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Time-changed Markov processes and coupled non-local equations

Giacomo Ascione, Enrico Scalas, Bruno Toaldo, Lorenzo Torricelli

TL;DR

This work develops a rigorous framework for coupled fully non-local time-space equations realized by time-changing Markov processes through the undershoot of a subordinator. Existence and uniqueness are established via a stochastic representation and a maximum principle, with detailed regularity analysis of the non-local solution and its boundary behavior. The theory is then applied to financial pricing problems, deriving a non-local Black–Scholes equation for derivatives when price dynamics feature dependent trade durations and jumps, and providing a renewal-type pricing structure for positive sojourn times. The results connect time-changed Feller processes, Bernstein-function based operators, and semi-Markov embeddings, offering a broad toolkit for anomalous diffusion models and intraday option pricing with dependent durations.

Abstract

In this paper we study coupled fully non-local equations, where a linear non-local operator jointly acts on the time and space variables. We establish existence and uniqueness of the solution. A maximum principle is proved and used to derive uniqueness. Existence is established by providing a stochastic representation based on anomalous processes constructed as a time change via the undershooting of an independent subordinator. This leads to general non-stepped processes with intervals of constancy representing a sticky or trapping effect. Our theory allows these intervals to be dependent on the immediately subsequent jump. These processes include scaling limit of suitable coupled continuous time random walks previously studied in applications, in particular in the context of anomalous diffusion and option pricing. Here we exploit our general theory to obtain a non-local analog of the Black and Scholes equation, addressing the problem of determining the seasoned price of a derivative security, in case the price fluctuations are described by a process whose jumps are dependent on the previous interval.

Time-changed Markov processes and coupled non-local equations

TL;DR

This work develops a rigorous framework for coupled fully non-local time-space equations realized by time-changing Markov processes through the undershoot of a subordinator. Existence and uniqueness are established via a stochastic representation and a maximum principle, with detailed regularity analysis of the non-local solution and its boundary behavior. The theory is then applied to financial pricing problems, deriving a non-local Black–Scholes equation for derivatives when price dynamics feature dependent trade durations and jumps, and providing a renewal-type pricing structure for positive sojourn times. The results connect time-changed Feller processes, Bernstein-function based operators, and semi-Markov embeddings, offering a broad toolkit for anomalous diffusion models and intraday option pricing with dependent durations.

Abstract

In this paper we study coupled fully non-local equations, where a linear non-local operator jointly acts on the time and space variables. We establish existence and uniqueness of the solution. A maximum principle is proved and used to derive uniqueness. Existence is established by providing a stochastic representation based on anomalous processes constructed as a time change via the undershooting of an independent subordinator. This leads to general non-stepped processes with intervals of constancy representing a sticky or trapping effect. Our theory allows these intervals to be dependent on the immediately subsequent jump. These processes include scaling limit of suitable coupled continuous time random walks previously studied in applications, in particular in the context of anomalous diffusion and option pricing. Here we exploit our general theory to obtain a non-local analog of the Black and Scholes equation, addressing the problem of determining the seasoned price of a derivative security, in case the price fluctuations are described by a process whose jumps are dependent on the previous interval.

Paper Structure

This paper contains 21 sections, 38 theorems, 473 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Uniqueness classes and semigroup actions
  4. Bernstein functions
  5. The non-local operator
  6. The undershooting of a subordinator and related time-changed processes
  7. A coupled fully non-local equation for time-changed Feller processes
  8. Regularity of $q$
  9. Proof of Theorem \ref{['thm:main1']}
  10. Another uniqueness criterion
  11. Option pricing with the undershooting of the $\alpha$-stable subordinator
  12. The price process
  13. The coupled non-local Black-Scholes equation
  14. Regularity of $q$
  15. Existence of the solution
  16. ...and 6 more sections

Key Result

Theorem 2.10

For any $\phi \in \mathfrak{B}$ there exists a unique triplet $(a_\phi,b_\phi,\nu_\phi)$, where $a_\phi,b_\phi \in \mathop{\mathrm{\mathbb{R}}}\nolimits^+$ and $\nu_\phi$ is a Borel measure on $(0,+\infty)$ with the property such that Vice versa, any triplet $(a_\phi,b_\phi,\nu_\phi)$ with $a_\phi,b_\phi>0$ and $\nu_\phi$ a Borel measure on $(0,+\infty)$ satisfying eq:Levydef defines a Bernstein

Theorems & Definitions (89)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 79 more