Concentration-Dependent Diffusion Conversion

Why the Drift Depends on the Diffusion Amplitude

A concentration-dependent diffusion does more than determine the size of random fluctuations. Its dependence on concentration also affects the drift assigned to the same process when the SDE is written using Ito rather than Stratonovich integration. This matters when translating a proposed PK model between conventions: retaining the same displayed drift in both forms generally changes the model.

The central calculation is local. If $C_t$ is concentration and $b(C_t)$ is the diffusion amplitude multiplying a Brownian motion, the conversion adjustment is one-half the product of that amplitude and its derivative. In particular, square-root and proportional amplitudes produce different concentration patterns in the adjustment.

Definition: Suppose $C_t$ is a continuous semimartingale satisfying a one-dimensional Stratonovich SDE $$ dC_t=a_S(C_t)\,dt+b(C_t)\circ dW_t. $$ On an interval where $b$ is twice continuously differentiable and the coefficients satisfy local existence and uniqueness conditions, its Ito form is $$ dC_t=\left(a_S(C_t)+\frac{1}{2}b(C_t)b'(C_t)\right)dt+b(C_t)\,dW_t. $$ Conversely, an Ito drift $a_I$ corresponds to the Stratonovich drift $a_S=a_I-\frac{1}{2}bb'$. The subscript identifies the convention used for the drift.

The correction concerns the diffusion amplitude $b$, not its square. For a scalar Brownian driver, the instantaneous variance rate is $b(C)^2$, but the conversion term is $\frac{1}{2}b(C)b'(C)$. The distinction is essential when the amplitude depends on concentration.

Deriving the Conversion Rule

The Stratonovich integral is the symmetric-integral convention. For a continuous semimartingale integrand, its relation to the Ito integral is $$ \int_0^t b(C_s)\circ dW_s = \int_0^t b(C_s)\,dW_s+\frac{1}{2}[b(C),W]_t, $$ where brackets denote quadratic covariation. The next derivation identifies that covariation for a concentration-dependent amplitude.

Result: If $C$ is a continuous semimartingale with diffusion term $b(C_t)dW_t$, and $b$ is twice continuously differentiable on the state interval under consideration, then $$ [b(C),W]_t=\int_0^t b'(C_s)b(C_s)\,ds. $$ Consequently, replacing the Stratonovich integral by the Ito integral adds $\frac{1}{2}bb'$ to the drift.

Proof. Write the semimartingale decomposition as $dC_t=u_t\,dt+b(C_t)dW_t$, where $u_tdt$ has finite variation. Ito's formula gives $$ db(C_t)=b'(C_t)u_t\,dt+b'(C_t)b(C_t)\,dW_t+\frac{1}{2}b''(C_t)b(C_t)^2\,dt. $$ The finite-variation terms have zero quadratic covariation with $W$. The covariation of the stochastic-integral term with $W$ is the integral of its integrand, so $$ [b(C),W]_t=\left[\int_0^\cdot b'(C_s)b(C_s)\,dW_s,W\right]_t =\int_0^t b'(C_s)b(C_s)\,ds. $$ Substituting this identity into the Stratonovich-to-Ito integral relation yields the stated drift adjustment. If coefficients are only locally regular, the argument applies up to exit times from compact subintervals on which the required derivatives and bounds hold. $\square$

The adjustment has a direct interpretation: when $b$ varies with concentration, the symmetric integral samples the noise amplitude in a way that contributes a finite-variation term. If $b$ is constant, then $b'=0$, and there is no conversion adjustment.

Square-Root Diffusion

A square-root amplitude is common when the local variance rate is proportional to concentration. Let $b(C)=\sigma\sqrt{C}$, with $\sigma>0$ and $C>0$. Then $$ b'(C)=\frac{\sigma}{2\sqrt{C}}, \qquad \frac{1}{2}b(C)b'(C) =\frac{1}{2}\left(\sigma\sqrt{C}\right)\left(\frac{\sigma}{2\sqrt{C}}\right) =\frac{\sigma^2}{4}. $$ Thus the conversion adjustment is constant over the positive-concentration interval, even though the diffusion amplitude depends on concentration.

Worked Example: Converting a Square-Root Concentration Model

Suppose a concentration is specified in Stratonovich form by $$ dC_t=(\alpha-kC_t)\,dt+\sigma\sqrt{C_t}\circ dW_t, $$ where $\alpha=0.8\ \mathrm{mg\,L^{-1}\,h^{-1}}$, $k=0.25\ \mathrm{h^{-1}}$, and $\sigma=0.6\ (\mathrm{mg\,L^{-1}})^{1/2}\mathrm{h^{-1/2}}$. Convert it to Ito form and evaluate both drifts at $C=4\ \mathrm{mg\,L^{-1}}$.

Here $b(C)=0.6\sqrt{C}$, so $b'(C)=0.3/\sqrt{C}$. The adjustment is $$ \frac{1}{2}b(C)b'(C) =\frac{1}{2}(0.6\sqrt{C})(0.3/\sqrt{C}) =0.09\ \mathrm{mg\,L^{-1}\,h^{-1}}. $$ The Ito drift is therefore $$ \alpha-kC+\frac{\sigma^2}{4} =0.8-0.25C+0.09 =0.89-0.25C. $$ At $C=4$, the Stratonovich drift is $0.8-0.25(4)=-0.20\ \mathrm{mg\,L^{-1}\,h^{-1}}$. The Ito drift is $-0.20+0.09=-0.11\ \mathrm{mg\,L^{-1}\,h^{-1}}$. Both equations use the same diffusion amplitude $0.6\sqrt{C}$; the drift difference is precisely the conversion adjustment.

The adjustment's units are concentration per time: $\sigma^2$ has units $\mathrm{mg\,L^{-1}\,h^{-1}}$. This unit check agrees with the drift units.

Proportional Diffusion

For a proportional amplitude, $b(C)=\sigma C$, the derivative is $b'(C)=\sigma$, and the adjustment is concentration-dependent: $$ \frac{1}{2}b(C)b'(C)=\frac{\sigma^2}{2}C. $$ The adjustment is proportional to concentration, unlike the constant adjustment from square-root noise.

Worked Example: Converting Proportional Noise

Consider the Stratonovich model $dC_t=-kC_t\,dt+\sigma C_t\circ dW_t$, with $k=0.30\ \mathrm{h^{-1}}$, $\sigma=0.40\ \mathrm{h^{-1/2}}$, and $C=5\ \mathrm{mg\,L^{-1}}$. Find the Ito drift coefficient and its value at the stated concentration.

Here $b'(C)=\sigma$, so the adjustment is $$ \frac{1}{2}b(C)b'(C)=\frac{1}{2}(\sigma C)\sigma =\frac{\sigma^2}{2}C =\frac{0.16}{2}C =0.08C. $$ Adding it to the Stratonovich drift gives the Ito drift $(-0.30+0.08)C=-0.22C$, in concentration per hour. At $C=5$, this is $-0.22(5)=-1.10\ \mathrm{mg\,L^{-1}\,h^{-1}}$. The Stratonovich drift at that concentration is $-0.30(5)=-1.50\ \mathrm{mg\,L^{-1}\,h^{-1}}$; the difference, $0.40\ \mathrm{mg\,L^{-1}\,h^{-1}}$, equals $0.08(5)$.

The adjustment does not mean that the model has acquired an additional random driver. The Brownian term is unchanged. The drift changes so the two written equations describe the same process under different integral conventions.

Converting in Reverse and Checking Equivalence

Conversion also works from Ito to Stratonovich, but the sign reverses. For example, if an Ito model has drift $a_I(C)=q-kC$ and proportional amplitude $b(C)=\sigma C$, then its Stratonovich drift is $$ a_S(C)=q-kC-\frac{\sigma^2}{2}C. $$ This subtraction is not optional: using $q-kC$ unchanged as the Stratonovich drift would specify a different process.

Worked Example: Recovering a Stratonovich Drift

An Ito model is $dC_t=(0.5-0.12C_t)\,dt+0.30C_t\,dW_t$, with concentration in $\mathrm{mg\,L^{-1}}$, time in hours, and $C=3\ \mathrm{mg\,L^{-1}}$. Determine the equivalent Stratonovich drift at $C=3$.

For $b(C)=0.30C$, the conversion adjustment is $$ \frac{1}{2}bb'=\frac{(0.30)^2}{2}C=0.045C. $$ Subtract it from the Ito drift to obtain $a_S(C)=0.5-0.12C-0.045C=0.5-0.165C$. At $C=3$, $a_I=0.5-0.36=0.14$, while $a_S=0.5-0.495=0.005$. Their difference is $0.135=0.045(3)$, as required.

Result: The conversion preserves the represented process when the initial value and Brownian driver are the same and the drift is adjusted by $\frac{1}{2}bb'$. Indeed, the Stratonovich integral identity shows that the finite-variation contribution $\frac{1}{2}\int_0^t b(C_s)b'(C_s)\,ds$ is exactly the difference between the two drift integrals. Thus the two equations have identical integral forms after conversion, up to the applicable existence interval.

Assumptions, Diagnostics, and Model Choice

The calculations above apply on a state interval where the diffusion amplitude is differentiable and the SDE has a well-defined solution. For proportional noise $b(C)=\sigma C$, the amplitude is smooth. For square-root noise $b(C)=\sigma\sqrt{C}$, the derivative is unbounded as $C$ approaches zero, and the amplitude is not differentiable at zero. The formula is valid for $C>0$, or up to the first time the process reaches a boundary where the required regularity fails. Behavior at zero requires a model-specific boundary analysis; the interior conversion formula alone does not establish existence, uniqueness, or nonnegativity there.

Several checks help detect a faulty conversion:

Differentiate the amplitude $b(C)$, not the variance rate $b(C)^2$.
Use a plus sign for Stratonovich-to-Ito conversion and a minus sign for Ito-to-Stratonovich conversion.
Check that the adjustment has the units of concentration per time.
At a concentration where $b'(C)=0$, the local adjustment must vanish.
Compare the two drifts at a chosen concentration; their difference must equal $\frac{1}{2}bb'$, with the sign determined by the direction of conversion.

A common mistake is to write the same drift beside the same diffusion amplitude in both Ito and Stratonovich form and call the equations equivalent. That is only correct when $bb'=0$ on the relevant state interval. The correction is also not a new biological input or a separate source of process noise: it is the drift adjustment required to preserve the chosen stochastic process under a change of convention.

The modeling choice remains important. One may specify a model directly in Ito form or in Stratonovich form, but the displayed drift then has meaning relative to that convention. Conversion is a translation between specifications, not a rule for deciding which convention is scientifically preferable. In particular, the adjustment does not by itself resolve whether a proposed diffusion amplitude or boundary behavior is physiologically appropriate.

Check Your Understanding

Answer each question, then check the short solutions.

For $b(C)=0.8\sqrt{C}$, calculate the Stratonovich-to-Ito drift adjustment.
For $b(C)=0.5C$, calculate the adjustment at $C=4$.
An Ito model has drift $a_I(C)$ and amplitude $b(C)$. What is its Stratonovich drift?
Why should the square-root conversion formula be applied cautiously near zero?

Solutions. (1) The adjustment is $\sigma^2/4=0.8^2/4=0.16$, in the units of the drift. (2) It is $\frac{1}{2}(0.5C)(0.5)=0.125C$, so at $C=4$ it equals $0.50$. (3) It is $a_S(C)=a_I(C)-\frac{1}{2}b(C)b'(C)$. (4) The square-root amplitude is not differentiable at zero and its derivative is unbounded there, so the interior conversion calculation does not settle the boundary behavior.