feat: a continuous local martingale of locally finite variation is almost surely constant - #490
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feat: a continuous local martingale of locally finite variation is almost surely constant#490CoolRmal wants to merge 4 commits into
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CoolRmal
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Could you extract a PR with the definition and basic API for |
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I'll do that later today. |
RemyDegenne
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Jul 29, 2026
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 #490 for review; the file depends only on Mathlib. Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
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This PR adds a new file
BrownianMotion/Continuity/FiniteVariationProcess.lean, which starts the formalization of some results about finite-variation martingales. The main theorem proved is that a continuous local martingale of locally finite variation is almost surely constant.