itoIntegralTerminal_norm

Plain-English statement

The terminal Ito integral is the unique L2 completion of elementary terminal sums and preserves the integrand norm.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

Mathematical statement

||I_T(eta)||_L2(P) = ||eta||_L2(P times dt).

Intuition

The elementary isometry turns every Cauchy sequence of integrands into a Cauchy sequence of terminal random variables, so completeness supplies the general integral.

Conditions

positive terminal time
progressive product-space L2 integrand
filtration-relative Brownian motion

Why these conditions cannot be dropped

positive time is used by the canonical dyadic density sequence
the L2 assumption supplies convergence
the Brownian law proves the elementary isometry

Proof route

choose canonical elementary approximants
transfer Cauchy control through the heterogeneous-grid isometry
take the Lp limit
prove universality against any convergent elementary sequence

Lean interface notes

itoIntegralTerminal is an actual Lp element
linearity is proved through universal approximation rather than compatibility of choice functions

Lean learning studio · mathematics → formal proof

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Lean statement

theorem itoIntegralTerminal_norm
    (eta : ProgressiveL2Integrand filtration mu T) (hT : 0 < T)
    (hB : IsBrownianMotionWithFiltration B filtration mu) :
    ‖itoIntegralTerminal eta hT hB‖ = ‖integrandToLp eta hB‖ := by
  let _ : IsProbabilityMeasure mu := hB.isProbabilityMeasure
  have hterminal := (tendsto_terminalApprox eta hT hB).norm
  have hprocess := (tendsto_processApprox eta hT hB).norm
  have heq : (fun n ↦ ‖terminalApprox eta hT hB n‖) =
      fun n ↦ ‖processApprox eta hT hB n‖ := by
    funext n
    exact norm_terminalToLp_eq_processToLp
      (canonicalElementaryApprox eta hT n) hB
  rw [heq] at hterminal
  exact tendsto_nhds_unique hterminal hprocess

/-- Same-grid sum after refining both operands to their least common dyadic
level. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/ItoTerminalCompletion.lean:159published source at 7bcd37294df1