Bounded Fractional Effects With Jacobi Diffusion

Why Use a Bounded Effect Diffusion?

A pharmacodynamic response expressed as a fraction—such as fractional inhibition, receptor occupancy, or the fraction of a maximal response—must lie between zero and one. An additive-noise model can leave that interval, even when its deterministic trajectory stays inside it. A Jacobi diffusion addresses this by making the random fluctuation shrink to zero at both endpoints while the drift pulls the effect toward an exposure-dependent target.

Let $Y_t$ denote the fractional effect at time $t$, and let $C_t$ denote drug concentration. The model below uses a mean-reverting drift with a drug-dependent target and a state-dependent diffusion. For clarity, the main results take the rate and noise scale to be constant; the exposure path may vary over time.

Definition: The exposure-driven Jacobi diffusion is the SDE $$ dY_t=\kappa\{\mu(C_t)-Y_t\}\,dt+\sigma\sqrt{Y_t(1-Y_t)}\,dW_t, $$ where $W_t$ is Brownian motion, $\kappa>0$, $\sigma\geq 0$, and the target function satisfies $0\leq\mu(c)\leq1$ for every concentration $c$ used by the model. The initial condition must satisfy $Y_0\in[0,1]$. The drift relaxes the effect toward $\mu(C_t)$; the diffusion is largest in the interior and vanishes at zero and one.

The coefficient $\sigma$ has units of inverse square-root time, while $\kappa$ has units of inverse time. The target $\mu(C_t)$ is dimensionless. One possible exposure-response target is $\mu(C)=\mu_0+\mu_{\max}C/(EC_{50}+C)$, with $EC_{50}>0$ and parameters chosen so that $0\leq\mu_0\leq\mu_0+\mu_{\max}\leq1$. This choice places the drug dependence in the drift; it does not make the entire effect an instantaneous deterministic function of concentration.

Interval Invariance and Boundary Behavior

The vanishing diffusion at the endpoints is important, but it is not enough on its own to ensure the effect stays in the interval: the drift must also point inward there. Here, at $Y=0$, the drift is $\kappa\mu(C_t)\geq0$; at $Y=1$, it is $\kappa\{\mu(C_t)-1\}\leq0$. The following result makes that boundary argument precise.

Theorem: Suppose $\mu(t)$ is a measurable deterministic function taking values in $[0,1]$, $\kappa>0$, $\sigma\geq0$ are finite constants, and $Y_0\in[0,1]$. The SDE $$ dY_t=\kappa\{\mu(t)-Y_t\}\,dt+\sigma\sqrt{Y_t(1-Y_t)}\,dW_t $$ has a pathwise unique nonexplosive solution, and $P(0\leq Y_t\leq1\text{ for all }t\geq0)=1$.

Justification. To apply standard existence and uniqueness results without presupposing that the solution remains in the interval, extend the diffusion coefficient to the real line as $g(y)=\sigma\sqrt{y_+(1-y)_+}$, where $x_+=\max(x,0)$. This coefficient is globally $1/2$-Hölder continuous and has at most linear growth; the drift $b(t,y)=\kappa\{\mu(t)-y\}$ is measurable in time, globally Lipschitz in $y$, and has linear growth. The standard existence and pathwise uniqueness theorem for an SDE with Lipschitz drift and a $1/2$-Hölder diffusion therefore gives a unique global solution.

For the interval claim, take smooth nonnegative approximations $\phi_n$ to $y^-=\max(-y,0)$, with $\phi_n=0$ on $y\geq0$, $\phi_n'\leq0$ on $y<0$, and $\phi_n''$ supported on $y<0$. On that region, $g(y)=0$ and $b(t,y)=\kappa\{\mu(t)-y\}\geq0$. Itô's formula therefore has no stochastic term for $\phi_n(Y_t)$, no second-derivative term, and a nonpositive drift term $\phi_n'(Y_t)b(t,Y_t)$. After localization if needed, $\phi_n(Y_t)\leq\phi_n(Y_0)=0$. Since $\phi_n\geq0$, passage to the approximated negative part gives $Y_t\geq0$. Apply the same argument to smooth approximations of $(y-1)^+$: above one, $g(y)=0$, $b(t,y)\leq0$, and the derivative of the upper-part approximation is nonnegative. This gives $Y_t\leq1$. Continuity of the solution extends the bounds from each fixed time to all times simultaneously. $\square$

In this model the interval is invariant, but an endpoint need not be unreachable. Under constant exposure, write $\theta=\mu(C)$. If $0<\theta<1$, the classical Feller boundary conditions for the Jacobi diffusion say that zero is inaccessible from the interior when $2\kappa\theta\geq\sigma^2$, and one is inaccessible when $2\kappa(1-\theta)\geq\sigma^2$. If either condition fails, that endpoint is attainable; with an interior target, the standard Jacobi solution is not made absorbing there and is pushed back inward by the drift. If $\theta=0$, zero is absorbing once reached, and if $\theta=1$, one is absorbing once reached, since both drift and diffusion vanish at the corresponding endpoint. These constant-target classifications should not be applied unchanged when exposure, and hence the target, varies over time.

Worked Example: Mean Effect After a Change in Exposure

Suppose exposure is held constant at a level for which $\theta=0.70$. Let $\kappa=0.8$ per hour, $\sigma=0.30$ per square-root hour, and $Y_0=0.20$. Find the mean effect after two hours.

For this constant target, take expectations in the SDE. The stochastic integral has mean zero: on any finite interval its integrand is square-integrable because $\sigma^2Y_t(1-Y_t)\leq\sigma^2/4$. Thus $m(t)=E[Y_t]$ satisfies $m'(t)=\kappa(\theta-m(t))$, with $m(0)=0.20$. Solving this linear equation gives $$ m(t)=\theta+\{m(0)-\theta\}e^{-\kappa t}. $$

At $t=2$, $m(2)=0.70+(0.20-0.70)e^{-0.8(2)} =0.70-0.50e^{-1.6}$. Using $e^{-1.6}\approx0.2019$, the mean is approximately $0.5991$. It lies between the initial effect and the target, as the mean-reverting drift suggests. This calculation is for the mean; it does not say that every simulated path moves monotonically toward $0.70$.

Moment Behavior Under Constant Exposure

For constant exposure and an interior target $\theta$, the Jacobi diffusion has a stationary beta distribution. The moment calculation below gives its mean and variance directly, without requiring a transition-density calculation. In particular, the noise level affects the stationary spread even though the long-run mean equals the drift target.

Result: If $0<\theta<1$, $\kappa>0$, and $\sigma>0$, the stationary distribution is $Y\sim\operatorname{Beta}(\alpha,\beta)$, where $\alpha=2\kappa\theta/\sigma^2$ and $\beta=2\kappa(1-\theta)/\sigma^2$. Consequently, $$ E[Y]=\theta,\qquad \operatorname{Var}(Y)=\frac{\theta(1-\theta)\sigma^2}{2\kappa+\sigma^2}. $$

Derivation. For a twice differentiable function $f$, the generator is $\mathcal L f(y)=\kappa(\theta-y)f'(y)+\frac{\sigma^2}{2}y(1-y)f''(y)$. At stationarity, the first-moment equation gives $E[Y]=\theta$. Applying Itô's formula to $Y_t^2$, using the same bounded-integrand argument on finite intervals, gives $$ \frac{d}{dt}E[Y_t^2] =2\kappa\theta E[Y_t]-(2\kappa+\sigma^2)E[Y_t^2]+\sigma^2E[Y_t]. $$ At stationarity, substitute $E[Y]=\theta$ and set the derivative to zero. This yields $E[Y^2]=\theta(2\kappa\theta+\sigma^2)/(2\kappa+\sigma^2)$. Subtracting $\theta^2$ gives the stated variance. To identify the distribution, the zero-flux stationary density on $(0,1)$ is proportional to $y^{2\kappa\theta/\sigma^2-1}(1-y)^{2\kappa(1-\theta)/\sigma^2-1}$, which is the beta density with the stated parameters. The parameters are positive because $0<\theta<1$, so the density is normalizable. $\square$

Worked Example: Checking Boundary Conditions and Stationary Spread

Let $\kappa=1$ per hour, $\sigma=0.60$ per square-root hour, and hold exposure so that $\theta=0.20$. Check whether either boundary is inaccessible from the interior, then calculate the stationary variance.

Here $\sigma^2=0.36$. At zero, $2\kappa\theta=2(1)(0.20)=0.40\geq0.36$, so zero is inaccessible from the interior. At one, $2\kappa(1-\theta)=2(1)(0.80)=1.60\geq0.36$, so one is also inaccessible. The beta parameters are $\alpha=0.40/0.36\approx1.111$ and $\beta=1.60/0.36\approx4.444$, both positive.

The stationary variance is $$ \frac{(0.20)(0.80)(0.36)}{2(1)+0.36} =\frac{0.0576}{2.36} \approx0.02441. $$ The corresponding stationary standard deviation is approximately $\sqrt{0.02441}=0.1562$. Inaccessibility of the endpoints does not mean the effect has little variability: the variance is appreciable relative to the unit interval.

Worked Example: A Bounded Drug-Dependent Target

Suppose concentration $C$, measured in mg/L, determines the target $\mu(C)=0.10+0.60C/(2+C)$. Take $C=3$ mg/L, $\kappa=0.50$ per hour, $\sigma=0.40$ per square-root hour, and $Y_0=0.20$. Calculate the target and the mean after one hour, and check the constant-target boundary conditions.

The target is $\theta=0.10+0.60(3)/(2+3)=0.10+0.36=0.46$. For all $C\geq0$, the fraction $C/(2+C)$ is between zero and one, so this target lies between $0.10$ and $0.70$; it stays within the required interval.

Using the mean formula, $E[Y_1]=0.46+(0.20-0.46)e^{-0.50} =0.46-0.26(0.6065)\approx0.3023$. For the boundary checks, $\sigma^2=0.16$, $2\kappa\theta=2(0.50)(0.46)=0.46\geq0.16$, and $2\kappa(1-\theta)=2(0.50)(0.54)=0.54\geq0.16$. Both endpoints are inaccessible from the interior under this fixed concentration and these parameters.

Diagnostics, Alternatives, and Limitations

A Jacobi diffusion is useful when a response is genuinely fractional and stochastic variation should diminish near its physical limits. It does not automatically make a proposed exposure-response model adequate. Check that the target remains in $[0,1]$ over the full concentration range used for prediction; inspect whether the estimated $\kappa$ and $\sigma$ imply plausible time scales and variability; and assess predictive calibration and residual structure using the diagnostics appropriate for SDE models. If simulated paths or fitted predictions concentrate at a boundary unexpectedly, examine the target, parameter estimates, and boundary classification rather than treating the pattern as harmless numerical behavior.

The continuous-time model stays bounded under the theorem's assumptions, but a basic Euler step need not. A discretized update can jump below zero or above one, especially with large time steps or near a boundary. Silently clipping such values to zero or one changes the simulated dynamics and can create artificial boundary mass. Use a simulation method appropriate for the diffusion, assess time-step sensitivity, and report any numerical boundary corrections.

Approach	What it offers	Main caution
Jacobi diffusion	State-dependent noise and a continuous-time effect that remains in $[0,1]$ under the stated conditions.	Boundary behavior and simulation require care; the diffusion form may not fit the observed variability.
Additive-noise SDE	A simple, often convenient fluctuation model.	It can produce impossible fractional effects outside the interval.
Transforming an unconstrained process	A transformation such as a logistic map can produce a bounded response.	The induced drift and diffusion on the response scale differ from the Jacobi specification.
Clipping an unconstrained update	A quick numerical patch.	It is not equivalent to the continuous-time Jacobi model and can distort boundary frequencies.

The Jacobi specification also has limits. Its diffusion is symmetric in the sense that its local variance is proportional to $Y(1-Y)$; data may call for a different state-dependent noise structure. A single target and relaxation rate may not capture delayed response or tolerance. The stationary beta result assumes constant exposure and an interior target, so it should not be read as the stationary distribution under a changing concentration profile.

Common Mistake

Assuming vanishing noise alone prevents boundary crossing. The diffusion coefficient is zero at the endpoints, but the model's boundary argument also uses the inward drift signs and a target in $[0,1]$.
Calling an attainable boundary absorbing. For an interior constant target, an endpoint can be attainable when the Feller condition fails, yet the inward drift does not make it absorbing.
Using a stationary variance formula during changing exposure. The formula describes the constant-target stationary distribution, not a general transient or time-varying-exposure variance.
Treating clipped Euler simulations as exact Jacobi paths. Clipping changes the numerical law; check step-size effects and use an appropriate simulation method.

Key takeaway: A bounded fractional effect can be modeled by a Jacobi diffusion when its target stays in $[0,1]$, the drift points inward at the endpoints, and the diffusion vanishes there. Invariance, boundary attainability, and numerical behavior are distinct questions and should be checked separately.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

For $\mu(t)\in[0,1]$, what are the drift values at $Y=0$ and $Y=1$, and why do their signs matter?
With constant $\theta=0.30$, $\kappa=0.50$ per hour, and $\sigma=0.50$ per square-root hour, are the boundaries inaccessible from the interior?
For a constant target $\theta=0.40$, $\kappa=1$, $\sigma=0.50$, and $Y_0=0.10$, find $E[Y_1]$.
Why is clipping an Euler update that falls below zero not the same as simulating the Jacobi diffusion?

Solutions. (1) At zero the drift is $\kappa\mu(t)\geq0$; at one it is $\kappa\{\mu(t)-1\}\leq0$. Together with zero diffusion at the endpoints, these inward signs support interval invariance. (2) Here $\sigma^2=0.25$. At zero, $2\kappa\theta=2(0.50)(0.30)=0.30\geq0.25$; at one, $2\kappa(1-\theta)=2(0.50)(0.70)=0.70\geq0.25$. Both boundaries are inaccessible from the interior. (3) $E[Y_1]=0.40+(0.10-0.40)e^{-1}=0.40-0.30(0.3679)\approx0.2896$. (4) Clipping changes the update rule and can add artificial mass at zero; it does not reproduce the continuous-time diffusion's law.