File size: 3,355 Bytes
b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 | import OAI.Probability.FreeIsing.Brownian.IntegralProcess
namespace OAI
noncomputable section
open MeasureTheory
open scoped NNReal
namespace Problem367.RandomStoppedSimpleIntegral
variable {Ω I : Type*} [LinearOrder I] [Nonempty I]
theorem stoppedProcess_min (B : I → Ω → ℝ) (τ : Ω → WithTop I) (a t : I) (ω : Ω) :
stoppedProcess B τ (min t a) ω =
B (min (min (t : WithTop I) (τ ω)).untopA a) ω := by
rcases eq_or_ne (τ ω) ⊤ with hτ | hτ
· simp only [stoppedProcess, hτ, min_eq_left le_top]
rfl
· obtain ⟨s, hs⟩ := WithTop.ne_top_iff_exists.mp hτ
simp only [stoppedProcess, ← hs, ← WithTop.coe_min]
change B (min (min t a) s) ω = B (min (min t s) a) ω
rw [min_right_comm t a s]
theorem stopped_intervalSum (B H : I → Ω → ℝ) (τ : Ω → WithTop I)
{a b : I} (hab : a ≤ b) (t : I) (ω : Ω) :
stoppedProcess (StoppedSimpleIntegral.intervalSum B (H a) a b) τ t ω =
StoppedSimpleIntegral.intervalSum (stoppedProcess B τ)
(stoppedProcess H τ a) a b t ω := by
rw [StoppedSimpleIntegral.intervalSum, stoppedProcess_min, stoppedProcess_min]
change H a ω * (B (min (min (t : WithTop I) (τ ω)).untopA b) ω -
B (min (min (t : WithTop I) (τ ω)).untopA a) ω) = _
by_cases ha : (a : WithTop I) ≤ τ ω
· rw [stoppedProcess_eq_of_le ha]
· have hτa : τ ω ≤ (a : WithTop I) := le_of_not_ge ha
have hcap : (min (t : WithTop I) (τ ω)).untopA ≤ a := by
exact WithTop.untopA_le ((min_le_right _ _).trans hτa)
rw [min_eq_left hcap, min_eq_left (hcap.trans hab), sub_self, mul_zero, mul_zero]
theorem stopped_intervalSum_integrator (B : I → Ω → ℝ) (H : Ω → ℝ)
(τ : Ω → WithTop I) (a b t : I) (ω : Ω) :
stoppedProcess (StoppedSimpleIntegral.intervalSum B H a b) τ t ω =
StoppedSimpleIntegral.intervalSum (stoppedProcess B τ) H a b t ω := by
rw [StoppedSimpleIntegral.intervalSum, stoppedProcess_min, stoppedProcess_min]
rfl
theorem stopped_finite_intervalSum {J : Type*} (s : Finset J)
(B : I → Ω → ℝ) (H : J → Ω → ℝ) (a b : J → I) (τ : Ω → WithTop I) :
stoppedProcess (fun t ω => ∑ i ∈ s,
StoppedSimpleIntegral.intervalSum B (H i) (a i) (b i) t ω) τ =
fun t ω => ∑ i ∈ s,
StoppedSimpleIntegral.intervalSum (stoppedProcess B τ) (H i) (a i) (b i) t ω := by
funext t ω
change (∑ i ∈ s, stoppedProcess
(StoppedSimpleIntegral.intervalSum B (H i) (a i) (b i)) τ t ω) = _
exact Finset.sum_congr rfl fun i hi =>
stopped_intervalSum_integrator B (H i) τ (a i) (b i) t ω
theorem stopped_uniformSum (B H : ℝ≥0 → Ω → ℝ) (T : ℝ≥0)
(n : ℕ) (τ : Ω → WithTop ℝ≥0) :
stoppedProcess (BrownianIntegralProcess.stoppedUniformSum B H T n) τ =
BrownianIntegralProcess.stoppedUniformSum
(stoppedProcess B τ) (stoppedProcess H τ) T n := by
funext t ω
change (∑ i ∈ Finset.range n, stoppedProcess
(StoppedSimpleIntegral.intervalSum B (H (BrownianGrid.uniformGrid T n i))
(BrownianGrid.uniformGrid T n i) (BrownianGrid.uniformGrid T n (i + 1)))
τ t ω) = _
exact Finset.sum_congr rfl fun i hi => stopped_intervalSum B H τ
(BrownianGrid.uniformGrid_mono T n (Nat.le_succ i)) t ω
end Problem367.RandomStoppedSimpleIntegral
end
end OAI
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