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namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
open Problem367.ConditionalCharacteristic
namespace Problem367.BrownianCharacterization
variable {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω]
{μ : Measure Ω} [IsProbabilityMeasure μ]
theorem condExp_character_of_setIntegral
(hm : m ≤ mΩ) {X : Ω → ℝ} (hX : Measurable X) (c : ℂ) (u : ℝ)
(hchar : ∀ A : Set Ω, MeasurableSet[m] A →
(∫ ω in A, character X u ω ∂μ) = μ.real A * c) :
μ[character X u | m] =ᵐ[μ] fun _ => c := by
symm
apply ae_eq_condExp_of_forall_setIntegral_eq hm (integrable_character hX u)
· intro A _ _
exact (integrable_const c).integrableOn
· intro A hA _
rw [hchar A hA]
simp
· exact stronglyMeasurable_const.aestronglyMeasurable
theorem hasIndepIncrements_of_past_events_general
{T E : Type*} [Preorder T] [MeasurableSpace E] [Sub E] [MeasurableSub₂ E]
{B : T → Ω → E} (F : Filtration T mΩ) (hB : Adapted F B)
(hind : ∀ s t : T, s ≤ t → ∀ A : Set Ω, MeasurableSet[F s] A →
∀ S : Set E, MeasurableSet S →
μ (A ∩ (fun ω => B t ω - B s ω) ⁻¹' S) =
μ A * μ ((fun ω => B t ω - B s ω) ⁻¹' S)) :
HasIndepIncrements B μ := by
intro n t ht
rw [iIndepFun_iff_measure_inter_preimage_eq_mul]
intro S sets
induction S using Finset.induction_on_max with
| empty => simp
| insert a S hlt ih =>
intro hsets
have ha : a ∉ S := by
intro ha
exact lt_irrefl a (hlt a ha)
have hmsets : ∀ i ∈ S, MeasurableSet (sets i) :=
fun i hi => hsets i (Finset.mem_insert_of_mem hi)
have hA : MeasurableSet[F (t a.castSucc)]
(⋂ i ∈ S, (fun ω => B (t i.succ) ω - B (t i.castSucc) ω) ⁻¹' sets i) := by
apply S.measurableSet_biInter
intro i hi
have hi1 : t i.succ ≤ t a.castSucc := ht (by
change i.val + 1 ≤ a.val
exact Nat.succ_le_of_lt (hlt i hi))
have hi0 : t i.castSucc ≤ t a.castSucc := ht (by
change i.val ≤ a.val
exact (hlt i hi).le)
exact ((hB.measurable_le hi1).sub (hB.measurable_le hi0)) (hmsets i hi)
have h := hind (t a.castSucc) (t a.succ) (ht (by change a.val ≤ a.val + 1; omega)) _ hA
(sets a) (hsets a (Finset.mem_insert_self a S))
rw [Finset.prod_insert ha]
rw [Finset.set_biInter_insert]
rw [Set.inter_comm, h, ih hmsets, mul_comm]
theorem hasIndepIncrements_of_past_events
{B : ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ)
(hB : StronglyAdapted F B)
(hind : ∀ s t : ℝ≥0, s ≤ t → ∀ A : Set Ω, MeasurableSet[F s] A →
∀ S : Set ℝ, MeasurableSet S →
μ (A ∩ (fun ω => B t ω - B s ω) ⁻¹' S) =
μ A * μ ((fun ω => B t ω - B s ω) ⁻¹' S)) :
HasIndepIncrements B μ :=
hasIndepIncrements_of_past_events_general F hB.adapted hind
theorem hasIndepIncrements_of_conditional_increment_laws
{T E : Type*} [Preorder T] [MeasurableSpace E] [Sub E] [MeasurableSub₂ E]
{B : T → Ω → E} (F : Filtration T mΩ) (hB : Adapted F B)
(ν : T → T → Measure E)
(hlaw : ∀ s t : T, s ≤ t → ∀ A : Set Ω, MeasurableSet[F s] A →
(μ.restrict A).map (fun ω => B t ω - B s ω) = μ A • ν s t) :
HasIndepIncrements B μ := by
apply hasIndepIncrements_of_past_events_general F hB
intro s t hst A hA S hS
have hm : Measurable (fun ω => B t ω - B s ω) :=
hB.measurable.sub hB.measurable
have huniv : μ.map (fun ω => B t ω - B s ω) = ν s t := by
simpa using hlaw s t hst Set.univ MeasurableSet.univ
have h := congrArg (fun κ : Measure E => κ S) (hlaw s t hst A hA)
rw [Measure.map_apply hm hS, Measure.restrict_apply (hm hS),
Measure.smul_apply, smul_eq_mul, ← huniv, Measure.map_apply hm hS] at h
simpa only [Set.inter_comm] using h
theorem isPreBrownianReal_of_conditional_charFun
{B : ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ)
(hB : StronglyAdapted F B) (hzero : ∀ᵐ ω ∂μ, B 0 ω = 0)
(hchar : ∀ s t : ℝ≥0, s ≤ t → ∀ u : ℝ,
μ[character (fun ω => B t ω - B s ω) u | F s]
=ᵐ[μ] fun _ => charFun (gaussianReal 0 (t - s)) u) :
IsPreBrownianReal B μ := by
have hm (t : ℝ≥0) : Measurable (B t) := ((hB t).mono (F.le t)).measurable
have hlaw (s t : ℝ≥0) (hst : s ≤ t) :
HasLaw (fun ω => B t ω - B s ω) (gaussianReal 0 (t - s)) μ :=
hasLaw_of_condExp_character (F.le s) ((hm t).sub (hm s)) _ (hchar s t hst)
apply HasIndepIncrements.isPreBrownianReal_of_hasLaw
· intro t
have h := hlaw 0 t (by positivity)
simp only [tsub_zero] at h
apply h.congr
filter_upwards [hzero] with ω hω
simp [hω]
· apply hasIndepIncrements_of_past_events F hB
intro s t hst A hA S hS
have hmap := map_restrict_eq_smul_of_condExp_character (F.le s)
((hm t).sub (hm s)) (gaussianReal 0 (t - s)) (hchar s t hst) hA
have h := congrArg (fun κ : Measure ℝ => κ S) hmap
rw [Measure.map_apply ((hm t).sub (hm s)) hS,
Measure.restrict_apply (((hm t).sub (hm s)) hS),
Measure.smul_apply, smul_eq_mul] at h
have hmass : μ ((fun ω => B t ω - B s ω) ⁻¹' S) = gaussianReal 0 (t - s) S := by
rw [← (hlaw s t hst).map_eq]
exact (Measure.map_apply (show Measurable (fun ω => B t ω - B s ω) from
(hm t).sub (hm s)) hS).symm
rw [hmass, Set.inter_comm]
exact h
theorem isPreBrownianReal_of_conditional_exponential
{B : ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ)
(hB : StronglyAdapted F B) (hzero : ∀ᵐ ω ∂μ, B 0 ω = 0)
(hchar : ∀ s t : ℝ≥0, s ≤ t → ∀ u : ℝ,
μ[character (fun ω => B t ω - B s ω) u | F s] =ᵐ[μ]
fun _ => Complex.exp (-(((t - s : ℝ≥0) : ℝ) : ℂ) * (u : ℂ)^2 / 2)) :
IsPreBrownianReal B μ := by
apply isPreBrownianReal_of_conditional_charFun F hB hzero
intro s t hst u
simpa only [charFun_gaussianReal, Complex.ofReal_zero, mul_zero, zero_mul,
zero_sub, neg_mul, neg_div] using hchar s t hst u
theorem isPreBrownianReal_of_setIntegral_character
{B : ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ)
(hB : StronglyAdapted F B) (hzero : ∀ᵐ ω ∂μ, B 0 ω = 0)
(hchar : ∀ s t : ℝ≥0, s ≤ t → ∀ A : Set Ω, MeasurableSet[F s] A →
∀ u : ℝ, (∫ ω in A, character (fun ω => B t ω - B s ω) u ω ∂μ) =
μ.real A * Complex.exp (-(((t - s : ℝ≥0) : ℝ) : ℂ) * (u : ℂ)^2 / 2)) :
IsPreBrownianReal B μ := by
apply isPreBrownianReal_of_conditional_exponential F hB hzero
intro s t hst u
exact condExp_character_of_setIntegral (F.le s)
(((hB t).mono (F.le t)).measurable.sub ((hB s).mono (F.le s)).measurable)
_ u (fun A hA => hchar s t hst A hA u)
theorem isBrownianReal_of_setIntegral_character
{B : ℝ≥0 → Ω → ℝ} (F : Filtration ℝ≥0 mΩ)
(hB : StronglyAdapted F B) (hzero : ∀ᵐ ω ∂μ, B 0 ω = 0)
(hcontinuous : ∀ᵐ ω ∂μ, Continuous (fun t => B t ω))
(hchar : ∀ s t : ℝ≥0, s ≤ t → ∀ A : Set Ω, MeasurableSet[F s] A →
∀ u : ℝ, (∫ ω in A, character (fun ω => B t ω - B s ω) u ω ∂μ) =
μ.real A * Complex.exp (-(((t - s : ℝ≥0) : ℝ) : ℂ) * (u : ℂ)^2 / 2)) :
IsBrownianReal B μ :=
⟨isPreBrownianReal_of_setIntegral_character F hB hzero hchar, hcontinuous⟩
end Problem367.BrownianCharacterization
end
end OAI
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