Dyadic $L^2$ Maximal Bound for a Martingale

This is a dyadic $L^2$ maximal inequality for a continuous-time martingale sampled at dyadic times.

Let $(\Omega,\mathcal F,P)$ be a finite probability space, let $M:\mathbb R\to\Omega\to\mathbb R$ be a martingale relative to a filtration $\mathcal F$, and let $T\ge0$. For every $n\in\mathbb N$, let dyadicRunMax M T n be the running maximum along the dyadic partition of $[0,T]$ at level $n$.

The integrals are ENNReal lower integrals, and the norm is represented by the nonnegative norm coercion. The estimate is intended for dyadic maximal-control arguments.

Original Lean declaration header

Within namespace LevyStochCalc.Probability. Shown in its original module context. Proof omitted.

theorem lintegral_sq_dyadicRunMax_le {μ : Measure Ω} [IsFiniteMeasure μ]
    {ℱ : Filtration ℝ mΩ} (hM : MeasureTheory.Martingale M ℱ μ) (hT : 0 ≤ T) (n : ℕ) :
    ∫⁻ ω, (‖dyadicRunMax M T n ω‖₊ : ℝ≥0∞) ^ 2 ∂μ
      ≤ 4 * ∫⁻ ω, (‖M T ω‖₊ : ℝ≥0∞) ^ 2 ∂μ

Module navigation

Module: LevyStochCalc/Probability/DoobContinuous.

Supporting theorems: maximal_ineq_Lp.

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Source excerpt, lines 268–271 · Frozen source 408915f.