Nonparametric estimation of trawl processes: Theory and applications

TL;DR

This paper introduces a novel nonparametric estimator for trawl processes, providing a comprehensive asymptotic theory, bias-variance estimation, and demonstrating practical applications in finance and queueing systems.

Contribution

It presents the first nonparametric estimator for the trawl function and develops a detailed asymptotic theory including bias and variance estimation.

Findings

Estimator shows good finite sample performance in simulations

Method effectively models high-frequency financial data

Applicable to queueing theory and model misspecification testing

Abstract

Trawl processes belong to the class of continuous-time, strictly stationary, infinitely divisible processes; they are defined as Levy bases evaluated over deterministic trawl sets. This article presents the first nonparametric estimator of the trawl function characterising the trawl set and the serial correlation of the process. Moreover, it establishes a detailed asymptotic theory for the proposed estimator, including a law of large numbers and a central limit theorem for various asymptotic relations between an in-fill and a long-span asymptotic regime. In addition, it develops consistent estimators for both the asymptotic bias and variance, which are subsequently used for establishing feasible central limit theorems which can be applied to data. A simulation study shows the good finite sample performance of the proposed estimators. The new methodology is applied to model misspecification testing, forecasting high-frequency financial spread data from a limit order book and to estimating the busy-time distribution of a stochastic queue.