Reading Scalar Conversion Examples
For a scalar stochastic differential equation, conversion changes the drift but leaves the diffusion coefficient unchanged. The correction depends on both the diffusion coefficient and its derivative with respect to the state. This dependence is especially visible when the noise is multiplicative: even if the Stratonovich drift is zero, the corresponding Itô drift need not be.
Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). On a fixed finite time interval \([0,T]\), an Itô integrand \(H\) is assumed progressively measurable and in \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). Thus its Itô integral is square-integrable. In particular, its increments have conditional mean zero: \(\mathbb{E}[\int_s^t H_u\,dB_u\mid\mathcal{F}_s]=0\) for \(s\le t\). Stratonovich integrals below are defined by the Itô integral and its quadratic covariation correction; the integrands in the examples satisfy the same finite-horizon integrability requirement.
We use \(dB_t\,dB_t=dt\), \(dB_t\,dt=0\), and \(dt\,dt=0\). These are multiplication rules for stochastic differentials, not ordinary algebraic identities between finite increments. The drift is the coefficient of \(dt\), and the diffusion coefficient is the coefficient of \(dB_t\).
The correction comes from the state dependence of the integrand \(\sigma(t,X_t)\). In the scalar case its contribution is one half of the covariation of that integrand with \(B\). The following calculation records why this gives the coefficient shown above.
Proof. Write the equation first in Itô form with drift \(b\): \(dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\). Apply Itô's formula to the state function \(x\mapsto\sigma(t,x)\), or localize and apply it when the coefficients are only locally bounded. The terms involving \(dt\) have zero covariation with \(B\), while the Brownian term has coefficient \(\partial_x\sigma(t,X_t)\sigma(t,X_t)\). Therefore $$ d[\sigma(t,X_t),B]_t =\sigma(t,X_t)\partial_x\sigma(t,X_t)\,dt. $$ By the definition of the Stratonovich integral, $$ \int_0^t \sigma(s,X_s)\circ dB_s =\int_0^t \sigma(s,X_s)\,dB_s +\tfrac12[\sigma(\,\cdot\,,X),B]_t. $$ Substitution into the equation adds precisely \(\tfrac12\sigma\partial_x\sigma\) to the Itô drift. Solving this same coefficient identity for the Stratonovich drift gives the inverse formula. The calculation is local, so stopping at bounded regions and then removing the stops gives the result whenever the stated integrals exist. \(\square\)
This conversion concerns the form of an equation, not a change of probability measure: all equations here are under \(\mathbb{P}\). Nor does converting coefficients by itself assert a particular existence or uniqueness theorem. A strong solution is adapted to the given Brownian motion on the given probability space; a weak solution permits the probability space and Brownian motion to be part of the construction. Pathwise uniqueness compares solutions driven by the same Brownian motion, whereas uniqueness in law compares their distributions. The examples below focus on coefficient conversion and make no additional uniqueness claim.
Geometric Brownian Motion: Multiplicative Noise
Worked Example: Converting a Geometric Brownian Motion
Convert the Stratonovich equation \(dS_t=\tfrac14 S_t\,dt+\tfrac12 S_t\circ dB_t\), with \(S_0=2\), into Itô form. Then verify a candidate solution directly.
Here \(\mu(x)=\tfrac14x\), \(\sigma(x)=\tfrac12x\), and \(\partial_x\sigma(x)=\tfrac12\). The correction is $$ \tfrac12\sigma(x)\partial_x\sigma(x) =\tfrac12\left(\tfrac12x\right)\left(\tfrac12\right) =\tfrac18x. $$ Consequently the Itô drift is \(\tfrac14x+\tfrac18x=\tfrac38x\), while the diffusion coefficient remains \(\tfrac12x\): $$ dS_t=\tfrac38S_t\,dt+\tfrac12S_t\,dB_t. $$
For an independent verification, set \(f(t,b)=2\exp(\tfrac14t+\tfrac12b)\) and \(S_t=f(t,B_t)\). Its derivatives are \(f_t=\tfrac14f\), \(f_b=\tfrac12f\), and \(f_{bb}=\tfrac14f\). Itô's formula, including its second-derivative correction, gives $$ dS_t=f_t\,dt+f_b\,dB_t+\tfrac12f_{bb}\,dt =\left(\tfrac14+\tfrac18\right)S_t\,dt+\tfrac12S_t\,dB_t =\tfrac38S_t\,dt+\tfrac12S_t\,dB_t. $$ Also \(S_0=2\exp(0)=2\), as required.
Check: Since \(B_t\) has distribution \(N(0,t)\), its exponential moment is \(\mathbb{E}[e^{B_t/2}]=e^{t/8}\). Thus \(\mathbb{E}[S_t]=2e^{t/4}e^{t/8}=2e^{3t/8}\). This is consistent with the Itô drift: the expectation of the displayed candidate has derivative \(\tfrac38(2e^{3t/8})\). All coefficients have at most linear growth, so the diffusion integrand is square-integrable on finite time intervals for this solution.
A Nonlinear Diffusion with a Trigonometric Coefficient
Worked Example: Converting a Sine Diffusion
Convert $$ dX_t=1\,dt+\sin(X_t)\circ dB_t,\qquad X_0=\frac{\pi}{6}, $$ to Itô form and check the correction at the initial state.
The Stratonovich drift is \(\mu(x)=1\), the diffusion coefficient is \(\sigma(x)=\sin x\), and its state derivative is \(\partial_x\sigma(x)=\cos x\). Hence $$ \tfrac12\sigma(x)\partial_x\sigma(x)=\tfrac12\sin x\cos x, $$ so the Itô equation is $$ dX_t=\left(1+\tfrac12\sin(X_t)\cos(X_t)\right)dt+\sin(X_t)\,dB_t. $$ At \(X_0=\pi/6\), \(\sin(\pi/6)=\tfrac12\) and \(\cos(\pi/6)=\tfrac{\sqrt3}{2}\). The correction at that state is \(\tfrac12\cdot\tfrac12\cdot\tfrac{\sqrt3}{2}=\tfrac{\sqrt3}{8}\). Thus the initial Itô drift coefficient is \(1+\tfrac{\sqrt3}{8}\), and the initial diffusion coefficient is \(\tfrac12\).
Check: Subtract the correction from the Itô drift function: \(1+\tfrac12\sin x\cos x-\tfrac12\sin x\cos x=1\). This recovers the original Stratonovich drift at every state, not just at \(X_0\). The coefficients and their derivatives are bounded, so the correction is well-defined and the diffusion integrand is square-integrable on every finite interval.
Converting in the Reverse Direction
Proof. If \(\mu\) is a Stratonovich drift for the given diffusion coefficient, the scalar conversion formula requires \(b=\mu+\tfrac12\sigma\partial_x\sigma\). Rearranging gives the displayed value of \(\mu\). Conversely, substituting that value into the conversion formula gives \(b-\tfrac12\sigma\partial_x\sigma+\tfrac12\sigma\partial_x\sigma=b\). Any drift producing the same Itô drift must satisfy the same equation and therefore must equal this value. This proves existence of the coefficient conversion and its uniqueness. \(\square\)
Worked Example: Recovering a Stratonovich Drift
Start with the Itô equation $$ dX_t=(1+X_t)\,dt+\sqrt{1+X_t^2}\,dB_t,\qquad X_0=2. $$ Find its Stratonovich drift and check both coefficients at the initial state.
Here \(b(x)=1+x\), \(\sigma(x)=\sqrt{1+x^2}\), and $$ \partial_x\sigma(x)=\frac{x}{\sqrt{1+x^2}}. $$ The correction is $$ \tfrac12\sigma(x)\partial_x\sigma(x) =\tfrac12\sqrt{1+x^2}\frac{x}{\sqrt{1+x^2}} =\tfrac12x. $$ The Stratonovich drift is therefore \(b(x)-\tfrac12x=1+\tfrac12x\), giving $$ dX_t=\left(1+\tfrac12X_t\right)dt+\sqrt{1+X_t^2}\circ dB_t. $$
At \(X_0=2\), the Itô drift is \(1+2=3\), the diffusion coefficient is \(\sqrt{1+2^2}=\sqrt5\), and the correction is \(\tfrac12\cdot2=1\). The Stratonovich drift at that state is \(1+\tfrac12\cdot2=2\). Converting back gives \(2+1=3\), while the diffusion coefficient remains \(\sqrt5\), as in the given Itô equation. The drift and diffusion are globally Lipschitz and have at most linear growth, so the standard existence theorem for Itô SDEs gives a strong solution with finite second moments on finite time intervals; in particular, its diffusion integrand satisfies the required \(\mathcal{L}^2\) condition.
A Common Conversion Error
The correction is not \(\tfrac12\partial_x\sigma\), and it is not \(\tfrac12\sigma^2\partial_x\sigma\). It is exactly \(\tfrac12\sigma\partial_x\sigma\). Omitting the factor \(\sigma\) can produce a wrong drift even when the derivative is easy to calculate.
For the sine diffusion above, the incorrect calculation \(\tfrac12\partial_x\sigma=\tfrac12\cos x\) would give a correction of \(\tfrac{\sqrt3}{4}\) at \(x=\pi/6\). The correct correction is \(\tfrac12\sin(\pi/6)\cos(\pi/6)=\tfrac{\sqrt3}{8}\). The discrepancy arises because the covariation depends on the Brownian coefficient \(\sigma(X_t)\) as well as on the derivative of that coefficient. A reliable check is to convert back: subtracting the correct correction from the Itô drift must recover the stated Stratonovich drift, and adding it to a proposed Stratonovich drift must recover the Itô drift.
Check Your Understanding
For each conversion, identify the drift, diffusion coefficient, and state derivative before applying the correction.
- Convert \(dX_t=2X_t\,dt+3X_t\circ dB_t\) to Itô form.
- An Itô equation has drift \(b(x)=x\) and diffusion coefficient \(\sigma(x)=\sin x\). What is its Stratonovich drift?
- For \(\sigma(x)=\sqrt{4+x^2}\), calculate \(\tfrac12\sigma(x)\partial_x\sigma(x)\).
- Explain why the diffusion coefficient is unchanged by scalar Stratonovich–Itô conversion.
- For the sine diffusion example, substitute the correction at \(x=\pi/6\) to check the initial Itô drift.