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Weak convergence of stochastic integrals

Xavier Bardina, Salim Boukfal

TL;DR

This paper addresses the weak convergence of multiparameter stochastic integrals driven by a Brownian sheet, showing that $X_n(t)=\int_{[0,t]} f_n(u)\theta_n(u)\,du$ converges in law in $\mathcal{C}([0,T])$ to $X(t)=\int_{[0,t]} f(u)W(du)$ when $\theta_n$ approximate the Brownian sheet $W$ and $f_n\to f$ in $L^{2q}([0,T])$ for some $q>1$ and $m>2q$ controlling increments. The proof uses tightness via a Bickel–Wichura criterion and finite‑dimensional convergence by showing $J^n(g)\to J(g)$ for simple $g$ and extending by density. The paper then verifies the key increment‑moment condition for two concrete kernels, the Donsker and Kac‑Stroock kernels, providing explicit multiparameter approximations to $\int f(u)W(du)$. Overall, it extends known one‑parameter results to multiparameter stochastic integrals and offers practical schemes for simulating Brownian‑sheet driven phenomena.

Abstract

In this paper we provide sufficient conditions for sequences of stochastic processes of the form $\int_{[0,t]} f_n(u) θ_n(u) du$, to weakly converge, in the space of continuous functions over a closed interval, to integrals with respect to the Brownian motion, $\int_{[0,t]} f(u)W(du)$, where $\{f_n\}_n$ is a sequence satisfying some integrability conditions converging to $f$ and $\{θ_n\}_n$ is a sequence of stochastic processes whose integrals $\int_{[0,t]}θ_n(u)du$ converge in law to the Brownian motion (in the sense of the finite dimensional distribution convergence), in the multidimensional parameter set case.

Weak convergence of stochastic integrals

TL;DR

This paper addresses the weak convergence of multiparameter stochastic integrals driven by a Brownian sheet, showing that converges in law in to when approximate the Brownian sheet and in for some and controlling increments. The proof uses tightness via a Bickel–Wichura criterion and finite‑dimensional convergence by showing for simple and extending by density. The paper then verifies the key increment‑moment condition for two concrete kernels, the Donsker and Kac‑Stroock kernels, providing explicit multiparameter approximations to . Overall, it extends known one‑parameter results to multiparameter stochastic integrals and offers practical schemes for simulating Brownian‑sheet driven phenomena.

Abstract

In this paper we provide sufficient conditions for sequences of stochastic processes of the form , to weakly converge, in the space of continuous functions over a closed interval, to integrals with respect to the Brownian motion, , where is a sequence satisfying some integrability conditions converging to and is a sequence of stochastic processes whose integrals converge in law to the Brownian motion (in the sense of the finite dimensional distribution convergence), in the multidimensional parameter set case.

Paper Structure

This paper contains 6 sections, 9 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Statement and proof of the main result
  4. Convergence for the Donsker kernels
  5. Convergence for the Kac-Stroock kernels
  6. Appendix

Key Result

Theorem 2.1

Let $\{Y_n\}_{n \in \mathbb{N}}$ be a sequence of real valued continuous processes over $[0,T]$ vanishing along the axes. Suppose that there exist $\beta > 1$, $\gamma > 0$ and finite nonnegative measures $\mu$ and $\{\mu_{n}\}_{n \in \mathbb{N}}$ on $[0,T]$ with continuous marginals such that $\mu_ Then the sequence of laws associated to the processes $\{Y_n\}_{n\in \mathbb{N}}$ is tight.

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['main theorem']}
  • Proposition 3.1
  • ...and 10 more