Theorems · Theorem · probability

MeasureTheory.submartingale_iff_condExp_sub_nonneg

∀ {Ω : Type u_1} {E : Type u_2} {ι : Type u_3} [inst : Preorder ι] {m0 : MeasurableSpace Ω}
  {μ : MeasureTheory.Measure Ω} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E]
  {ℱ : MeasureTheory.Filtration ι m0} [CompleteSpace E] [inst_4 : PartialOrder E] [IsOrderedAddMonoid E]
  [MeasureTheory.SigmaFiniteFiltration μ ℱ] {f : ι → Ω → E},
  MeasureTheory.Submartingale f ℱ μ ↔
    MeasureTheory.StronglyAdapted ℱ f ∧
      (∀ (i : ι), MeasureTheory.Integrable (f i) μ) ∧ ∀ (i j : ι), i ≤ j → 0 ≤ᵐ[μ] μ[f j - f i | ↑ℱ i]

Defined in

Mathlib.Probability.Martingale.Basic

Cited by

0 results in Mathlib

Foundations

Depth 301 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PreorderNormedAddCommGroupNormedSpaceCompleteSpacePartialOrderIsOrderedAddMonoidMeasureTheory.SigmaFiniteFiltration