Power Transformations of Concentration States

Why a Power Transformation Changes the Drift

A pharmacokinetic model may describe the concentration \(C_t\), while a calculation or summary uses its square or cube. Such a transformation changes both the random fluctuations and their average trend. The reason is that an Itô process has nonzero quadratic variation: the curvature of the transformation contributes an additional drift term. Ignoring that contribution gives a different stochastic model.

Consider a scalar Itô SDE driven by a standard Brownian motion \(W_t\):

dC_t=a(t,C_t)\,dt+b(t,C_t)\,dW_t.

Here \(a\) is the drift coefficient and \(b\) is the diffusion coefficient. For a precise local statement, assume the coefficients are measurable in time and locally Lipschitz in the state, with conditions ensuring a nonexplosive solution on the time interval considered. The Itô formula applies to a twice continuously differentiable function of this solution. For real powers, we additionally work where \(C_t>0\), so the power and its derivatives are defined. If positivity is not guaranteed for all time, the calculation is valid up to the first time the process leaves a positive region.

Definition: For a positive concentration state \(C_t\) and a power \(p>0\), the power-transformed state is \(X_t=C_t^p\). Its Itô drift is the coefficient multiplying \(dt\) in its SDE; its Itô diffusion is the coefficient multiplying \(dW_t\).

The Power-Transformation Formula

For \(f(c)=c^p\), the derivatives on \(c>0\) are \(f'(c)=p c^{p-1}\) and \(f''(c)=p(p-1)c^{p-2}\). The scalar Itô formula says that \(df(C_t)=f'(C_t)dC_t+\tfrac12 f''(C_t)b(t,C_t)^2dt\). Substituting the SDE and collecting the terms gives the general power rule.

Theorem: Under the conditions above, while \(C_t>0\), the transformed process \(X_t=C_t^p\) satisfies

dX_t= \left[ pC_t^{p-1}a(t,C_t) +\frac{p(p-1)}{2}C_t^{p-2}b(t,C_t)^2 \right]dt + pC_t^{p-1}b(t,C_t)\,dW_t.

This follows directly from the derivatives: the first derivative multiplies both the original drift and diffusion, while the second derivative contributes the extra drift. The result is local if the process can approach zero or become unbounded. More formally, one can stop the process on leaving any compact interval contained in \((0,\infty)\), apply Itô’s formula there, and use the formula up to that exit time. This avoids assuming that the derivatives or coefficients remain bounded over the entire state space.

For the square and cube, the coefficients take especially useful forms:

d(C_t^2)= \left[2C_ta(t,C_t)+b(t,C_t)^2\right]dt +2C_tb(t,C_t)\,dW_t,

d(C_t^3)= \left[3C_t^2a(t,C_t)+3C_tb(t,C_t)^2\right]dt +3C_t^2b(t,C_t)\,dW_t.

The correction is \(b^2\) for the square and \(3Cb^2\) for the cube. It is nonnegative when \(C>0\), but the total transformed drift need not be positive: the original drift may offset or exceed it.

Worked Example: Squaring a Concentration at a Given State

Suppose \(dC_t=(0.4-0.2C_t)dt+0.3C_t\,dW_t\), and consider the instant when \(C_t=2\). Find the drift and diffusion of \(X_t=C_t^2\) at that state.

The original drift is \(a=0.4-0.2(2)=0\), and the diffusion is \(b=0.3(2)=0.6\). The square formula gives transformed drift \(2C_ta+b^2\) and diffusion \(2C_tb\). Thus

2(2)(0)+(0.6)^2=0.36,\qquad 2(2)(0.6)=2.4.

At this state, \(dX_t\) therefore has drift coefficient \(0.36\) and diffusion coefficient \(2.4\). Although the concentration’s instantaneous drift is zero, its square has positive drift because of the quadratic-variation correction.

Consequences for Common Concentration Models

If the diffusion is proportional to concentration, \(b(t,C_t)=\sigma C_t\), the square’s correction is \(\sigma^2C_t^2\). A power transformation can therefore preserve a multiplicative form of the diffusion while changing its drift. The next example checks this both from the transformation formula and from the explicit solution.

Worked Example: Squaring a Geometric Concentration Process

Consider \(dC_t=-0.5C_t\,dt+0.2C_t\,dW_t\), with \(C_0=10\). Find the SDE for \(X_t=C_t^2\), then check its mean using the explicit solution.

Here \(a=-0.5C_t\) and \(b=0.2C_t\). The square formula gives

2C_ta+b^2 =-C_t^2+0.04C_t^2 =-0.96X_t, \qquad 2C_tb=0.4C_t^2=0.4X_t.

Hence \(dX_t=-0.96X_t\,dt+0.4X_t\,dW_t\). The original geometric SDE has solution \(C_t=10\exp(-0.52t+0.2W_t)\), so

X_t=100\exp(-1.04t+0.4W_t).

Since \(W_t\sim N(0,t)\), the normal exponential moment gives \(E[\exp(0.4W_t)]=\exp(0.08t)\). Therefore \(E[X_t]=100\exp(-0.96t)\), agreeing with the transformed drift. In particular, the exponent in the pathwise expression is \(-1.04t\), while the mean decays at rate \(0.96\); these are not interchangeable quantities.

For any power \(p>0\), the transformed drift also gives an expectation identity when the relevant integrals are integrable. Specifically, suppose on \([0,T]\) that

E\int_0^T \left|pC_s^{p-1}a(s,C_s) +\frac{p(p-1)}{2}C_s^{p-2}b(s,C_s)^2\right|ds<\infty, \qquad E\int_0^T \left|pC_s^{p-1}b(s,C_s)\right|^2ds<\infty.

Then the stochastic integral in the transformed equation is square-integrable and has expectation zero. Taking expectations in its integral form yields

E[C_t^p]=E[C_0^p]+ E\int_0^t\left[ pC_s^{p-1}a(s,C_s) +\frac{p(p-1)}{2}C_s^{p-2}b(s,C_s)^2 \right]ds,\qquad 0\le t\le T.

This is an integral identity, not permission to replace a random state by its mean inside a nonlinear coefficient. Such a replacement generally changes the expectation.

Worked Example: Cubing with Additive Diffusion

Suppose locally \(dC_t=(1-0.1C_t)dt+0.5\,dW_t\), and the concentration is \(C_t=4\) at the instant of interest. Calculate the drift and diffusion of \(Y_t=C_t^3\).

For the cube, the drift is \(3C_t^2a+3C_tb^2\), and the diffusion is \(3C_t^2b\). At \(C_t=4\), \(a=1-0.1(4)=0.6\), \(b=0.5\), and \(b^2=0.25\). Thus

3(4)^2(0.6)+3(4)(0.25)=28.8+3=31.8, \qquad 3(4)^2(0.5)=24.

The transformed drift coefficient is \(31.8\) and the diffusion coefficient is \(24\). Additive Brownian noise does not ensure that \(C_t\) stays positive. The calculation is valid while the path remains in the positive region; for instance, it can be applied up to the first time \(C_t\) reaches zero.

Common Errors, Diagnostics, and Alternatives

A frequent mistake is to use the ordinary chain rule and write \(d(C_t^2)=2C_t\,dC_t\). That omits \(b(t,C_t)^2dt\). The correction is present even if the drift in the original equation is zero, as the first worked example shows. For a cube, omitting the correction loses \(3C_tb^2dt\). A practical check is to set \(b=0\): the correction should disappear and the ordinary chain rule should be recovered. Another check is to verify that the diffusion coefficient is \(f'(C_t)b(t,C_t)\), not \(f(C_t)b(t,C_t)\).

The formula is for an Itô SDE. It also depends on the state being in the domain of the chosen power; real noninteger powers require \(C_t>0\). If a model does not guarantee positivity, do not silently extend a fractional power through zero. For integer powers the algebraic function is defined at zero, but a concentration interpretation may still require stopping or otherwise addressing the possibility of negative states.

One alternative is to keep the SDE in the original concentration state and calculate a transformed quantity only when needed. This avoids introducing a new state equation, though it does not remove the correction when calculating its stochastic dynamics. Another is to derive the transformed SDE directly with Itô’s formula, as above. In either case, comparing the resulting coefficients against the formula is a useful diagnostic. A missing correction, incorrect power of \(C_t\), or sign error will often show up in that comparison.

Key takeaway: For \(X_t=C_t^p\), the transformed diffusion is \(pC_t^{p-1}b\), and the transformed drift includes the curvature term \(\tfrac12p(p-1)C_t^{p-2}b^2\). A power transformation changes the stochastic dynamics, not just the units or scale of the state.

Check Your Understanding

Work through each question, then check the concise solutions below.

If \(dC_t=a_tdt+b_tdW_t\), write the drift and diffusion of \(C_t^2\).
For \(dC_t=-kC_tdt+\sigma C_tdW_t\), what is the drift of \(C_t^2\) in terms of \(C_t^2\)?
For \(dC_t=0.2dt+0.3dW_t\), calculate the drift correction in the SDE for \(C_t^3\) at \(C_t=2\).
Why is \(d(C_t^2)=2C_tdC_t\) generally incorrect for an Itô process?

Solutions. (1) The drift is \(2C_ta_t+b_t^2\), and the diffusion is \(2C_tb_t\). (2) The drift is \((-2k+\sigma^2)C_t^2\), since \(2C_t(-kC_t)+(\sigma C_t)^2=(-2k+\sigma^2)C_t^2\). (3) The cube correction is \(3C_tb_t^2\), so it equals \(3(2)(0.3)^2=0.54\). (4) It omits the quadratic-variation term \(b_t^2dt\); Itô’s formula gives \(d(C_t^2)=2C_tdC_t+b_t^2dt\).