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probability.martingale.optional_stopping

Optional stopping theorem (fair game theorem) #

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The optional stopping theorem states that an adapted integrable process f is a submartingale if and only if for all bounded stopping times τ and π such that τ ≤ π, the stopped value of f at τ has expectation smaller than its stopped value at π.

This file also contains Doob's maximal inequality: given a non-negative submartingale f, for all ε : ℝ≥0, we have ε • μ {ε ≤ f* n} ≤ ∫ ω in {ε ≤ f* n}, f n where f* n ω = max_{k ≤ n}, f k ω.

Main results #

theorem measure_theory.submartingale.expected_stopped_value_mono {Ω : Type u_1} {m0 : measurable_space Ω} {μ : measure_theory.measure Ω} {𝒢 : measure_theory.filtration m0} {f : Ω } {τ π : Ω } [measure_theory.sigma_finite_filtration μ 𝒢] (hf : measure_theory.submartingale f 𝒢 μ) (hτ : measure_theory.is_stopping_time 𝒢 τ) (hπ : measure_theory.is_stopping_time 𝒢 π) (hle : τ π) {N : } (hbdd : (ω : Ω), π ω N) :

Given a submartingale f and bounded stopping times τ and π such that τ ≤ π, the expectation of stopped_value f τ is less than or equal to the expectation of stopped_value f π. This is the forward direction of the optional stopping theorem.

The converse direction of the optional stopping theorem, i.e. an adapted integrable process f is a submartingale if for all bounded stopping times τ and π such that τ ≤ π, the stopped value of f at τ has expectation smaller than its stopped value at π.

The optional stopping theorem (fair game theorem): an adapted integrable process f is a submartingale if and only if for all bounded stopping times τ and π such that τ ≤ π, the stopped value of f at τ has expectation smaller than its stopped value at π.

@[protected]

The stopped process of a submartingale with respect to a stopping time is a submartingale.

theorem measure_theory.smul_le_stopped_value_hitting {Ω : Type u_1} {m0 : measurable_space Ω} {μ : measure_theory.measure Ω} {𝒢 : measure_theory.filtration m0} {f : Ω } [measure_theory.is_finite_measure μ] (hsub : measure_theory.submartingale f 𝒢 μ) {ε : nnreal} (n : ) :
ε μ {ω : Ω | ε (finset.range (n + 1)).sup' finset.nonempty_range_succ (λ (k : ), f k ω)} ennreal.of_real ( (ω : Ω) in {ω : Ω | ε (finset.range (n + 1)).sup' finset.nonempty_range_succ (λ (k : ), f k ω)}, measure_theory.stopped_value f (measure_theory.hitting f {y : | ε y} 0 n) ω μ)
theorem measure_theory.maximal_ineq {Ω : Type u_1} {m0 : measurable_space Ω} {μ : measure_theory.measure Ω} {𝒢 : measure_theory.filtration m0} {f : Ω } [measure_theory.is_finite_measure μ] (hsub : measure_theory.submartingale f 𝒢 μ) (hnonneg : 0 f) {ε : nnreal} (n : ) :
ε μ {ω : Ω | ε (finset.range (n + 1)).sup' finset.nonempty_range_succ (λ (k : ), f k ω)} ennreal.of_real ( (ω : Ω) in {ω : Ω | ε (finset.range (n + 1)).sup' finset.nonempty_range_succ (λ (k : ), f k ω)}, f n ω μ)

Doob's maximal inequality: Given a non-negative submartingale f, for all ε : ℝ≥0, we have ε • μ {ε ≤ f* n} ≤ ∫ ω in {ε ≤ f* n}, f n where f* n ω = max_{k ≤ n}, f k ω.

In some literature, the Doob's maximal inequality refers to what we call Doob's Lp inequality (which is a corollary of this lemma and will be proved in an upcomming PR).