Theorems · Theorem · probability

MeasureTheory.Martingale.stoppedValue_ae_eq_condExp_of_le_const_of_countable_range

∀ {Ω : Type u_1} {E : Type u_2} {m : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} [inst : NormedAddCommGroup E]
  [inst_1 : NormedSpace ℝ E] [CompleteSpace E] {ι : Type u_3} [inst_3 : LinearOrder ι] [inst_4 : TopologicalSpace ι]
  [OrderTopology ι] [FirstCountableTopology ι] {ℱ : MeasureTheory.Filtration ι m}
  [MeasureTheory.SigmaFiniteFiltration μ ℱ] {τ : Ω → WithTop ι} {f : ι → Ω → E} {n : ι} [inst_8 : Nonempty ι],
  MeasureTheory.Martingale f ℱ μ →
    ∀ (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hτ_le : ∀ (x : Ω), τ x ≤ ↑n),
      (Set.range τ).Countable →
        ∀ [MeasureTheory.SigmaFinite (μ.trim ⋯)], MeasureTheory.stoppedValue f τ =ᵐ[μ] μ[f n | hτ.measurableSpace]

The value of a martingale f at a stopping time τ bounded by n is the conditional expectation of f n with respect to the σ-algebra generated by τ.

Defined in

Mathlib.Probability.Martingale.OptionalSampling

Cited by

2 results in Mathlib

Foundations

Depth 304 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceLinearOrderTopologicalSpaceOrderTopologyFirstCountableTopologyMeasureTheory.SigmaFiniteFiltrationNonemptyMeasureTheory.SigmaFinite