The localized zero-rate error martingale #
Positive-rate martingales are stopped first. Simultaneous compact-state domination then passes the centered zero-rate limit through deterministic conditional expectations.
References #
- J. Bertoin, Lévy Processes, Cambridge University Press, 1996, Chapter IV.
theorem
BGW.StableTrees.martingale_errorMartingale_stopped
{θ : ℝ}
(hθ : 1 < θ ∧ θ ≤ 2)
{ε : ℝ}
(hε : 0 < ε)
(n : ℕ)
{σ : Skorokhod.DLoc → ENNReal}
(hσ : MeasureTheory.IsStoppingTime Skorokhod.DLoc.filtration σ)
:
MeasureTheory.Martingale (fun (t : NNReal) (f : Skorokhod.DLoc) => errorMartingale θ ε (errStopTime n σ t f) f)
Skorokhod.DLoc.filtration ↑(stableLocalLaw θ)
The zero-rate error stopped at an arbitrary stopping time and the localizing cutoff is a martingale. Localization of the positive-rate processes precedes the centered limit.