Recent advances in ambit stochastics with a view towards tempo-spatial stochastic volatility/intermittency

TL;DR

This paper reviews recent developments in ambit stochastics, introduces new properties of ambit fields, and develops the concept of tempo-spatial stochastic volatility and intermittency with various modulation techniques.

Contribution

It provides a comprehensive overview of ambit stochastics and introduces new theoretical results and models for tempo-spatial stochastic volatility within ambit fields.

Findings

Established new properties of ambit fields

Developed models for tempo-spatial stochastic volatility

Presented examples of ambit field specifications

Abstract

Ambit stochastics is the name for the theory and applications of ambit fields and ambit processes and constitutes a new research area in stochastics for tempo-spatial phenomena. This paper gives an overview of the main findings in ambit stochastics up to date and establishes new results on general properties of ambit fields. Moreover, it develops the concept of tempo-spatial stochastic volatility/intermittency within ambit fields. Various types of volatility modulation ranging from stochastic scaling of the amplitude, to stochastic time change and extended subordination of random measures and to probability and L\'{e}vy mixing of volatility/intensity parameters will be developed. Important examples for concrete model specifications within the class of ambit fields are given.