# Claim 6 — weak Itô formula

## Anchored claim

> A novel 'weak Itô formula' for Lipschitz-smooth convex functions under Lévy integrators is proved to enable the analysis despite discontinuous sample paths and potentially infinite noise variance when p<2 (Section 2).

**Verdict: VERIFIED.**

## Paper quantity and protocol

The paper’s exact inequality is evaluated pathwise. For a 1-smooth convex
Huber function `psi`, the code measures the margin

`RHS-LHS = integral(grad psi(X[t-]) dX^c) + large-jump increments + 1/2 integral(g^2 dt) - (psi(X_T)-psi(X_0))`.

The continuous component is Brownian with `sigma=0.6`; the discontinuous
component is a raw compound-Poisson Pareto large-jump process with tail index
`alpha=1.45`, so jumps are unbounded in the simulated law. Each branch has 12
independent paths, `T=2`, `dt=0.00025`, and thresholds `a={1.0,1.5,2.0}`.
The pass tolerance is a numerical `-0.02` margin. The destructive control
removes the paper’s `L/2` continuous correction while retaining the same paths,
objective, and jumps.

| function branch | seeds/trials | correction-on margin mean ± SE | pass | correction-off margin mean ± SE | destructive pass |
|---|---:|---:|---:|---:|---:|
| Huber(a=1.0) | 12 / 12 | 0.107 ± 0.020 | 12/12 | -0.253 ± 0.020 | 0/12 |
| Huber(a=1.5) | 12 / 12 | 0.174 ± 0.020 | 12/12 | -0.186 ± 0.020 | 0/12 |
| Huber(a=2.0) | 12 / 12 | 0.204 ± 0.018 | 12/12 | -0.156 ± 0.018 | 0/12 |

A quadratic equality baseline under the same correction has margin
`0.0070 ± 0.0052`, confirming the numerical stochastic-integral scale. The
Huber threshold sweep materially changes the measured margin while every
correction-on path satisfies the weak-Itô inequality. Removing the correction
fails every path in every branch, which is the destructive test of the formula’s
non-vacuous correction term.

This is a direct pathwise test of the paper’s Lipschitz-smooth convex inequality
with Brownian and unbounded-jump components; it does not rely on a paper table,
figure digitization, or a proxy retention statistic.