Second-Moment Bound for a Stochastic-Integral Increment

This is a technical moment estimate for increments of a scalar Itô integral.

Let $(\Omega,P)$ be a probability space, let $W$ be a Brownian motion with Brownian filtration $\mathcal F$, and let $H:\Omega\times\mathbb R\to\mathbb R$ be measurable and progressively measurable. Assume that for every $T>0$ its time-space squared norm has finite $\mathrm{ENNReal}$ integral, that $C\ge 0$, and that $|H(\omega,s)|\le C$ for all $\omega,s$. For $0\le a<b$, write $M_t=\texttt{stochasticIntegralBrownian}(W,\mathcal F,H,t)$.

Then $M_b-M_a$ is square-integrable and

The result supplies the second-moment estimate used to derive first- and higher-moment bounds for stochastic-integral increments.

Formalization Note The hypotheses use an $\mathrm{ENNReal}$ finiteness condition for the integrated squared norm of $H$; the conclusion is an ordinary real integral and includes integrability explicitly.

Original Lean declaration header

Within namespace LevyStochCalc.Brownian.Ito. Shown in its original module context. Proof omitted.

theorem integral_sub_sq_le {a b : ℝ} (ha : 0 ≤ a) (hab : a < b) :
    MeasureTheory.Integrable (fun ω => (stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
        - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω) ^ 2) P
      ∧ ∫ ω, (stochasticIntegralBrownian W ℱ hℱ H hm hp hq b ω
          - stochasticIntegralBrownian W ℱ hℱ H hm hp hq a ω) ^ 2 ∂P
        ≤ C ^ 2 * (b - a)

Module navigation

Module: LevyStochCalc/Brownian/ItoIncrementMomentBounds.

Definition modules: LevyStochCalc/Brownian/ItoL2CompletionMaster, LevyStochCalc/Brownian/Filtered, LevyStochCalc/Brownian/Construction, LevyStochCalc/Probability/Progressive.

Supporting theorems: lintegral_stochasticIntegralBrownian_sub_sq_le.

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Source excerpt, lines 443–448 · Frozen source 408915f.