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Additive subordination of multiparameter Markov processes

Giuseppe D'Onofrio, Alessandro Mutti, Patrizia Semeraro

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 Lévy-Khintchine representation. In the specific case of multiparameter Ornstein-Uhlenbeck processes, we obtain explicit expression of the symbol, along with the associated characteristic Lévy 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.

Additive subordination of multiparameter Markov 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 Lévy-Khintchine representation. In the specific case of multiparameter Ornstein-Uhlenbeck processes, we obtain explicit expression of the symbol, along with the associated characteristic Lévy 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.

Paper Structure

This paper contains 10 sections, 13 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Markov processes, characteristic function, and symbol
  3. Multiparameter Markov process
  4. Multiparameter Additive subordination
  5. Additive subordination of M-OU processes
  6. Multiparameter Ornstein-Uhlenbeck process
  7. Construction of a M-OU process
  8. The Sato subordinator
  9. Sato subordination of M-OU processes
  10. Future works

Key Result

Proposition 2.1

If $(T_{\boldsymbol{s}})_{\boldsymbol{s}\succeq 0}$ is a $k$-parameter strongly continuous semigroup on $\mathfrak{B}_b(\mathbb{R}^d)$ then it is the direct product of $k$ one-parameter strongly continuous semigroups $(T_{s}^{(j)})_{s\geq 0}$, $j=1,\ldots, k$, on $\mathfrak{B}_b(\mathbb{R}^d)$ with

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2: Definition 1.6 jacob2001levy
  • Definition 2.3
  • Definition 2.4: khoshnevisan2006multiparameter
  • Remark 2.1
  • Example 1
  • Remark 2.2
  • Proposition 2.1: mendoza2016multivariate
  • Proposition 2.2
  • Corollary 2.1
  • ...and 31 more