Window Bound for Differences of Clamped Itô Versions

This bounds the window supremum of the difference between two clamped vector Itô versions.

Let $P$ be a probability measure, $W$ a $d$-dimensional Brownian motion, and $\mathcal F$ a filtration satisfying the stated Brownian-coordinate assumptions. Let $H$ and $b$ be measurable, progressively measurable coefficient and drift families with finite squared energy on every positive window. For clamping levels $j,j'\in\mathbb N$, let $X_j$ and $X_{j'}$ be vector Itô versions driven by the clamped coefficients and drifts, with common initial value.

For every $T>0$,

The right-hand side is an ENNReal quantity. The result is a technical estimate used to control convergence of clamped versions on bounded time windows.

Original Lean declaration header

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

theorem lintegral_iSup_sq_norm_clamp_version_sub_le
    {H : Fin n → Fin d → Ω → ℝ → ℝ}
    {hm : ∀ p k, Measurable (Function.uncurry (H p k))}
    {hpg : ∀ p k, Probability.ProgressivelyMeasurable ℱ (H p k)}
    {X₀ : Ω → Fin n → ℝ} {b : Fin n → Ω → ℝ → ℝ}
    (hbm : ∀ p, Measurable (Function.uncurry (b p)))
    {j j' : ℕ} {Xj Xj' : ℝ → Ω → Fin n → ℝ}
    (hXj : IsVectorItoVersion W ℱ hcoord (clampCoeff H j)
      (fun p k => measurable_clampCoeff hm j p k)
      (fun p k => progressivelyMeasurable_clampCoeff hpg j p k)
      (fun p k T' hT' => energy_clampCoeff_lt_top j p k T' hT')
      X₀ (clampDrift b j) Xj)
    (hXj' : IsVectorItoVersion W ℱ hcoord (clampCoeff H j')
      (fun p k => measurable_clampCoeff hm j' p k)
      (fun p k => progressivelyMeasurable_clampCoeff hpg j' p k)
      (fun p k T' hT' => energy_clampCoeff_lt_top j' p k T' hT')
      X₀ (clampDrift b j') Xj')
    {T : ℝ} (hT : 0 < T) :
    ∫⁻ ω, (⨆ t : Set.Icc (0 : ℝ) T, (‖Xj' (t : ℝ) ω - Xj (t : ℝ) ω‖₊ : ℝ≥0∞)) ^ 2 ∂P
      ≤ 32 * (n : ℝ≥0∞) ^ 2 * (clampMesh P H b T j + clampMesh P H b T j')

Module navigation

Module: LevyStochCalc/Ito/VectorItoVersionUnbounded.

Definition modules: LevyStochCalc/Ito/ItoFormulaUnboundedClamp, LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Multidim, LevyStochCalc/Probability/Progressive, LevyStochCalc/Brownian/VectorItoProcessVersion.

Supporting theorems: lintegral_energy_sub_le, lintegral_iSup_sq_norm_version_sub_le.

Source excerpt, lines 41–60 · Frozen source 408915f.