AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppedItoOverlap

import AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.EnergyStoppingL2Bridge
import AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.ItoIntegralProcessCongruence
import AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.RandomStoppingProcessConsistency
import AutoSamplingTheory.TechnicalLemmas.StochasticProcesses.CanonicalStoppedItoIntegral

/-!
# Overlap of canonical energy-stopped Itô martingales

The energy truncations form a nested family.  At the integrand level this was
proved in `EnergyStoppingL2Bridge`: the lower energy truncation is exactly the
closed stop of every higher truncation at the lower hitting time, in completed
product `L²`.

Here we transport that identity through the completed Itô map and the general
process-level random-stopping theorem.  The result is the consistency relation
needed to glue the canonical martingales into one local martingale.
-/

namespace AutoSamplingTheory
namespace TechnicalLemmas
namespace StochasticProcesses
namespace EnergyStoppedItoOverlap

open Filter MeasureTheory Set
open scoped NNReal Topology

open BrownianMotion CanonicalEnergyLocalizer CanonicalLocalizationTheorem
  CanonicalStoppedItoIntegral EnergyStoppedProgressiveL2 EnergyStoppingL2Bridge
  ItoIntegralProcess ItoIntegralProcessCongruence LocalProgressiveL2 ProgressiveL2
  ProgressiveL2Stopping RandomStoppingProcessConsistency StoppingTime

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

/-- A lower energy-stopped Itô process is the stopped version of every higher
energy-stopped Itô process, at every deterministic time in the common finite
horizon. -/
theorem energyStoppedItoProcess_overlap_ae
    [IsProbabilityMeasure mu]
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : LocalProgressiveL2Integrand filtration mu T)
    (hT : 0 < T)
    (hB : IsBrownianMotionWithFiltration B filtration mu)
    {c d : ℝ} (hc : 0 ≤ c) (hcd : c ≤ d)
    {t : ℝ≥0} (htT : t ≤ T) :
    itoIntegralProcess (stoppedProgressiveL2 hUsual eta hc) hT hB hUsual t =ᵐ[mu]
      stoppedProcess
        (itoIntegralProcess
          (stoppedProgressiveL2 hUsual eta (hc.trans hcd)) hT hB hUsual)
        (fun omega =>
          (canonicalEnergyLocalizer hUsual eta c omega : WithTop ℝ≥0)) t := by
  let low : ProgressiveL2Integrand filtration mu T :=
    stoppedProgressiveL2 hUsual eta hc
  let high : ProgressiveL2Integrand filtration mu T :=
    stoppedProgressiveL2 hUsual eta (hc.trans hcd)
  let tau : Omega → ℝ≥0 := fun omega =>
    canonicalEnergyLocalizer hUsual eta c omega
  have htau : IsChewiStoppingTime filtration
      (fun omega => (tau omega : WithTop ℝ≥0)) := by
    simpa only [tau] using
      CanonicalEnergyStoppingTime.canonicalEnergyLocalizer_isChewiStoppingTime
        hUsual eta hc
  have htauT : ∀ omega, tau omega ≤ T := by
    intro omega
    simpa only [tau] using
      canonicalEnergyLocalizer_le_terminal hUsual eta c omega
  let stoppedHigh : ProgressiveL2Integrand filtration mu T :=
    stop high (fun omega => (tau omega : WithTop ℝ≥0)) htau
  have hLp : low.toLp = stoppedHigh.toLp := by
    simpa only [low, high, tau, stoppedHigh] using
      stoppedProgressiveL2_toLp_eq_stop_larger hUsual eta hc hcd
  have hcongr :
      itoIntegralProcess low hT hB hUsual t =ᵐ[mu]
        itoIntegralProcess stoppedHigh hT hB hUsual t :=
    itoIntegralProcess_congr_toLp_ae low stoppedHigh hT hB hUsual hLp htT
  have hstop :
      itoIntegralProcess stoppedHigh hT hB hUsual t =ᵐ[mu]
        stoppedProcess (itoIntegralProcess high hT hB hUsual)
          (fun omega => (tau omega : WithTop ℝ≥0)) t := by
    simpa only [stoppedHigh] using
      itoIntegralProcess_stop_eq_stoppedProcess_ae
        high hT tau htau htauT hB hUsual htT
  simpa only [low, high, tau] using hcongr.trans hstop

/-- Natural-number form matching Chewi's canonical levels `n+1`.  For `n ≤ m`,
the `n`-th canonical stopped Itô martingale agrees with the `m`-th one stopped
at the `n`-th canonical energy localizer. -/
theorem canonicalStoppedItoProcess_overlap_ae
    [IsProbabilityMeasure mu]
    (hUsual : SatisfiesUsualConditions filtration mu)
    (eta : LocalProgressiveL2Integrand filtration mu T)
    (hT : 0 < T)
    (hB : IsBrownianMotionWithFiltration B filtration mu)
    {n m : ℕ} (hnm : n ≤ m)
    {t : ℝ≥0} (htT : t ≤ T) :
    canonicalStoppedItoProcess hUsual eta hT hB n t =ᵐ[mu]
      stoppedProcess
        (canonicalStoppedItoProcess hUsual eta hT hB m)
        (fun omega =>
          (canonicalLocalizingTime hUsual eta n omega : WithTop ℝ≥0)) t := by
  have hcd : (n + 1 : ℝ) ≤ (m + 1 : ℝ) := by
    exact_mod_cast Nat.add_le_add_right hnm 1
  simpa only [canonicalStoppedItoProcess, canonicalStoppedProgressiveL2,
    canonicalLocalizingTime] using
    energyStoppedItoProcess_overlap_ae hUsual eta hT hB
      (c := (n + 1 : ℝ)) (d := (m + 1 : ℝ)) (by positivity) hcd htT

end EnergyStoppedItoOverlap
end StochasticProcesses
end TechnicalLemmas
end AutoSamplingTheory