[Merged by Bors] - feat(probability/martingale): the discrete stochastic integral of a submartingale is a submartingale #14909
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…rover-community/mathlib into JasonKYi/disc_stoch_int
…rover-community/mathlib into JasonKYi/disc_stoch_int
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…rty of the conditional expectation (#15274) We prove this result: ```lean lemma condexp_strongly_measurable_mul {f g : α → ℝ} (hf : strongly_measurable[m] f) (hfg : integrable (f * g) μ) (hg : integrable g μ) : μ[f * g | m] =ᵐ[μ] f * μ[g | m] := ``` This could be extended beyond multiplication, to any bounded bilinear map, but we leave this to a future PR. For now we only prove the real multiplication case, which is needed for #14909 and #14933.
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Co-authored-by: Rémy Degenne <remydegenne@gmail.com>
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…ubmartingale is a submartingale (#14909) This PR proves that the discrete stochastic integral of a predictable process with a submartingale is a submartingale. Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
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…rty of the conditional expectation (leanprover-community#15274) We prove this result: ```lean lemma condexp_strongly_measurable_mul {f g : α → ℝ} (hf : strongly_measurable[m] f) (hfg : integrable (f * g) μ) (hg : integrable g μ) : μ[f * g | m] =ᵐ[μ] f * μ[g | m] := ``` This could be extended beyond multiplication, to any bounded bilinear map, but we leave this to a future PR. For now we only prove the real multiplication case, which is needed for leanprover-community#14909 and leanprover-community#14933.
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…ubmartingale is a submartingale (leanprover-community#14909) This PR proves that the discrete stochastic integral of a predictable process with a submartingale is a submartingale. Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
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…rty of the conditional expectation (#15274) We prove this result: ```lean lemma condexp_strongly_measurable_mul {f g : α → ℝ} (hf : strongly_measurable[m] f) (hfg : integrable (f * g) μ) (hg : integrable g μ) : μ[f * g | m] =ᵐ[μ] f * μ[g | m] := ``` This could be extended beyond multiplication, to any bounded bilinear map, but we leave this to a future PR. For now we only prove the real multiplication case, which is needed for #14909 and #14933.
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…ubmartingale is a submartingale (#14909) This PR proves that the discrete stochastic integral of a predictable process with a submartingale is a submartingale. Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
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…rty of the conditional expectation (#15274) We prove this result: ```lean lemma condexp_strongly_measurable_mul {f g : α → ℝ} (hf : strongly_measurable[m] f) (hfg : integrable (f * g) μ) (hg : integrable g μ) : μ[f * g | m] =ᵐ[μ] f * μ[g | m] := ``` This could be extended beyond multiplication, to any bounded bilinear map, but we leave this to a future PR. For now we only prove the real multiplication case, which is needed for #14909 and #14933.
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…ubmartingale is a submartingale (#14909) This PR proves that the discrete stochastic integral of a predictable process with a submartingale is a submartingale. Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
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This PR proves that the discrete stochastic integral of a predictable process with a submartingale is a submartingale.