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Lean gate passed 2026-08-19T06:32:39.895922+00:00 · 7bcd37294df1
Reviewed teaching declaration

clipped

CompiledCompiled

A progressive square-integrable process can be clipped at a deterministic level without losing progressiveness or product-space L2 control.

Plain-English statement

A progressive square-integrable process can be clipped at a deterministic level without losing progressiveness or product-space L2 control.

Mathematical statement

eta^M(t,omega) = max(-M,min(M,eta(t,omega))) and eta^M converges to eta in L2 as M tends to infinity.

Intuition

Clipping isolates a bounded approximation problem while dominated convergence later removes the artificial bound.

Conditions

  • a progressive process
  • finite probability-time L2 energy
  • a finite construction horizon

Why these conditions cannot be dropped

  • progressiveness must survive scalar clipping
  • L2 integrability provides the dominating function
  • the finite horizon fixes the product measure

Proof route

  • apply the measurable scalar clipping map
  • prove the pointwise bound
  • retain the original square-integrable process as a dominator
  • pass to the product-space L2 class
Lean learning studio · mathematics → formal proof

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Lean statement

noncomputable def clipped
    (eta : ProgressiveL2Integrand filtration mu T) (n : ℕ) :
    ProgressiveL2Integrand filtration mu T where
  process := clipProcess n eta.process
  progressive := clipProcess_stronglyProgressive eta n
  memLp := clipProcess_memLp eta n

Lean interface notes

  • the clipped process remains a ProgressiveL2Integrand
  • tendsto_clipped_toLp proves removal of the truncation