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.
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 $1$, $\mathcal{L}_0 u_1 = \partial_t u_0 - \mathcal{L}_1 u_0$. 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 solution $u_1$ of that Poisson equation is the corrector. It depends on the fast variable, and it is where the average's error lives.
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 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 first order in the square root of the fast time scale, through a few group parameters
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 not computed; the certificate holds without it, and the sharpness lost is priced in nats
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 regime-switching example is worked out in full, with a numerical solution to check against and demonstrations. The others link to where they are developed. The bibliography and literature map currently cover the regime-switching example and the classical theory.
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.