Choosing a Convention Is Part of Specifying the Model
An SDE is not fully specified by writing down a drift and a diffusion coefficient. With state-dependent diffusion, the same displayed drift can describe different stochastic processes under Itô and Stratonovich interpretations. A defensible PK specification therefore states both the convention and what its drift is intended to mean.
The choice should be guided by three questions: what mechanism is represented by the noise, which state variables are being modeled, and what interpretation is intended for the drift? These questions work together. For example, an idealized rapidly fluctuating environmental input can motivate a Stratonovich model, while a drift intended to equal the infinitesimal conditional mean change naturally points to an Itô specification. Neither statement is a universal rule: the modeler must make the assumptions explicit.
For notation, let \(X_t\) be a state in an open subset of \(\mathbb{R}^d\), and let \(W_t\) be an \(m\)-dimensional standard Brownian motion with independent components. Write \(B_{ij}(x)\) for the effect of Brownian component \(j\) on state component \(i\). Assume coefficients are sufficiently regular for the equations and conversions below to be defined.
This conversion relationship is established in the earlier material on Itô and Stratonovich interpretations. Its practical implication here is that matching the written drift values is not the same as matching the modeled dynamics. First decide which drift meaning is wanted; then specify the coefficient in the chosen convention or convert it appropriately.
What Does the Drift Mean?
In an Itô model, the drift has a precise local-mean interpretation under ordinary regularity conditions. Suppose \(a_I\) and \(B\) are globally bounded and globally Lipschitz, and start the process at \(X_0=x\). Then $$ \lim_{h\downarrow 0}\frac{E[X_h-x]}{h}=a_I(x). $$ Thus if the intended drift is the local conditional mean rate of change, the Itô coefficient is the direct quantity to specify. The Stratonovich drift does not generally have that interpretation; its corresponding Itô drift includes the correction above.
Proof. The integral form of the Itô equation is $$ X_h-x=\int_0^h a_I(X_s)\,ds+\int_0^h B(X_s)\,dW_s. $$ The stochastic integral has mean zero because the bounded integrand is square-integrable. Therefore $$ \frac{E[X_h-x]}{h}=\frac{1}{h}\int_0^h E[a_I(X_s)]\,ds. $$ If \(M_a\) bounds \(|a_I|\), and \(M_B\) bounds the matrix norm of \(B\), the Itô isometry and the drift bound give, for \(0\leq s\leq h\), $$ E|X_s-x|^2\leq 2M_a^2s^2+2M_B^2s. $$ The right-hand side tends to zero as \(s\downarrow0\). If \(L_a\) is a Lipschitz constant for \(a_I\), then $$ |E[a_I(X_s)]-a_I(x)|\leq L_a E|X_s-x| \leq L_a\sqrt{E|X_s-x|^2}, $$ which also tends to zero. Its average over \(0\leq s\leq h\) consequently tends to zero. Taking \(h\downarrow0\) in the integral expression proves the claim. \(\square\)
State Variables Also Constrain the Choice
The same physical model can be written using different state variables, such as an amount or its logarithm. A convention should be documented alongside the state because a drift coefficient can change meaning under a coordinate transformation. For Stratonovich equations, a smooth change of variables obeys the ordinary chain rule. For Itô equations, the second-derivative term in Itô’s formula must be included.
Derivation. The Stratonovich chain rule gives the displayed equation directly. To see why this agrees with the Itô representation, first convert the equation for \(X\) to Itô form, whose drift is \(a_S+c\), where $$ c_i=\frac{1}{2}\sum_{j=1}^{m}\sum_{k=1}^{d}B_{kj}\partial_kB_{ij}. $$ Itô’s formula gives the drift of \(Y_r=\phi_r(X)\) as $$ \sum_i\partial_i\phi_r(a_S+c)_i+ \frac{1}{2}\sum_{i,k,j}B_{ij}B_{kj}\partial_{ik}\phi_r. $$ The transformed diffusion coefficient is \(\widetilde B_{rj}=\sum_i\partial_i\phi_r B_{ij}\). Its Stratonovich-to-Itô correction is $$ \widetilde c_r=\frac{1}{2}\sum_{j,q}\widetilde B_{qj}\partial_{y_q}\widetilde B_{rj}. $$ By the chain rule for derivatives in the new coordinates, expanding this expression gives $$ \widetilde c_r= \sum_i\partial_i\phi_r c_i+ \frac{1}{2}\sum_{i,k,j}B_{ij}B_{kj}\partial_{ik}\phi_r. $$ Subtracting this correction from the Itô drift of \(Y\) leaves \(\sum_i\partial_i\phi_r(a_S)_i\), exactly the stated Stratonovich drift. The coordinate transformation therefore preserves the ordinary chain-rule interpretation, provided the corresponding drift and diffusion are transformed together. \(\square\)
Worked Decisions in PK-Style Models
Worked Example: Relative Noise With a Smooth-Forcing Interpretation
Suppose positive concentration \(C_t\) is modeled with proportional noise, and the assumed mechanism is an idealized rapidly fluctuating input whose limiting interpretation is Stratonovich. The proposed model is $$ dC_t=-0.4C_t\,dt+0.3C_t\circ dW_t. $$ The stated Stratonovich drift is \(-0.4C\). To compare it with the local mean rate, calculate the conversion term. Here \(b(C)=0.3C\) and \(b'(C)=0.3\), so $$ \frac{1}{2}b(C)b'(C)=\frac{1}{2}(0.3C)(0.3)=0.045C. $$ The equivalent Itô equation is therefore $$ dC_t=(-0.4C_t+0.045C_t)\,dt+0.3C_t\,dW_t =(-0.355C_t)\,dt+0.3C_t\,dW_t. $$ At \(C=1\), the Stratonovich drift is \(-0.4\), but the infinitesimal conditional mean rate is \(-0.355\). This is not a contradiction: the coefficient \(-0.4C\) was selected as the Stratonovich drift, not the Itô local-mean drift. A specification should record the forcing assumption and this distinction rather than call \(-0.4C\) simply “the mean elimination rate.”
Worked Example: A Drift Intended as the Local Mean Rate
Suppose instead that the modeling target is an Itô local mean drift of \(-0.2C\), with a smooth state-dependent noise coefficient \(b(C)=0.6\sqrt{C+1}\), for \(C>-1\). At \(C=4\), $$ b(4)=0.6\sqrt{5},\qquad b'(4)=\frac{0.6}{2\sqrt{5}}=\frac{0.3}{\sqrt{5}}. $$ The conversion correction is $$ \frac{1}{2}b(4)b'(4) =\frac{1}{2}(0.6\sqrt{5})\left(\frac{0.3}{\sqrt{5}}\right) =0.09. $$ The intended Itô drift at \(C=4\) is \(-0.2(4)=-0.8\). If the same dynamics are written in Stratonovich form, its drift must be the Itô drift minus the correction: $$ (a_S)(4)=-0.8-0.09=-0.89. $$ More generally, the correction is \(0.09\) at every \(C>-1\), so an equivalent Stratonovich drift is \(-0.2C-0.09\). Choosing Itô is particularly direct when the stated drift is intended to equal the local conditional mean rate; choosing Stratonovich remains possible if the coefficient is adjusted and the intended meaning is documented.
Worked Example: Recording the State Transformation
Take the positive-concentration model from the first example and define \(Y_t=\log C_t\). The Stratonovich chain rule gives $$ dY_t=\frac{1}{C_t}(-0.4C_t)\,dt+\frac{1}{C_t}(0.3C_t)\circ dW_t =-0.4\,dt+0.3\circ dW_t. $$ The diffusion of \(Y\) is constant, so its conversion correction is zero. Its Itô form is therefore \(dY_t=-0.4\,dt+0.3\,dW_t\). This agrees with applying Itô’s formula to the concentration equation: the Itô drift of \(C\) is \(-0.355C\), and the logarithm contributes the second-derivative term \(-\tfrac12(0.3)^2=-0.045\), giving \(-0.355-0.045=-0.4\).
A clear record would say that the original model uses concentration with a Stratonovich specification, and that the log-concentration representation has drift \(-0.4\) and constant diffusion \(0.3\). Merely copying the original drift into a log-state equation would not be a justified transformation.
A Practical Convention-Selection Record
A convention is a modeling decision, not a label to infer after fitting. A useful specification records the state and units, the source and structure of each noise driver, the convention, and what the drift coefficient is meant to represent. It should also identify whether a displayed equation is the original specification or an equivalent equation converted to the other convention.
State whether the process represents concentration, amount, log concentration, or another quantity, and specify its units and admissible range.
Say whether the Brownian term represents an idealized rapidly varying external input, an effective random perturbation, or another explicitly defined mechanism. A mechanistic story alone does not prove which limiting convention is appropriate.
For example, specify that the Itô drift is the infinitesimal conditional mean rate, or that the Stratonovich drift is the coefficient associated with the assumed smooth-noise limit.
For state-dependent diffusion, calculate the correction. If the state is transformed, derive the new equation under the chosen convention instead of carrying over drift coefficients unchanged.
Comparison, Diagnostics, and Limits
| Modeling consideration | Itô specification | Stratonovich specification |
|---|---|---|
| Drift interpretation | Directly gives the local conditional mean rate, under regularity conditions | Requires the conversion correction to obtain the Itô local mean rate |
| Noise interpretation | Natural for models specified directly through conditional increments or martingale noise | Often used for limits of suitably regularized, rapidly varying noise |
| Change of smooth state coordinates | Use Itô’s formula, including its second-derivative term | Transforms by the ordinary chain rule |
| Constant diffusion | Conversion correction is zero | Equivalent drift coefficient when diffusion is constant |
A common mistake is to read a Stratonovich coefficient as the expected instantaneous change because it appears beside \(dt\). That reasoning overlooks the correction: when \(B\) depends on the state, the Stratonovich coefficient need not equal the Itô local-mean drift. The correction can be checked by computing \(\frac12\sum_{j,k}B_{kj}\partial_kB_{ij}\) for each state component, with the chosen Brownian channels held fixed.
Other warning signs include a model document that omits the convention, a drift interpretation that changes between equations without explanation, or a transformed state equation whose drift was copied without applying the appropriate calculus rule. If a calculated correction is unexpectedly large, verify the derivatives, channel indices, and units. The correction has units of state per time, just like a drift; a mismatch there is evidence of an error.
These conventions are idealizations. A real environmental driver may have a finite correlation time, in which case neither white-noise convention necessarily captures all relevant dynamics. The Stratonovich interpretation can be motivated by particular smooth-noise limits, but the mechanism and limiting assumptions must support that choice. Likewise, an Itô specification is not automatically a claim that the underlying physical fluctuations are discontinuous; it can simply define a model through its conditional dynamics. The convention does not replace biological justification, and it does not resolve uncertainty about whether the chosen diffusion structure is appropriate.
Check Your Understanding
For each question, identify the convention issue and compare your reasoning with the solutions.
- A one-dimensional model has \(b(x)=0.5x\). If its Stratonovich drift at \(x=2\) is \(-0.6\), what is its Itô drift there?
- A model is specified in Itô form with drift \(a_I(x)=-0.3x\) and diffusion \(b(x)=0.4x\). What Stratonovich drift gives the same process at \(x=1\)?
- Why is it not enough to write “the drift is the elimination rate” when specifying a model with state-dependent diffusion?
- A Stratonovich model has constant diffusion and is changed to a smooth one-to-one state coordinate. What chain rule applies to its drift, and what must be checked before using it?
Solutions. 1. The correction is \(\tfrac12b(2)b'(2)=\tfrac12(1)(0.5)=0.25\). Thus \(a_I(2)=-0.6+0.25=-0.35\). 2. The correction at \(x=1\) is \(\tfrac12(0.4)(0.4)=0.08\). Hence \(a_S(1)=a_I(1)-0.08=-0.3-0.08=-0.38\). 3. The coefficient could be an Itô drift or a Stratonovich drift; with state-dependent diffusion those generally differ. The specification must identify the convention and whether the intended rate is a local conditional mean or another drift interpretation. 4. Apply the ordinary chain rule to the Stratonovich drift, alongside the transformed diffusion. Check that the coordinate map is sufficiently smooth and one-to-one on the state region in use; if instead rewriting in Itô form, include Itô’s second-derivative term.