Stochastic Taylor expansion via Poisson point processes

TL;DR

This paper introduces a stochastic generalization of Taylor's theorem using Poisson point processes, leading to a new non-linear regression method with proven convergence and practical applications in finance.

Contribution

It presents a novel stochastic Taylor expansion framework based on Poisson processes, with theoretical convergence proofs and real-world data applications.

Findings

The estimator converges uniformly almost surely to the true function.

The methodology is effective in both univariate and multivariate cases.

Applications demonstrate the approach's utility in stock market data analysis.

Abstract

We generalize Taylor's theorem by introducing a stochastic formulation based on an underlying Poisson point process model. We utilize this approach to propose a novel non-linear regression framework and perform statistical inference of the model parameters. Theoretical properties of the proposed estimator are also proven, including its convergence, uniformly almost surely, to the true function. The theory is presented for the univariate and multivariate cases, and we exemplify the proposed methodology using several examples via simulations and an application to stock market data.