Homogenization
Averaging a fast hidden variable, and computing what the average leaves out.
Many models carry a slow quantity of interest and a fast one that nobody observes: a regime that switches, a volatility that mean-reverts quickly, a coefficient that oscillates on a small spatial scale.
The first step is always the same. Replace the fast variable by its average. Homogenization treats that averaged model as the leading term and computes the next one, the corrector, as a series in the ratio of the two scales.
If this is new, start with how averaging works, which derives the first corrections of a small example by hand and explains what each one means. The two ideas that make the higher orders work are solvability and initial layers.
The mechanism
Let the fast variable have generator $\mathcal{L}_0/\varepsilon$ and the slow variable generator $\mathcal{L}_1$, so that an expectation $u^\varepsilon$ solves
$$\partial_t u^\varepsilon = \varepsilon^{-1}\mathcal{L}_0 u^\varepsilon + \mathcal{L}_1 u^\varepsilon.$$Expand $u^\varepsilon = u_0 + \varepsilon u_1 + \dots$ and match powers of $\varepsilon$.
- At order $\varepsilon^{-1}$, $\mathcal{L}_0 u_0 = 0$, so $u_0$ does not depend on the fast variable.
- At order $1$, $\mathcal{L}_0 u_1 = \partial_t u_0 - \mathcal{L}_1 u_0$. By the Fredholm alternative this has a solution only if the right side averages to zero under the stationary law of the fast variable. That solvability condition is the averaged equation $\partial_t u_0 = \langle \mathcal{L}_1 \rangle u_0$.
- The Poisson equation determines only the part of $u_1$ that depends on the fast variable. The part shared by all fast states is fixed by the solvability condition at the next order.
- Near the start the expansion needs an initial layer as well.
The same steps apply with a diffusive scaling,
$$\varepsilon^{-2}\mathcal{L}_0 + \varepsilon^{-1}\mathcal{L}_{\mathrm{mix}} + \mathcal{L}_1,$$which is the setting of periodic media and of fast mean-reverting volatility. The averaged coefficient is then not always the plain average.
Examples
Regime-switching survival
fast a two-state chain switching the mean level and volatility of an Ornstein–Uhlenbeck hazard
averaged Vasicek's bond price with the regime averages
corrector every order in $1/\lambda$, in closed form, checked against a numerical solution
Fast switching in affine models
fast any finite chain modulating a model whose state factors out of an exponential-affine formula
averaged the model with stationary-weighted parameters
corrector one recursion to all orders, applied to several regimes, two-factor rates and credit, CIR, jumps, Poisson counts and Heston
Hidden regimes in a yield curve
fast a chain of any size switching the level and volatility of a Vasicek short rate
averaged Vasicek with averaged level and variance
corrector three numbers from a Green–Kubo covariance matrix, bound by an inequality; different chains sharing them give the same curve
Fast mean-reverting volatility
fast a mean-reverting factor driving the volatility of an asset or a short rate
averaged constant volatility at its effective level
corrector a series in the square root of the fast time scale; the first term needs correlation
Periodic media
fast a coefficient oscillating on a small spatial scale
averaged an effective coefficient; in one dimension the harmonic mean, not the arithmetic mean
corrector the solution of the cell problem
Choice among racing processes
fast hidden kinetics of competing correlated Cox processes
averaged Luce's choice axiom, the softmax
corrector a first correction of Green–Kubo type, uniform over blocked subsets
Conformal prediction
fast the state on which the forecast error depends
averaged residuals pooled across states, and a quantile of the pool
corrector the conditional coverage error: exponential in the horizon for a terminal score, algebraic for a path average, and set by the predictor's own memory for a filtered state
Pooled residuals under a fast regime · without the corrector
Forecast calibration
fast latent regimes behind a forecaster's residuals
averaged one effective width for the predictive distribution
corrector a hedged two-point Gaussian scale mixture, learned online
The model-by-model pricing pages (Black–Scholes, Heston, Merton, Bates, variance gamma, CIR, Vasicek with jumps, Hull–White, G2++, CEV, equity with stochastic rates, options on bonds and the other instruments) have moved to regimelib.org, where each model's mathematics sits beside the Python that evaluates it. Their old addresses here redirect. This site keeps the general theory and the examples that are not about one pricing model.
All of the finance examples follow one first-order rule: a fast chain adds a Green–Kubo-weighted second-order term to the pricing operator, and an irreversible chain enters through commutators. See fast regimes in any pricing equation.
The same correction appears in statistical physics, spectroscopy, chromatography, gene expression, population biology, queues, choice and machine learning. The survey collects that literature and the prior art in homogenization.
The regime-switching example is worked out in full, and the same recursion covers the affine models listed under Finance. Each is checked against a numerical solution, and each model's reduction against a Monte Carlo simulation of the switching model. The others link to where they are developed.
The bibliography covers every field above. The literature map and the timeline mark the regime-switching homogenization papers.
Cite
Cotton, P. (2001). An Analytic Approach to Ornstein–Uhlenbeck Processes with Fluctuating Parameters and Applications in the Modeling of Fixed Income Securities. PhD thesis, Stanford University.