Cross-Diffusion Terms in Two-Compartment Transformations

Why a Two-State Transformation Has a Cross Term

A transformation of one stochastic compartment state has a curvature correction involving its second derivative. With two states, there can also be a mixed second derivative: it describes how the transformation curves jointly in the two state directions. That term matters when the states have nonzero instantaneous covariance. In a pharmacokinetic model, shared process noise or correlated noise sources can make this covariance nonzero, even when the states are tracked separately.

The key calculation is to pair the mixed derivative of a function with the instantaneous cross-variation of the states. A mixed derivative alone does not guarantee a correction, and correlated noise alone does not guarantee one either: both must be nonzero. The calculations below keep these two ingredients distinct.

Definition: Let $B_1$ and $B_2$ be Brownian motions with $d[B_1,B_2]_t=\rho\,dt$, where $-1\leq\rho\leq1$. Suppose $dX_t=a_X(t)\,dt+s_X(t)\,dB_1(t)$ and $dY_t=a_Y(t)\,dt+s_Y(t)\,dB_2(t)$. Their instantaneous cross-variation rate is $c_{XY}(t)=\rho s_X(t)s_Y(t)$, meaning $d[X,Y]_t=c_{XY}(t)\,dt$.

Here $a_X,a_Y,s_X,s_Y$ may depend on time and the states, provided they are adapted and locally integrable in the senses needed to define the stochastic integrals. The Brownian motions are defined relative to the same filtration. The correlation parameter must lie in $[-1,1]$. For a transformation $f$, assume $f$ is twice continuously differentiable on an open region containing the states of interest; for logarithms, that region must also have positive states. As usual, the formula can be applied up to exit from that region.

The Two-State Ito Formula

For $Z_t=f(X_t,Y_t)$, the mixed contribution to the drift is $f_{xy}(X_t,Y_t)c_{XY}(t)\,dt$. There is no extra factor of one-half on this term. The one-half in Ito's formula multiplies the full Hessian quadratic form, in which the two equal mixed contributions combine.

Theorem: Under the conditions above, the Ito differential is $$ \begin{aligned} df(X_t,Y_t) ={}&\left[ f_xa_X+f_ya_Y +\frac{1}{2}f_{xx}s_X^2 +f_{xy}\rho s_Xs_Y +\frac{1}{2}f_{yy}s_Y^2 \right]dt\\ &+f_xs_X\,dB_1+f_ys_Y\,dB_2. \end{aligned} $$ All derivatives and coefficients on the right are evaluated at the current time and states.

To see why the mixed term has this coefficient, write the diffusion increments as $dX=s_XdB_1$ and $dY=s_YdB_2$ when calculating quadratic variations; finite-variation drift increments do not contribute to these variations. The second-order part of the two-variable Ito formula is $\frac12 f_{xx}\,d[X]+f_{xy}\,d[X,Y]+\frac12 f_{yy}\,d[Y]$. Substituting $d[X]=s_X^2dt$, $d[Y]=s_Y^2dt$, and $d[X,Y]=\rho s_Xs_Ydt$ gives the displayed drift terms. Equivalently, the Hessian contribution is one-half of $f_{xx}s_X^2+2f_{xy}\rho s_Xs_Y+f_{yy}s_Y^2$; the two off-diagonal entries explain why the mixed term appears once, without a one-half. The first-order terms are $f_xdX+f_ydY$, which give the displayed drift and diffusion terms by substitution into the Ito formula. This is the two-state formula; its specific cross-diffusion contribution follows directly from the quadratic covariation.

Result: For the stated model, $d[X,Y]_t=\rho s_Xs_Y\,dt$. If instead the states are driven by independent Brownian motions $W_1,\ldots,W_m$, with diffusion coefficients $\sigma_{Xk}$ and $\sigma_{Yk}$ on driver $W_k$, then $$ d[X,Y]_t=\left(\sum_{k=1}^{m}\sigma_{Xk}\sigma_{Yk}\right)dt. $$

For the first identity, the finite-variation parts of $X$ and $Y$ have zero quadratic covariation with continuous semimartingales. The remaining parts are $s_XdB_1$ and $s_YdB_2$; bilinearity of quadratic covariation gives $d[X,Y]=s_Xs_Yd[B_1,B_2]=\rho s_Xs_Ydt$. For independent drivers, expand the covariation of the two stochastic-integral parts. Terms on distinct drivers have zero quadratic covariation, while each same-driver term contributes $\sigma_{Xk}\sigma_{Yk}dt$. Summing those terms proves the second identity. In particular, a shared driver with coefficients $s_X,s_Y$ gives rate $s_Xs_Y$, including its sign. $\square$

Worked Transformations

Consider a local two-compartment model with states $X,Y$ and current values $X=10$, $Y=4$. Suppose the drift rates at this instant are $a_X=-1$, $a_Y=0.2$, and the diffusion terms are $2\,dB_1$ and $1.2\,dB_2$, with $\rho=0.5$. Thus $s_X=2$, $s_Y=1.2$, and the cross-variation rate is $0.5(2)(1.2)=1.2$. These are local instantaneous calculations; they do not assert that the rates are constant along a whole trajectory.

Worked Example: Transforming the Product of Two States

Let $Z=XY$. The derivatives are $f_x=Y$, $f_y=X$, $f_{xx}=f_{yy}=0$, and $f_{xy}=1$. At $X=10,Y=4$, the first-order drift contribution is $4(-1)+10(0.2)=-4+2=-2$. The mixed correction is $1(1.2)=1.2$. Therefore the drift rate of $Z$ is $-2+1.2=-0.8$.

The diffusion terms are $f_xs_XdB_1=4(2)dB_1=8dB_1$ and $f_ys_YdB_2=10(1.2)dB_2=12dB_2$. Hence $dZ=-0.8\,dt+8\,dB_1+12\,dB_2$ at this instant. As a check on the cross contribution, $d[X,Y]/dt=1.2$, and the product's mixed derivative is exactly one, so their product adds $1.2$ to the drift. Omitting it would give the incorrect drift rate $-2$.

Worked Example: Transforming the Ratio of Two States

Let $R=X/Y$, which is defined here because $Y=4>0$. Its derivatives are $f_x=1/Y=1/4$, $f_y=-X/Y^2=-10/16=-0.625$, $f_{xx}=0$, $f_{xy}=-1/Y^2=-1/16=-0.0625$, and $f_{yy}=2X/Y^3=20/64=0.3125$.

The drift contributions are $f_xa_X=(1/4)(-1)=-0.25$, $f_ya_Y=(-0.625)(0.2)=-0.125$, $\frac12f_{yy}s_Y^2=\frac12(0.3125)(1.2^2)=0.225$, and $f_{xy}\rho s_Xs_Y=(-0.0625)(1.2)=-0.075$. The $f_{xx}$ contribution is zero. Adding them gives $-0.25-0.125+0.225-0.075=-0.225$.

The diffusion coefficients are $f_xs_X=(1/4)(2)=0.5$ and $f_ys_Y=(-0.625)(1.2)=-0.75$. Thus $dR=-0.225\,dt+0.5\,dB_1-0.75\,dB_2$ at the specified state and rates. The mixed correction is negative because the ratio has $f_{xy}<0$ while the cross-variation rate is positive.

Worked Example: A Sum Has No Mixed Drift Correction

Let $S=X+Y$. Then $f_x=f_y=1$ and all second derivatives are zero, so the drift rate is $a_X+a_Y=-1+0.2=-0.8$. The diffusion is $2dB_1+1.2dB_2$. There is no mixed second-derivative correction to the drift of $S$.

The sum's instantaneous variance rate is nevertheless $2^2+1.2^2+2(2)(1.2)(0.5)=4+1.44+2.4=7.84$. The final term is the covariance contribution between the two diffusion terms. It appears when calculating the quadratic variation of $S$, even though $f_{xy}=0$. Thus covariance in a transformed process and a mixed Ito drift correction are related but distinct calculations.

Checks, Alternatives, and Limitations

A reliable calculation separates three questions: what are the derivatives of the transformation, what is the cross-variation rate of the states, and how does the formula combine them? For example, under independent noises $\rho=0$, the mixed drift correction vanishes, even if $f_{xy}\ne0$. Under shared noise it need not vanish. If the noise coefficients have opposite signs, the cross-variation rate can be negative; correlation is not automatically a positive covariance contribution.

A practical alternative to working with correlated Brownian motions is to express the diffusion in terms of independent drivers and use the sum of coefficient products for $d[X,Y]$. Both calculations give the same cross-variation rate. For multiple drivers, this sum is often less error-prone than trying to identify a single correlation parameter. In either representation, it is the covariance structure of the state increments that enters the mixed term.

Common diagnostics are to check that the proposed correlation lies in $[-1,1]$, that the signs of both diffusion coefficients are retained, and that the transformation is twice differentiable on the states being considered. A missing factor of two in the covariance expansion or an erroneous one-half on $f_{xy}$ can be detected by expanding the Hessian quadratic form. For a product, $f_{xy}=1$, which makes the cross term especially easy to check. For a separable transformation $f(X,Y)=g(X)+h(Y)$, $f_{xy}=0$, so there can be no mixed drift correction, although the transformed diffusion components may still be correlated.

The formula is an Ito formula: its drift correction is tied to quadratic variation under the Ito interpretation. It is a local statement and requires the stated regularity and integrability conditions; transformations such as a ratio or logarithm also require the state to remain in their domains. It does not by itself specify a pharmacokinetic model or establish that a particular correlation structure is biologically appropriate. Those are modeling decisions, separate from the calculation.

Key takeaway: The mixed Ito drift contribution is $f_{xy}\,d[X,Y]/dt$. Calculate the cross-variation rate first, then multiply by the mixed derivative. Do not confuse that drift correction with the covariance contribution in the quadratic variation of the transformed state.

Check Your Understanding

Answer each question, then compare with the solutions.

If $d[X,Y]/dt=0.3$ and $f_{xy}=2$, what is the mixed drift contribution?
For independent Brownian drivers, state the cross-variation rate when the diffusion coefficients on the shared driver $W_1$ are $\sigma_{X1}=0.4$ and $\sigma_{Y1}=-0.5$, and all other driver products are zero.
Why does $f(X,Y)=g(X)+h(Y)$ have no mixed Ito drift correction?
In the worked product example, what drift rate would result from incorrectly omitting the cross term?
Can a transformation have zero mixed drift correction while its diffusion components remain correlated? Give the sum transformation as an example.

Solutions. (1) The contribution is $f_{xy}d[X,Y]/dt=2(0.3)=0.6$. (2) The rate is $\sigma_{X1}\sigma_{Y1}=0.4(-0.5)=-0.2$. (3) Its mixed derivative is zero because differentiating $g(X)+h(Y)$ once in each variable gives zero. (4) Omitting the correction gives the first-order drift $-2$, instead of the correct $-0.8$. (5) Yes. For $X+Y$, $f_{xy}=0$, so the mixed drift correction is zero, but correlated Brownian drivers still contribute $2s_Xs_Y\rho$ to the quadratic-variation rate of the sum.

Cross-Diffusion Terms in Two-Compartment Transformations

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