[Paper Review] Stochastic Integrals and Abelian Processes

Claudio Albanese|ArXiv.org|Nov 19, 2007

Neural Networks and Applications4 citations

TL;DR

This paper presents a constructive, pathwise approach to defining stochastic integrals in the continuum limit using Fourier analysis and renormalization group techniques applied to triangulated diffusion processes with uniformly continuous coefficients. It establishes convergence in the graph norm of the Fourier-transformed Markov generator, proving smoothness and analyticity of the joint kernel and extending the method to non-resonant Abelian processes such as stochastic integrals, the sup process, and discrete-time summations.

ABSTRACT

We study triangulation schemes for the joint kernel of a diffusion process with uniformly continuous coefficients and an adapted, non-resonant Abelian process. The prototypical example of Abelian process to which our methods apply is given by stochastic integrals with uniformly continuous coeffcients. The range of applicability includes also a broader class of processes of practical relevance, such as the sup process and certain discrete time summations we discuss. We discretize the space coordinate in uniform steps and assume that time is either continuous or finely discretized as in a fully explicit Euler method and the Courant condition is satisfied. We show that the Fourier transform of the joint kernel of a diffusion and a stochastic integral converges in a uniform graph norm associated to the Markov generator. Convergence also implies smoothness properties for the Fourier transform of the joint kernel. Stochastic integrals are straightforward to define for finite triangulations and the convergence result gives a new and entirely constructive way of defining stochastic integrals in the continuum. The method relies on a reinterpretation and extension of the classic theorems by Feynman-Kac, Girsanov, Ito and Cameron-Martin, which are also re-obtained. We make use of a path-wise analysis without relying on a probabilistic interpretation. The Fourier representation is needed to regularize the hypo-elliptic character of the joint process of a diffusion and an adapted stochastic integral. The argument extends as long as the Fourier analysis framework can be generalized. This condition leads to the notion of non-resonant Abelian process.

Motivation & Objective

• To develop a constructive, non-probabilistic method for defining stochastic integrals in the continuum limit without relying on measure-theoretic compactness arguments.
• To establish convergence of triangulation schemes for the joint distribution of a diffusion process and an adapted stochastic integral under uniform continuity of coefficients.
• To extend the framework to a broader class of path-dependent processes—termed non-resonant Abelian processes—beyond standard stochastic integrals.
• To derive convergence rates in the graph norm of the Fourier-transformed Markov generator, dependent on the Hölder differentiability of coefficients.
• To provide a pathwise, operator-algebraic derivation of classic results (e.g., Feynman-Kac, Girsanov, Cameron-Martin) via analytic continuation and block-diagonalization of joint kernels.

Proposed method

• Uses a triangulation scheme with uniform spatial discretization and either continuous time or fully explicit Euler time discretization, satisfying the Courant condition.
• Applies a Fourier transform to the joint kernel to regularize the hypo-elliptic structure and enable block-diagonalization of the Markov generator.
• Employs a renormalization group transformation to resum path contributions, treating decorated paths with time-ordered interactions.
• Introduces a path-wise analysis without probabilistic interpretation, relying on analytic continuation to complex weights for convergence proof.
• Leverages operator algebra techniques to define a joint Markov generator on finite triangulations, with convergence proven in the graph norm of the Fourier-transformed generator.
• Extends the method to discrete-time processes by lifting the elementary propagator and using partial Fourier transforms to achieve block-diagonalization.

Experimental results

Research questions

• RQ1 Under what conditions does the joint distribution of a diffusion process and a stochastic integral converge in the continuum limit under uniform spatial and time discretization?
• RQ2 How can stochastic integrals be defined constructively in the continuum without relying on non-constructive measure-theoretic methods?
• RQ3 What class of path-dependent processes beyond stochastic integrals allows for similar convergence and analyticity results via Fourier-based regularization?
• RQ4 What are the convergence rates in the graph norm of the Fourier-transformed Markov generator, and how do they depend on the smoothness of the coefficients?
• RQ5 Can classic formulas for characteristic functions of stochastic integrals (e.g., Girsanov, Cameron-Martin) be rederived through a pathwise, constructive framework?

Key findings

• The Fourier transform of the joint kernel of a diffusion and a stochastic integral is entire analytic in the conjugate variable of the stochastic integral, ensuring smoothness and convergence.
• Convergence of the triangulation scheme occurs in the graph norm of the Fourier-transformed Markov generator, with convergence rates depending on the Hölder differentiability of the coefficients.
• The method provides a constructive derivation of the Feynman-Kac, Girsanov, Cameron-Martin, and Ito formulas without compactness or probabilistic assumptions.
• The framework extends to non-resonant Abelian processes, including the sup process and discrete-time summations, with convergence established via similar Fourier and renormalization techniques.
• For discrete-time Euler schemes, convergence in the graph norm follows directly from the convergence of the one-period kernel, with bounds preserved under time discretization.
• The joint kernel remains well-regularized under absorbing boundary conditions (e.g., in the sup process), with bounds holding due to trivialized dynamics at absorption points.