Why the observation regime matters
A discretely observed diffusion contributes a transition density to the likelihood for each observation interval. The density approximation that works well for closely spaced observations in a smooth interior region may fail when intervals are long, the drift changes rapidly, or the process is close to a boundary. In pharmacokinetics, those distinctions arise, for example, between frequent concentration measurements in a well-populated range and sparse measurements near zero.
Consider a one-dimensional time-homogeneous Itô diffusion \(dX_t=b(X_t)\,dt+\sigma(X_t)\,dW_t\), with \(a(x)=\sigma^2(x)\). For an interval of length \(h>0\), the transition density \(p(h,y\mid x)\) describes the conditional distribution of \(X_{t+h}=y\) given \(X_t=x\). An approximation is useful only when its support, scale, and local dynamics are credible for the interval being analyzed.
The formulas below assume enough smoothness for the derivatives used, and positive diffusion where a Lamperti transformation is applied. The earlier material on Lamperti transforms gives the coordinate change; here the focus is how its density approximation compares with alternatives. None of these approximations repairs a misspecified continuous-time model.
Four approximations and what they assume
The Euler approximation is the simplest choice when \(h\) is small relative to the timescale of drift and diffusion changes. It uses the starting values \(b(x)\) and \(a(x)\) throughout the interval. If the drift is locally close to linear but its slope matters over the interval, the local-linear approximation retains that slope while still freezing the diffusion.
For positive states with state-dependent diffusion, the Lamperti coordinate can give a more natural local Gaussian approximation. Suppose \(\sigma(x)>0\), and define \(z=\Lambda(x)=\int^x du/\sigma(u)\). Itô's formula gives the unit-diffusion process \(dZ_t=\alpha(Z_t)\,dt+dW_t\), where \(\alpha(\Lambda(x))=b(x)/\sigma(x)-\sigma'(x)/2\). A Gaussian step in \(Z\), transformed back with the density Jacobian, gives $$ p_T(h,y\mid x)=\frac{1}{\sigma(y)\sqrt{2\pi h}} \exp\!\left[-\frac{(\Lambda(y)-\Lambda(x)-\alpha(\Lambda(x))h)^2}{2h}\right]. $$ This approximation still freezes the transformed drift at its starting value. The factor \(1/\sigma(y)\) is essential: omitting it does not give the density with respect to \(dy\).
A short-time expansion can add an endpoint-dependent correction to the transformed Gaussian kernel. For a twice differentiable transformed drift \(\alpha\), let \(A\) be any antiderivative of \(\alpha\), and put \(V(z)=\alpha(z)^2+\alpha'(z)\). The leading bridge-corrected approximation is $$ p_S(h,y\mid x)=\frac{\exp\!\left[-\frac{(\Lambda(y)-\Lambda(x))^2}{2h}+A(\Lambda(y))-A(\Lambda(x))-\frac{h}{4}\{V(\Lambda(x))+V(\Lambda(y))\}\right]}{\sigma(y)\sqrt{2\pi h}}. $$ This is a local short-time approximation, not necessarily a normalized density for finite \(h\). It is most defensible when the interval is short and the transformed drift and its derivatives remain controlled across the interval.
Worked comparisons
Worked Example: Euler versus Local-Linear in a Mean-Reverting Regime
Take \(dX_t=(2-X_t)\,dt+0.3\,dW_t\), with \(x=1\), \(h=0.5\), and endpoint \(y=1.4\). Here \(b(1)=1\), \(J=-1\), and \(a=0.09\).
The Euler mean and variance are \(1+1(0.5)=1.5\) and \(0.09(0.5)=0.045\). The endpoint is \(0.1\) below that mean, so $$ p_E=\frac{1}{\sqrt{2\pi(0.045)}}\exp\!\left[-\frac{0.1^2}{2(0.045)}\right] \approx 1.88063(0.89484)=1.683. $$
For the local-linear approximation, \(m_L=1+(e^{-0.5}-1)/(-1)=1.393469\), and \(v_L=0.09(e^{-1}-1)/(-2)=0.0284454\). Thus \(y-m_L=0.006531\), giving $$ p_L=\frac{1}{\sqrt{2\pi(0.0284454)}}\exp\!\left[-\frac{(1.4-1.393469)^2}{2(0.0284454)}\right] \approx 2.3654(0.99925)=2.364. $$
The local-linear value is more appropriate here: the drift is exactly linear, so the approximation captures the actual mean reversion over the interval. In fact, with constant diffusion this model has an exact Gaussian transition and the local-linear formula gives its exact mean and variance. The Euler value is not the exact transition density.
Worked Example: A Lamperti Density for Multiplicative Noise
For geometric Brownian motion \(dX_t=0.2X_t\,dt+0.4X_t\,dW_t\), take \(x=10\), \(y=11\), and \(h=0.25\). The log coordinate is \(\Lambda(x)=\log x\), and the transformed drift is \(\alpha=0.2-0.4^2/2=0.12\). The transformed Gaussian formula gives $$ p_T=\frac{1}{11(0.4)\sqrt{0.25}\sqrt{2\pi}} \exp\!\left[-\frac{(\log(1.1)-0.12(0.25))^2}{2(0.4^2)(0.25)}\right]. $$
Here \(\log(1.1)=0.0953102\), the difference in the numerator is \(0.0653102\), and the denominator of the exponent is \(0.08\). The exponent is therefore \(-0.05332\) (to five decimal places). The prefactor is \(1/(2.2\sqrt{2\pi})\approx0.18134\), so \(p_T\approx0.1719\). The density is with respect to concentration \(y\), which explains the factor \(1/y\) from the log-coordinate Jacobian.
In this example the transformed drift is constant, so the transformed Gaussian transition is exact. This is an especially favorable case for the transformation, not a general guarantee that transformed Gaussian steps are exact.
Worked Example: What the Short-Time Correction Adds
Consider the transformed unit-diffusion process \(dZ_t=0.3\,dt+dW_t\), from \(z=2\) to \(w=2.1\) over \(h=0.2\). Here \(A(w)-A(z)=0.3(0.1)=0.03\), and \(V=0.3^2=0.09\). The short-time formula gives $$ p_S=\frac{1}{\sqrt{2\pi(0.2)}}\exp\!\left[-\frac{0.1^2}{2(0.2)}+0.03-\frac{0.2}{4}(0.09+0.09)\right] =0.89206\exp(-0.004)\approx0.88850. $$
The constant-drift transition is exactly \(N(2.06,0.2)\), whose density at \(2.1\) is the same value: its exponent relative to the mean is \(-(0.04)^2/(2(0.2))=-0.004\). This checks the correction in a case where it is exact. With nonconstant transformed drift, the endpoint correction is only a short-time approximation.
Justification and limits of the formulas
The local-linear formula follows by solving the linearized SDE. Write \(U_s=X_{t+s}-x\), so \(dU_s=(b(x)+JU_s)\,ds+\sigma(x)\,dW_{t+s}\), with \(U_0=0\). Multiplying by \(e^{-Js}\) and integrating yields $$ U_h=b(x)\int_0^h e^{J(h-s)}\,ds+\sigma(x)\int_0^h e^{J(h-s)}\,dW_{t+s}. $$ The first integral is \((e^{Jh}-1)/J\) for \(J\ne0\). The stochastic integral is Gaussian with mean zero and variance \(a(x)\int_0^h e^{2J(h-s)}\,ds=a(x)(e^{2Jh}-1)/(2J)\). Taking limits as \(J\to0\) gives \(h\) in both ratios. Adding \(x\) to the mean proves the stated formula. \(\square\)
The short-time correction can be justified using the Brownian-bridge representation. Assume the transformed drift is sufficiently smooth and bounded, or satisfies suitable local conditions ensuring the change-of-measure and conditional expectations below exist. Under a Brownian reference process starting at \(z\), Girsanov's formula and Itô's formula give the likelihood ratio along a path ending at \(w\): $$ \exp\!\left\{A(w)-A(z)-\frac12\int_0^h[\alpha(Z_s)^2+\alpha'(Z_s)]\,ds\right\}. $$ Conditioning the reference process on its endpoint gives the exact transformed density as the Brownian Gaussian kernel times the bridge expectation of this ratio. For short \(h\) and endpoints with \(w-z=O(\sqrt h)\), the bridge is concentrated near the segment from \(z\) to \(w\). If \(V\) is smooth with bounded derivatives in this region, its bridge expectation satisfies $$ E\!\left[\int_0^h V(Z_s)\,ds\mid Z_0=z,Z_h=w\right] =\frac{h}{2}\{V(z)+V(w)\}+O(h^2). $$ The integral is \(O(h)\), so replacing the expectation of its exponential by the exponential of this trapezoidal approximation incurs a higher-order local error. Substitution gives the stated leading short-time correction. The argument is local; it does not establish accurate tails or normalization over a long interval. \(\square\)
Selecting a regime and checking failure
A practical choice begins with the interval length and the local scale of change, not with a preference for the most elaborate formula. The dimensionless ratio \(\sqrt{a(x)h}/d\), where \(d\) is distance to a relevant boundary, is a useful warning signal: if it is not small, a Gaussian step can place substantial probability outside the admissible state space. A large value of \(h|b'(x)|\) warns that the drift may change materially during the interval; appreciable curvature \(b''\) or rapidly changing \(\sigma\) adds further concern.
| Regime | Reasonable starting choice | Principal caution |
|---|---|---|
| Short interval, smooth interior, nearly constant coefficients | Euler Gaussian | Check for accumulated bias when many intervals are used |
| Drift locally linear with meaningful slope over the interval | Local-linear Gaussian | It still freezes diffusion and extrapolates the local drift line |
| Positive state with strongly state-dependent diffusion | Lamperti-transformed Gaussian | Check the transformed drift and the endpoint Jacobian |
| Short interval with smooth transformed drift but material local drift variation | Short-time expansion | It may be inaccurate or poorly normalized for long intervals |
These are comparisons, not a universal ranking. Euler is transparent and inexpensive; local-linear transitions improve the treatment of locally linear drift; the transformed approximation handles multiplicative diffusion in a coordinate with unit diffusion; and the short-time correction uses endpoint drift information beyond a frozen transformed drift. For a genuinely linear Gaussian diffusion, an exact transition is preferable to all four approximations. For strongly nonlinear dynamics, long intervals, or difficult boundary behavior, none should be accepted merely because its formula is available.
Useful diagnostics include evaluating each candidate over the observed state range, checking that variances are positive and densities finite, and comparing approximations after halving the interval where a finer grid or reliable benchmark is available. Check the implied probability below a physical boundary, not just whether the observed endpoint is positive. Large differences between approximations, unstable likelihood estimates under modest interval changes, or a candidate density that changes sharply under small perturbations of parameters are warnings to investigate. Numerical renormalization can conceal an inadequate approximation; it does not make the underlying transition approximation valid.
Common mistake
Check Your Understanding
Answer each question, then compare with the solutions.
- For \(b(x)=c\) and constant \(\sigma\), what does the local-linear approximation reduce to?
- Why must a Lamperti transition density include a factor depending on the endpoint \(y\)?
- If \(\sqrt{a(x)h}\) is comparable to distance from a zero boundary, what should you check before using an Euler Gaussian density?
- What additional feature does the short-time correction use compared with a Gaussian step with frozen transformed drift?
Solutions. (1) Since \(J=b'(x)=0\), it is \(N(x+ch,\sigma^2h)\), the Euler transition. (2) The change of variables from transformed endpoint to \(y\) has derivative \(d\Lambda(y)/dy=1/\sigma(y)\); without it, the expression is not a density with respect to \(dy\). (3) Check the Gaussian mass outside the admissible state space and whether boundary proximity makes the approximation unsuitable; positive observed endpoints alone do not settle this. (4) It includes an endpoint-dependent drift potential through \(A(\Lambda(y))-A(\Lambda(x))\) and the correction involving \(V=\alpha^2+\alpha'\), approximating short-time bridge effects.