AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessCongruence

import AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcess
import AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ProgressiveL2Algebra

/-!
# Congruence of completed Itô processes

The completed Itô process is represented by a particular pathwise-continuous
version, but its stochastic content depends only on the product-space `L²`
class of the integrand.  This module records both deterministic-time and
compact-path congruence principles needed for localization and gluing.
-/

namespace AutoSamplingTheory
namespace TechnicalLemmas
namespace StochasticProcesses
namespace ItoIntegralProcessCongruence

open Filter MeasureTheory Set
open scoped NNReal

open BrownianMotion ItoIntegralProcess ItoTerminalCompletion ProgressiveL2
  ProgressiveL2Algebra

variable {Omega : Type*} {m : MeasurableSpace Omega}
  {filtration : Filtration ℝ≥0 m} {mu : Measure Omega} {T : ℝ≥0}
  {B : ℝ≥0 → Omega → ℝ}

/-- Equal product-space `L²` integrands have almost-surely equal completed Itô
process values at every deterministic time in the construction horizon. -/
theorem itoIntegralProcess_congr_toLp_ae [IsFiniteMeasure mu]
    (eta xi : ProgressiveL2Integrand filtration mu T)
    (hT : 0 < T)
    (hB : IsBrownianMotionWithFiltration B filtration mu)
    (hUsual : SatisfiesUsualConditions filtration mu)
    (hEq : eta.toLp = xi.toLp)
    {t : ℝ≥0} (htT : t ≤ T) :
    itoIntegralProcess eta hT hB hUsual t =ᵐ[mu]
      itoIntegralProcess xi hT hB hUsual t := by
  have hrestrict : (eta.restrictAt t).toLp = (xi.restrictAt t).toLp :=
    restrictAt_toLp_eq_of_toLp_eq eta xi hEq t
  have hterminal :
      itoIntegralTerminal (eta.restrictAt t) hT hB =
        itoIntegralTerminal (xi.restrictAt t) hT hB := by
    apply itoIntegralTerminal_congr_toLp
    simpa only [integrandToLp] using hrestrict
  have heta := itoIntegralProcess_at_eq_terminal eta hT hB hUsual htT
  have hxi := itoIntegralProcess_at_eq_terminal xi hT hB hUsual htT
  filter_upwards [heta, hxi] with omega hetaOmega hxiOmega
  rw [hetaOmega, hxiOmega, hterminal]

/-- Equal product-space `L²` integrands determine the same continuous Itô
version simultaneously at every time of the finite construction horizon, on
one full-measure event.

This avoids intersecting an uncountable family of fixed-time almost-sure
identities: the continuous-version uniqueness theorem performs the upgrade. -/
theorem itoIntegralProcess_congr_toLp_pathwise_ae [IsFiniteMeasure mu]
    (eta xi : ProgressiveL2Integrand filtration mu T)
    (hT : 0 < T)
    (hB : IsBrownianMotionWithFiltration B filtration mu)
    (hUsual : SatisfiesUsualConditions filtration mu)
    (hEq : eta.toLp = xi.toLp) :
    ∀ᵐ omega ∂mu, ∀ t ∈ Icc (0 : ℝ≥0) T,
      itoIntegralProcess eta hT hB hUsual t omega =
        itoIntegralProcess xi hT hB hUsual t omega := by
  let J : ℝ≥0 → Omega → ℝ := itoIntegralProcess eta hT hB hUsual
  have hJadapted : StronglyAdapted filtration J := by
    simpa only [J] using
      itoIntegralProcess_stronglyAdapted eta hT hB hUsual
  have hJcontinuous : ∀ᵐ omega ∂mu,
      ContinuousOn (fun t => J t omega) (Icc (0 : ℝ≥0) T) := by
    filter_upwards [] with omega
    simpa only [J] using
      itoIntegralProcess_continuousOn eta hT hB hUsual omega
  have hJterminal : ∀ t ≤ T,
      J t =ᵐ[mu] fun omega =>
        itoIntegralTerminal (xi.restrictAt t) hT hB omega := by
    intro t ht
    have heta := itoIntegralProcess_at_eq_terminal eta hT hB hUsual ht
    have hrestrict : (eta.restrictAt t).toLp = (xi.restrictAt t).toLp :=
      restrictAt_toLp_eq_of_toLp_eq eta xi hEq t
    have hterminal :
        itoIntegralTerminal (eta.restrictAt t) hT hB =
          itoIntegralTerminal (xi.restrictAt t) hT hB := by
      apply itoIntegralTerminal_congr_toLp
      simpa only [integrandToLp] using hrestrict
    rw [hterminal] at heta
    simpa only [J] using heta
  simpa only [J] using
    itoIntegralProcess_unique xi hT hB hUsual
      J hJadapted hJcontinuous hJterminal

end ItoIntegralProcessCongruence
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory