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feat(probability_theory/martingale): define martingales (#10625)
Co-authored-by: RemyDegenne <remydegenne@gmail.com>
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/- | ||
Copyright (c) 2021 Rémy Degenne. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Rémy Degenne | ||
-/ | ||
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import probability_theory.stopping | ||
import measure_theory.function.conditional_expectation | ||
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/-! | ||
# Martingales | ||
A family of functions `f : ι → α → E` is a martingale with respect to a filtration `ℱ` if every | ||
`f i` is integrable, `f` is adapted with respect to `ℱ` and for all `i ≤ j`, | ||
`μ[f j | ℱ.le i] =ᵐ[μ] f i`. | ||
The definitions of filtration and adapted can be found in `probability_theory.stopping`. | ||
### Definitions | ||
* `measure_theory.martingale f ℱ μ`: `f` is a martingale with respect to filtration `ℱ` and | ||
measure `μ`. | ||
### Results | ||
* `measure_theory.martingale_condexp f ℱ μ`: the sequence `λ i, μ[f | ℱ i, ℱ.le i])` is a | ||
martingale with respect to `ℱ` and `μ`. | ||
-/ | ||
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open topological_space filter | ||
open_locale nnreal ennreal measure_theory | ||
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namespace measure_theory | ||
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variables {α E ι : Type*} [preorder ι] [measurable_space E] | ||
{m0 : measurable_space α} {μ : measure α} | ||
[normed_group E] [normed_space ℝ E] [complete_space E] [borel_space E] | ||
[second_countable_topology E] | ||
{f g : ι → α → E} {ℱ : filtration ι m0} [sigma_finite_filtration μ ℱ] | ||
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/-- A family of functions `f : ι → α → E` is a martingale with respect to a filtration `ℱ` if `f` | ||
is adapted with respect to `ℱ` and for all `i ≤ j`, `μ[f j | ℱ.le i] =ᵐ[μ] f i`. -/ | ||
def martingale (f : ι → α → E) (ℱ : filtration ι m0) (μ : measure α) | ||
[sigma_finite_filtration μ ℱ] : Prop := | ||
adapted ℱ f ∧ ∀ i j, i ≤ j → μ[f j | ℱ i, ℱ.le i] =ᵐ[μ] f i | ||
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variables (E) | ||
lemma martingale_zero (ℱ : filtration ι m0) (μ : measure α) [sigma_finite_filtration μ ℱ] : | ||
martingale (0 : ι → α → E) ℱ μ := | ||
⟨adapted_zero E ℱ, λ i j hij, by { rw [pi.zero_apply, condexp_zero], simp, }⟩ | ||
variables {E} | ||
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namespace martingale | ||
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lemma adapted (hf : martingale f ℱ μ) : adapted ℱ f := hf.1 | ||
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lemma measurable (hf : martingale f ℱ μ) (i : ι) : measurable[ℱ i] (f i) := hf.adapted i | ||
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lemma condexp_ae_eq (hf : martingale f ℱ μ) {i j : ι} (hij : i ≤ j) : | ||
μ[f j | ℱ i, ℱ.le i] =ᵐ[μ] f i := | ||
hf.2 i j hij | ||
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lemma integrable (hf : martingale f ℱ μ) (i : ι) : integrable (f i) μ := | ||
integrable_condexp.congr (hf.condexp_ae_eq (le_refl i)) | ||
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lemma add (hf : martingale f ℱ μ) (hg : martingale g ℱ μ) : martingale (f + g) ℱ μ := | ||
begin | ||
refine ⟨hf.adapted.add hg.adapted, λ i j hij, _⟩, | ||
exact (condexp_add (hf.integrable j) (hg.integrable j)).trans | ||
((hf.2 i j hij).add (hg.2 i j hij)), | ||
end | ||
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lemma neg (hf : martingale f ℱ μ) : martingale (-f) ℱ μ := | ||
⟨hf.adapted.neg, λ i j hij, (condexp_neg (f j)).trans ((hf.2 i j hij).neg)⟩ | ||
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lemma sub (hf : martingale f ℱ μ) (hg : martingale g ℱ μ) : martingale (f - g) ℱ μ := | ||
by { rw sub_eq_add_neg, exact hf.add hg.neg, } | ||
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lemma smul (c : ℝ) (hf : martingale f ℱ μ) : martingale (c • f) ℱ μ := | ||
begin | ||
refine ⟨hf.adapted.smul c, λ i j hij, _⟩, | ||
refine (condexp_smul c (f j)).trans ((hf.2 i j hij).mono (λ x hx, _)), | ||
rw [pi.smul_apply, hx, pi.smul_apply, pi.smul_apply], | ||
end | ||
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end martingale | ||
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lemma martingale_condexp (f : α → E) (ℱ : filtration ι m0) (μ : measure α) | ||
[sigma_finite_filtration μ ℱ] : | ||
martingale (λ i, μ[f | ℱ i, ℱ.le i]) ℱ μ := | ||
⟨λ i, measurable_condexp, λ i j hij, condexp_condexp_of_le (ℱ.mono hij) _⟩ | ||
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end measure_theory |