File: Mathlib/Probability/Martingale/Centering.lean

full_name: MeasureTheory.predictablePart
code: /-- Any `ℕ`-indexed stochastic process can be written as the sum of a martingale and a predictable
process. This is the predictable process. See `martingalePart` for the martingale. -/
noncomputable def predictablePart {m0 : MeasurableSpace Ω} (f : ℕ → Ω → E) (ℱ : Filtration ℕ m0)
    (μ : Measure Ω) : ℕ → Ω → E 
goal: ℕ → Ω → E

full_name: MeasureTheory.predictablePart_zero
code: @[simp]
theorem predictablePart_zero : predictablePart f ℱ μ 0 = 0 
goal: MeasureTheory.predictablePart f ℱ μ 0 = 0

full_name: MeasureTheory.adapted_predictablePart
code: theorem adapted_predictablePart : Adapted ℱ fun n => predictablePart f ℱ μ (n + 1) 
goal: MeasureTheory.Adapted ℱ fun n => MeasureTheory.predictablePart f ℱ μ (n + 1)

full_name: MeasureTheory.adapted_predictablePart'
code: theorem adapted_predictablePart' : Adapted ℱ fun n => predictablePart f ℱ μ n 
goal: MeasureTheory.Adapted ℱ fun n => MeasureTheory.predictablePart f ℱ μ n

full_name: MeasureTheory.martingalePart
code: /-- Any `ℕ`-indexed stochastic process can be written as the sum of a martingale and a predictable
process. This is the martingale. See `predictablePart` for the predictable process. -/
noncomputable def martingalePart {m0 : MeasurableSpace Ω} (f : ℕ → Ω → E) (ℱ : Filtration ℕ m0)
    (μ : Measure Ω) : ℕ → Ω → E 
goal: ℕ → Ω → E

full_name: MeasureTheory.martingalePart_add_predictablePart
code: theorem martingalePart_add_predictablePart (ℱ : Filtration ℕ m0) (μ : Measure Ω) (f : ℕ → Ω → E) :
    martingalePart f ℱ μ + predictablePart f ℱ μ = f 
goal: MeasureTheory.martingalePart f ℱ μ + MeasureTheory.predictablePart f ℱ μ = f

full_name: MeasureTheory.martingalePart_eq_sum
code: theorem martingalePart_eq_sum : martingalePart f ℱ μ = fun n =>
    f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|ℱ i]) 
goal: MeasureTheory.martingalePart f ℱ μ = fun n => f 0 + ∑ i ∈ Finset.range n, (f (i + 1) - f i - μ[f (i + 1) - f i|↑ℱ i])

full_name: MeasureTheory.adapted_martingalePart
code: theorem adapted_martingalePart (hf : Adapted ℱ f) : Adapted ℱ (martingalePart f ℱ μ) 
goal: MeasureTheory.Adapted ℱ (MeasureTheory.martingalePart f ℱ μ)

full_name: MeasureTheory.integrable_martingalePart
code: theorem integrable_martingalePart (hf_int : ∀ n, Integrable (f n) μ) (n : ℕ) :
    Integrable (martingalePart f ℱ μ n) μ 
goal: MeasureTheory.Integrable (MeasureTheory.martingalePart f ℱ μ n) μ

full_name: MeasureTheory.martingale_martingalePart
code: theorem martingale_martingalePart (hf : Adapted ℱ f) (hf_int : ∀ n, Integrable (f n) μ)
    [SigmaFiniteFiltration μ ℱ] : Martingale (martingalePart f ℱ μ) ℱ μ 
goal: MeasureTheory.Martingale (MeasureTheory.martingalePart f ℱ μ) ℱ μ

full_name: MeasureTheory.martingalePart_add_ae_eq
code: theorem martingalePart_add_ae_eq [SigmaFiniteFiltration μ ℱ] {f g : ℕ → Ω → E}
    (hf : Martingale f ℱ μ) (hg : Adapted ℱ fun n => g (n + 1)) (hg0 : g 0 = 0)
    (hgint : ∀ n, Integrable (g n) μ) (n : ℕ) : martingalePart (f + g) ℱ μ n =ᵐ[μ] f n 
goal: MeasureTheory.martingalePart (f + g) ℱ μ n =ᵐ[μ] f n

full_name: MeasureTheory.predictablePart_add_ae_eq
code: theorem predictablePart_add_ae_eq [SigmaFiniteFiltration μ ℱ] {f g : ℕ → Ω → E}
    (hf : Martingale f ℱ μ) (hg : Adapted ℱ fun n => g (n + 1)) (hg0 : g 0 = 0)
    (hgint : ∀ n, Integrable (g n) μ) (n : ℕ) : predictablePart (f + g) ℱ μ n =ᵐ[μ] g n 
goal: MeasureTheory.predictablePart (f + g) ℱ μ n =ᵐ[μ] g n

full_name: MeasureTheory.predictablePart_bdd_difference
code: theorem predictablePart_bdd_difference {R : ℝ≥0} {f : ℕ → Ω → ℝ} (ℱ : Filtration ℕ m0)
    (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, ∀ i, |predictablePart f ℱ μ (i + 1) ω - predictablePart f ℱ μ i ω| ≤ R 
goal: ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |MeasureTheory.predictablePart f ℱ μ (i + 1) ω - MeasureTheory.predictablePart f ℱ μ i ω| ≤ ↑R

full_name: MeasureTheory.martingalePart_bdd_difference
code: theorem martingalePart_bdd_difference {R : ℝ≥0} {f : ℕ → Ω → ℝ} (ℱ : Filtration ℕ m0)
    (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, ∀ i, |martingalePart f ℱ μ (i + 1) ω - martingalePart f ℱ μ i ω| ≤ ↑(2 * R) 
goal: ∀ᵐ (ω : Ω) ∂μ,
  ∀ (i : ℕ), |MeasureTheory.martingalePart f ℱ μ (i + 1) ω - MeasureTheory.martingalePart f ℱ μ i ω| ≤ ↑(2 * R)