Existence of a Continuous Adapted Vector Itô Version

This asserts existence of a continuous adapted version of a vector Itô process with unbounded coefficients.

Let $P$ be a probability measure, $W$ a $d$-dimensional Brownian motion, and $\mathcal F$ a filtration such that: (1) every Brownian coordinate has the required Brownian filtration; (2) $\mathcal F_0\le\mathcal F_t$ whenever $t\le0$; and (3) every measurable $P$-null set is measurable in $\mathcal F_0$. Let $H$ be measurable and progressively measurable, with squared coefficient energy strictly below $\top$ on every positive window. Let $X_0:\Omega\to\mathbb R^n$ be $\mathcal F_0$-measurable, and let the drift $b$ be measurable and progressively measurable with finite squared energy on every positive window.

Then there exists $X:\mathbb R\to\Omega\to\mathbb R^n$ such that

in the precise sense of IsVectorItoVersion. The theorem is an existence result for finite-dimensional vector processes; it does not assert uniqueness.

Original Lean declaration header

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

theorem exists_isVectorItoVersion_of_unbounded
    (W : Multidim.MultidimBrownianMotion P d)
    (ℱ : Filtration ℝ ‹MeasurableSpace Ω›)
    (hcoord : ∀ k : Fin d, IsBrownianFiltration (W.W k) ℱ)
    (H : Fin n → Fin d → Ω → ℝ → ℝ)
    (hHm : ∀ p k, Measurable (Function.uncurry (H p k)))
    (hHp : ∀ p k, Probability.ProgressivelyMeasurable ℱ (H p k))
    (hHs : ∀ (p : Fin n) (k : Fin d) (T : ℝ), 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖H p k ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤)
    (hℱ0 : ∀ t : ℝ, t ≤ 0 → ℱ 0 ≤ ℱ t)
    (hnull : ∀ s : Set Ω, MeasurableSet s → P s = 0 → MeasurableSet[ℱ 0] s)
    {X₀ : Ω → Fin n → ℝ} (hX₀ : ∀ p : Fin n, Measurable[ℱ 0] fun ω => X₀ ω p)
    (bdrift : Fin n → Ω → ℝ → ℝ) (hbm : ∀ p, Measurable (Function.uncurry (bdrift p)))
    (hbp : ∀ p, Probability.ProgressivelyMeasurable ℱ (bdrift p))
    (hbq : ∀ (p : Fin n) (T : ℝ), 0 < T → ∫⁻ ω, ∫⁻ s in Set.Icc (0 : ℝ) T,
      (‖bdrift p ω s‖₊ : ℝ≥0∞) ^ 2 ∂volume ∂P < ⊤) :
    ∃ X : ℝ → Ω → Fin n → ℝ,
      IsVectorItoVersion W ℱ hcoord H hHm hHp hHs X₀ bdrift X

Module navigation

Module: LevyStochCalc/Ito/VectorItoVersionLimit.

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

Selected supporting theorems: clampMesh_mono_window, exists_isVectorItoVersion, exists_seq_clampMesh_window_lt, le_add_inv_mul_sq, lintegral_iSup_sq_norm_clamp_version_sub_le.

Previous: vectorItoProcess_ae_eq_zero.

Source excerpt, lines 190–207 · Frozen source 408915f.