity and then verify candidate SDE solutions using Itô’s formula.
Throughout, work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying the Brownian motion or Brownian vector specified in each example. An Itô integrand \(H\) is progressively measurable and belongs to \(\mathcal{L}^2\) on \([0,T]\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). The Stratonovich integral of a continuous semimartingale integrand \(H\) is related to its Itô integral by the covariation correction below; in the examples, both integrals are well-defined and the integrands satisfy the stated square-integrability condition.
The sign and factor are already fixed by this definition. A useful diagnostic identity follows by taking \(H_s=B_s\), for which \([B,B]_t=t\). In differential shorthand, the corresponding rules are \(dB_t\,dB_t=dt\), \(dB_t\,dt=0\), and \(dt\,dt=0\). These rules summarize quadratic-variation calculations; they are not ordinary multiplication rules for finite increments.
A Brownian Identity That Detects Sign and Factor Errors
Proof. Apply Itô’s formula to \(f(x)=x^2\), whose derivatives are \(f'(x)=2x\) and \(f''(x)=2\). The second-derivative correction is \(\tfrac12 f''(B_s)\,ds=ds\), so $$ d(B_t^2)=2B_t\,dB_t+dt,\qquad B_T^2=2\int_0^T B_s\,dB_s+T. $$ Rearranging gives the Itô identity. By the definition of the Stratonovich integral, \(\int_0^T B_s\circ dB_s=\int_0^T B_s\,dB_s+\tfrac12[B,B]_T\), and \([B,B]_T=T\). Substitution gives \(\tfrac12(B_T^2-T)+\tfrac12T=\tfrac12B_T^2\), as required. \(\square\)
Worked Example: Finding a Sign or Factor Error
Take \(T=2\). Compare the correct Stratonovich integral with two common incorrect conversions: subtracting the correction and adding the full correction.
The correct conversion adds half the covariation: $$ \int_0^2 B_s\circ dB_s =\int_0^2 B_s\,dB_s+\frac12[B,B]_2 =\frac12(B_2^2-2)+\frac12(2) =\frac12B_2^2. $$ The sign is positive, and the correction is \(\tfrac12[B,B]_2=1\).
A sign error would instead produce \(\int_0^2 B_s\,dB_s-\tfrac12[B,B]_2=\tfrac12(B_2^2-2)-1=\tfrac12B_2^2-2\). A factor-of-two error would add \([B,B]_2=2\), giving \(\tfrac12(B_2^2-2)+2=\tfrac12B_2^2+1\). Neither equals the exact Stratonovich identity in general.
Check: The Itô integral has expectation zero: it is a square-integrable Itô integral, hence a martingale, and its conditional expectation given \(\mathcal{F}_0\) is its initial value, zero. Thus the formula \(\tfrac12(B_2^2-2)\) has mean zero because \(\mathbb{E}[B_2^2]=2\). The correct Stratonovich integral has mean \(\tfrac12\mathbb{E}[B_2^2]=1\). The sign-error expression has mean \(-1\), and the full-correction expression has mean \(2\); both checks expose the mistake.
The Scalar SDE Test: Differentiate the Diffusion
Let \(X_t\) be scalar and suppose it satisfies a Stratonovich equation driven by one Brownian motion: $$ dX_t=a(t,X_t)\,dt+b(t,X_t)\circ dB_t. $$ When \(b\) is continuously differentiable in its state variable, the corresponding Itô equation has drift \(a+\tfrac12 b\,\partial_x b\) and the same diffusion coefficient \(b\). The derivative is essential: the correction is not generally one half of \(b\) or one half of \(b^2\). The coefficients should have enough regularity for the formula and the solution to be defined; if they are unbounded, the calculation can be applied up to exit times from bounded regions.
Proof. By the definition of the Stratonovich integral, its correction is \(\tfrac12[b(t,X_t),B]_t\). The finite-variation drift contributes zero covariation with \(B\). The diffusion part of \(X\) is \(b(t,X_t)\,dB_t\); applying the covariation chain rule to \(b(t,X_t)\) and \(B_t\) gives $$ d[b(t,X_t),B]_t=\partial_xb(t,X_t)b(t,X_t)\,dt. $$ (The time dependence contributes finite variation and hence no covariation.) Therefore the integral correction is \(\tfrac12\int_0^t \partial_xb(s,X_s)b(s,X_s)\,ds\). Substituting it into the Stratonovich equation yields the stated Itô drift and leaves the diffusion coefficient unchanged. \(\square\)
Worked Example: The Linear Stratonovich Equation
Convert \(dX_t=X_t\circ dB_t\), \(X_0=1\), and check the answer with a candidate solution. The scalar diffusion is \(b(x)=x\), so \(\partial_xb(x)=1\). The correction is \(\tfrac12x\cdot1=\tfrac12x\), giving $$ dX_t=\frac12X_t\,dt+X_t\,dB_t. $$
To verify a candidate, take \(X_t=e^{B_t}\). Apply Itô’s formula to \(f(x)=e^x\), for which \(f'(x)=e^x\), \(f''(x)=e^x\), and the correction term is \(\tfrac12 f''(B_t)\,dt=\tfrac12e^{B_t}\,dt\). Hence $$ d(e^{B_t})=e^{B_t}\,dB_t+\frac12e^{B_t}\,dt, $$ which is exactly the converted equation, with initial value \(e^{B_0}=1\). This check is valid on each finite time interval: \(\mathbb{E}\int_0^T e^{2B_s}\,ds=\int_0^T e^{2s}\,ds<\infty\).
Check: The candidate has expectation \(\mathbb{E}[e^{B_t}]=e^{t/2}\), using the moment-generating function of \(N(0,t)\). This agrees with the drift contribution in the converted equation: \(e^{B_t}\) is not being claimed to be a martingale. Omitting the correction would give \(dX_t=X_t\,dB_t\); the candidate \(e^{B_t}\) fails that equation because its Itô formula contains the nonzero term \(\tfrac12e^{B_t}\,dt\).
Worked Example: A Missing-Derivative Error
Consider \(dX_t=X_t^2\circ dB_t\), starting from \(X_0=1\). To avoid any issue with unbounded coefficients, stop at \(\tau=\inf\{t\ge0:|B_t|\ge\tfrac12\}\) and consider the equation up to \(t\wedge\tau\). On this interval the candidate below and its diffusion coefficient are bounded, so the stopped integrands are progressively measurable and square-integrable on every finite horizon.
Here \(b(x)=x^2\) and \(\partial_xb(x)=2x\). The correction is $$ \frac12b(x)\partial_xb(x)=\frac12x^2(2x)=x^3, $$ so the Itô equation up to \(\tau\) has drift \(X_t^3\) and diffusion coefficient \(X_t^2\).
For an independent check, use the candidate \(X_t=(1-B_t)^{-1}\) before \(\tau\). With \(f(y)=(1-y)^{-1}\), its derivatives are \(f'(y)=(1-y)^{-2}\) and \(f''(y)=2(1-y)^{-3}\). Itô’s formula, including the term \(\tfrac12 f''(B_t)\,dt\), gives $$ dX_t=(1-B_t)^{-2}\,dB_t+(1-B_t)^{-3}\,dt=X_t^2\,dB_t+X_t^3\,dt. $$ Also \(X_0=1\), so the candidate verifies the converted equation. At \(B_t=\tfrac14\), for example, \(X_t=\tfrac43\) and the correction is \(X_t^3=\tfrac{64}{27}\). The incorrect rule “add half the diffusion” gives \(\tfrac12X_t^2=\tfrac89\), which is different. The derivative must be taken before multiplying by the diffusion coefficient.
Correlated Noise: Keep the Cross-Covariations
For a two-dimensional Brownian motion \(\mathbf{B}_t=(B_t^1,B_t^2)\) with constant covariance matrix \(C\), the convention is \([B^j,B^k]_t=C_{jk}t\). Each Itô integral below is understood for progressively measurable integrands \(H_j\) satisfying \(\mathbb{E}\int_0^T H_j(s)^2\,ds<\infty\). For a scalar state and Stratonovich noise fields \(V_1,V_2\), the correction to the drift is $$ \frac12\sum_{j=1}^2\sum_{k=1}^2 C_{jk}V_k(x)\,\partial_xV_j(x). $$ The diagonal terms alone are correct only when the noise components have zero cross-covariation.
Worked Example: Correlation Changes the Drift
Let \(C=\begin{pmatrix}1&\tfrac12\\[2pt]\tfrac12&1\end{pmatrix}\), and consider $$ dX_t=X_t\circ dB_t^1+2X_t\circ dB_t^2,\qquad X_0=1. $$ The fields are \(V_1(x)=x\) and \(V_2(x)=2x\), with derivatives \(V_1'(x)=1\) and \(V_2'(x)=2\). The full correction is $$ \frac12\left(C_{11}V_1'V_1+C_{12}V_1'V_2+C_{21}V_2'V_1+C_{22}V_2'V_2\right) =\frac12\left(x+x+x+4x\right)=\frac72x. $$ Thus the converted Itô equation is \(dX_t=\tfrac72X_t\,dt+X_t\,dB_t^1+2X_t\,dB_t^2\). Ignoring correlation keeps only the two diagonal terms and incorrectly gives \(\tfrac12(x+4x)=\tfrac52x\).
To check the cross terms independently, set \(Y_t=B_t^1+2B_t^2\). Its quadratic variation is $$ [Y]_t=[B^1]_t+4[B^2]_t+4[B^1,B^2]_t=t+4t+4\left(\frac12t\right)=7t. $$ The Stratonovich equation is \(dX_t=X_t\circ dY_t\). Its candidate solution is \(X_t=e^{Y_t}\). Apply Itô’s formula to \(f(y)=e^y\), with \(f'(y)=e^y\) and \(f''(y)=e^y\). The second-derivative correction is \(\tfrac12 f''(Y_t)\,d[Y]_t=\tfrac72e^{Y_t}\,dt\), so $$ d(e^{Y_t})=e^{Y_t}\,dB_t^1+2e^{Y_t}\,dB_t^2+\frac72e^{Y_t}\,dt. $$ This agrees with the conversion. The integrands are square-integrable on finite intervals because \(Y_t\) is centered normal with variance \(7t\), so \(\mathbb{E}[e^{2Y_t}]=e^{14t}\) and \(\int_0^T e^{14t}\,dt<\infty\). As a further check, \(\mathbb{E}[X_t]=e^{7t/2}\), from the normal moment-generating function. The independent-noise answer would instead give \(e^{5t/2}\), confirming that it misses the correlation contribution.
A Reliable Error-Checking Routine
For a proposed conversion, check the following in order. First, write down the defining identity between the Stratonovich and Itô integrals; this fixes the positive sign and the factor \(\tfrac12\). Second, identify the state-dependent diffusion field and differentiate it with respect to the state. Third, for multiple drivers, write the entire covariance matrix and include every cross term. Finally, verify the proposed Itô equation using an exact candidate or a Brownian integral identity, applying Itô’s formula with its second-derivative correction shown explicitly.
| Proposed conversion | Diagnostic question | Correction to check |
|---|---|---|
| Wrong sign | Does it agree with \([B,B]_t=t\)? | Add \(+\tfrac12[H,B]_t\) |
| Wrong factor | Is the correction half the covariation? | Use \(\tfrac12[H,B]_t\), not \([H,B]_t\) |
| Missing derivative | Was the diffusion field differentiated in the state? | Use \(\tfrac12b\,\partial_xb\) in the scalar case |
| Missing correlation | Were all cross-covariations included? | Use the full covariance matrix \(C\) |
A common mistake is to treat \(dB_t\,dB_t=dt\) as if it directly supplied the entire drift correction. It supplies the quadratic-variation factor, but the chain rule also differentiates the integrand, and a multidimensional calculation also uses cross-covariations. The correction belongs to the drift; the diffusion coefficient itself remains unchanged.
Check Your Understanding
For each question, state any covariance or stopping assumptions needed, then justify the conversion or check.
- For standard Brownian motion, what Itô integral equals \(\int_0^T B_s\circ dB_s\), and why is its correction positive?
- Convert \(dX_t=\sin(X_t)\circ dB_t\). Which derivative is required in the drift correction?
- For two Brownian components with covariance rate \(C_{12}=\rho\), where does \(\rho\) enter the conversion of \(\sum_jV_j(X_t)\circ dB_t^j\)?
- Apply Itô’s formula to \(e^{\lambda B_t}\), showing the second-derivative correction, and identify the drift in its Itô differential.
- Why does checking only diagonal terms give an incorrect conversion when the driving Brownian components are correlated?