Plain-English statement
- Diagonal term: an adapted coefficient factors from the squared future increment, whose second moment is the clipped interval length.
Read the mathematics first, then descend into Lean
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Proof architecture
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How to read the exact Lean declaration
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- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
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- Proof actionsAfter
by, ask what each tactic does to the mathematical goal—not only what syntax it uses.
Syntax used on this page
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Lean statement
theorem integral_weightedIncrement_sq
{Ω : Type*} {m : MeasurableSpace Ω}
{B : ℝ≥0 → Ω → ℝ} {filtration : Filtration ℝ≥0 m} {μ : Measure Ω}
{n : ℕ} (eta : ElementaryAdaptedProcess filtration n)
(hB : IsBrownianMotionWithFiltration B filtration μ) (T : ℝ≥0) (i : Fin n) :
∫ omega, weightedIncrement eta B T i omega ^ 2 ∂μ =
(∫ omega, eta.coeff i omega ^ 2 ∂μ) *
((min (eta.times i.succ) T - min (eta.times i.castSucc) T : ℝ≥0) : ℝ) := by
let _ : IsProbabilityMeasure μ := hB.isProbabilityMeasure
by_cases hT : T ≤ eta.times i.castSucc
· have hT' : T ≤ eta.times i.succ :=
hT.trans (eta.times_strictMono Fin.castSucc_lt_succ).le
simp [weightedIncrement, brownianIncrement, min_eq_right hT, min_eq_right hT']
· have haT : eta.times i.castSucc ≤ T := le_of_not_ge hT
have hab : eta.times i.castSucc ≤ min (eta.times i.succ) T :=
le_min (eta.times_strictMono Fin.castSucc_lt_succ).le haT
let increment : Ω → ℝ := fun omega =>
B (min (eta.times i.succ) T) omega - B (eta.times i.castSucc) omega
have hindep : IndepFun (eta.coeff i) increment μ :=
hB.indepFun_increment_of_stronglyMeasurable hab
(eta.coeff_stronglyMeasurable i)
have hindepSq :
IndepFun (fun omega => eta.coeff i omega ^ 2)
(fun omega => increment omega ^ 2) μ := by
simpa [Function.comp_def] using
hindep.comp (measurable_id.pow_const 2) (measurable_id.pow_const 2)
have hcoeffMeas : AEStronglyMeasurable (fun omega => eta.coeff i omega ^ 2) μ :=
(((eta.coeff_stronglyMeasurable i).mono (filtration.le _)).pow 2).aestronglyMeasurable
have hincMeas : AEStronglyMeasurable (fun omega => increment omega ^ 2) μ :=
((hB.increment_stronglyMeasurable _ _).pow 2).aestronglyMeasurable
have hfactor := hindepSq.integral_fun_mul_eq_mul_integral hcoeffMeas hincMeas
have hmoment := hB.integral_increment_sq hab
calc
∫ omega, weightedIncrement eta B T i omega ^ 2 ∂μ =
∫ omega, eta.coeff i omega ^ 2 * increment omega ^ 2 ∂μ := by
apply integral_congr_ae
filter_upwards [] with omega
simp [weightedIncrement, brownianIncrement, increment, min_eq_left haT]
ring
_ = (∫ omega, eta.coeff i omega ^ 2 ∂μ) *
∫ omega, increment omega ^ 2 ∂μ := hfactor
_ = (∫ omega, eta.coeff i omega ^ 2 ∂μ) *
((min (eta.times i.succ) T - eta.times i.castSucc : ℝ≥0) : ℝ) := by
rw [hmoment]
_ = (∫ omega, eta.coeff i omega ^ 2 ∂μ) *
((min (eta.times i.succ) T - min (eta.times i.castSucc) T : ℝ≥0) : ℝ) := by
rw [min_eq_left haT]Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ElementaryItoIsometry.lean:94published source at 7bcd37294df1
Proof architecture
factor each diagonal Ito-isometry term into coefficient energy and clipped interval length
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `have` creates a named intermediate mathematical fact.
- `calc` records an equality or inequality chain matching a paper calculation.
- `rw` rewrites by an established identity.
- `simp` normalizes through registered definitional and theorem rewrites.
- `apply` reduces the goal to the hypotheses of a reusable theorem.
- `simpa` closes the goal after a controlled simplification of a typed result.
- `filter_upwards` moves an almost-everywhere or eventual statement into a pointwise local context.
Why the statement has this shape
The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.
Hidden assumptions and non-claims
- Measurability is represented explicitly or must be supplied by a dependency.
- Integrability is an input or proved output; a displayed integral alone does not supply it.
- Almost-everywhere hypotheses depend on the stated measure and representative.