On the Goodness-of-Fit Tests for Some Continuous Time Processes

TL;DR

This paper reviews goodness-of-fit tests for various continuous time stochastic processes, including diffusion, Poisson, and self-exciting processes, focusing on their asymptotic properties and numerical performance.

Contribution

It introduces specific goodness-of-fit tests for multiple continuous time models and analyzes their asymptotic behavior and power under local alternatives.

Findings

Tests have asymptotic size α

Numerical simulations demonstrate test performance

Behavior under local alternatives analyzed

Abstract

We present a review of several results concerning the construction of the Cramer-von Mises and Kolmogorov-Smirnov type goodness-of-fit tests for continuous time processes. As the models we take a stochastic differential equation with small noise, ergodic diffusion process, Poisson process and self-exciting point processes. For every model we propose the tests which provide the asymptotic size $\alpha $ and discuss the behaviour of the power function under local alternatives. The results of numerical simulations of the tests are presented.