Additive subordination of multiparameter Markov processes

TL;DR

This paper extends Phillips theorem to multiparameter Markov processes subordinated by additive processes, characterizing their generators and symbols, with applications in finance involving explicit formulas for Ornstein-Uhlenbeck processes.

Contribution

It generalizes the theory of subordination to multiparameter Markov processes, providing explicit generator and symbol characterizations, and applies these results to finance-related models.

Findings

The subordinated process is a Feller evolution with a well-characterized generator.

The process's symbol admits a Lévy-Khintchine representation.

Explicit formulas are derived for multiparameter Ornstein-Uhlenbeck processes.

Abstract

In this work, we consider, in a general setting, multiparameter multidimensional Markov processes that are time-changed by an independent additive subordinator. By extending Phillips theorem, we show that the resulting process is a Feller evolution and we characterize its generator. We further derive its pseudo-differential representation and show that its symbol admits a L\'evy-Khintchine representation. In the specific case of multiparameter Ornstein-Uhlenbeck processes, we obtain explicit expression of the symbol, along with the associated characteristic L\'evy triplet. As an application, we consider a factor-based specification for the Ornstein-Uhlenbeck process subordinated by a Sato process. The constructive nature of this process is inspired by applications in finance.