feat: the pathwise variation process - #494
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Add `variationProcess X a`: at time `t` and outcome `ω`, the total variation of the path `s ↦ X s ω` on `[a, t]`, defined as `(eVariationOn (X · ω) (Set.Icc a t)).toReal`. Since `Set.Icc a t` is empty for `t < a`, the process vanishes below `a`, which makes it adapted without assuming the index type has a bottom element. API: * `variationProcess_nonneg`, `variationProcess_of_le`; * `monotone_variationProcess`; * `continuous_variationProcess`, with one-sided `continuousWithinAt_*`; * `MeasureTheory.StronglyAdapted.variationProcess` (adaptedness); * `isStronglyProgressive_variationProcess`. This is the process used in Kallenberg, Proposition 18.2, to reduce a continuous local martingale of locally finite variation to a bounded-variation stopping argument. Split out of RemyDegenne#490 for review; the file depends only on Mathlib. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Redefine `variationProcess X a t ω` as `variationOnFromTo (X · ω) Set.univ a t` (signed) instead of the clamped `(eVariationOn (Set.Icc a t)).toReal`. Consequences: * `variationProcess_nonneg` and `variationProcess_eq_toReal_eVariationOn_Icc` now require `a ≤ t`; * continuity and monotonicity become uniform (no `a`-vs-point case split); * adaptedness is stated pointwise as `MeasureTheory.StronglyAdapted.stronglyMeasurable_variationProcess` for `a ≤ t` (global `StronglyAdapted` no longer holds: below `a` the value depends on the path over `[t, a]`); * progressive measurability is deferred (see the TODO in the module doc). Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
`Set` is already open; remove `Finset` from the `open` list (it is only used qualified, as `Finset.range`/`Finset.sum`) so `Icc`, `univ`, `Iic`, … resolve to `Set` without the prefix. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
Since `variationProcess X a · ω = variationOnFromTo (X · ω) univ a`, right-continuity is exactly Mathlib's `BoundedVariationOn.continuousWithinAt_variationOnFromTo_Ici`, so `continuousWithinAt_variationProcess_Ici` becomes a one-liner. This requires the continuity hypotheses to be the global `BoundedVariationOn univ` that lemma takes (rather than `LocallyBoundedVariationOn`); the left-continuity proof then also simplifies to a single uniform argument (no interval-bracketing case split). Drops the now-unused `eVariationOn_Icc_ne_top` helper. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
This was referenced Jul 20, 2026
CoolRmal
marked this pull request as ready for review
July 26, 2026 03:42
…into variation-process
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`countable_setOf_isolated_right_within` was deprecated (2026-07-09) in favour of `countable_setOfPred_isolated_right_within`. Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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@RemyDegenne Sorry for the delay, I spent a lot of time golfing the AI-generated proof. The mathlib PR is here: leanprover-community/mathlib4#42102 |
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Can you split the file in two? One for results that are about variation of a function, and another one for the new variation process. |
Move the results about the variation of a function (`eVariationOn_*`, `measurable_eVariationOn_*` and the auxiliary `nhdsGT_subtype_eq_bot_iff`) into `BrownianMotion/Auxiliary/EVariationOn.lean`, leaving `BrownianMotion/StochasticIntegral/VariationProcess.lean` with only `variationProcess` and its properties. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
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I just split the file. |
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I moved the measurability statements to the file about the process. Thanks for this PR! |
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Adds the pathwise variation process
variationProcess X a: at timetand outcomeω, the signed variationvariationOnFromTo (X · ω) Set.univ a tof the paths ↦ X s ω.Main definition
variationProcessMain results
variationProcess_nonneg— nonnegativity, fora ≤ t;monotone_variationProcess— monotone in time, for paths of locally bounded variation;MeasureTheory.StronglyAdapted.measurable_variationProcess_of_continuous,..._of_continuousWithinAt_Ioi,..._of_continuousWithinAt_Iio— fora ≤ t, the value at timetof the variation process of a strongly adapted process is𝓕 t-measurable.Measurability: two sets of assumptions
The variation over a set of times is an uncountable supremum, so measurability is obtained by computing it over a countable dense set of times instead (
measurable_eVariationOn_of_countable). Which dense sets suffice depends on the regularity of the paths, and this file provides two independent results:eVariationOn_eq_comp_val_of_dense,measurable_eVariationOn_of_continuousWithinAt;eVariationOn_eq_comp_val_of_dense_Ioi,measurable_eVariationOn_of_continuousWithinAt_Ioi, together with the left-continuous counterpartseVariationOn_eq_comp_val_of_dense_Iioandmeasurable_eVariationOn_of_continuousWithinAt_Iio.Neither hypothesis implies the other, and the second genuinely needs second countability: a separable linearly ordered set can have uncountably many right isolated points (e.g. https://topology.pi-base.org/spaces/S000093)