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The Sticky Lévy Process as a solution to a Time Change Equation

Miriam Ramírez, Gerónimo Uribe Bravo

Abstract

Stochastic Differential Equations (SDEs) were originally devised by Itô to provide a pathwise construction of diffusion processes. A less explored approach to represent them is through Time Change Equations (TCEs) as put forth by Doeblin. TCEs are a generalization of Ordinary Differential Equations driven by random functions. We present a simple example where TCEs have some advantage over SDEs. We represent sticky Lévy processes as the unique solution to a TCE driven by a Lévy process with no negative jumps. The solution is adapted to the time-changed filtration of the Lévy process driving the equation. This is in contrast to the SDE describing sticky Brownian motion, which is known to have no adapted solutions as first proved by Chitashvili. A known consequence of such non-adaptability for SDEs is that certain natural approximations to the solution of the corresponding SDE do not converge in probability, even though they do converge weakly. Instead, we provide strong approximation schemes for the solution of our TCE (by adapting Euler's method for ODEs), whenever the driving Lévy process is strongly approximated.

The Sticky Lévy Process as a solution to a Time Change Equation

Abstract

Stochastic Differential Equations (SDEs) were originally devised by Itô to provide a pathwise construction of diffusion processes. A less explored approach to represent them is through Time Change Equations (TCEs) as put forth by Doeblin. TCEs are a generalization of Ordinary Differential Equations driven by random functions. We present a simple example where TCEs have some advantage over SDEs. We represent sticky Lévy processes as the unique solution to a TCE driven by a Lévy process with no negative jumps. The solution is adapted to the time-changed filtration of the Lévy process driving the equation. This is in contrast to the SDE describing sticky Brownian motion, which is known to have no adapted solutions as first proved by Chitashvili. A known consequence of such non-adaptability for SDEs is that certain natural approximations to the solution of the corresponding SDE do not converge in probability, even though they do converge weakly. Instead, we provide strong approximation schemes for the solution of our TCE (by adapting Euler's method for ODEs), whenever the driving Lévy process is strongly approximated.
Paper Structure (9 sections, 9 theorems, 71 equations)

This paper contains 9 sections, 9 theorems, 71 equations.

Table of Contents

  1. Introduction and statement of the results
  2. Deterministic analysis
  3. Monotonicity and Uniqueness
  4. Existence
  5. Approximation
  6. Application to sticky Lévy processes
  7. Existence, Uniqueness and Approximation
  8. Measurability details and the strong Markov property
  9. Stickiness and martingales

Key Result

Theorem 1

Let $X$ be a SPLP adapted to a right-continuous and complete filtration $(\mathscr{F}_t, t\geq 0)$. Assume that the sample paths of $X$ have unbounded variation. Given a parameter $\gamma > 0$ and a point $z\geq 0$, there exists a unique pair of stochastic processes $Z=(Z_t, t\geq 0)$ and $C=(C_t,t\ for every $t\geq 0$. For the unique pair $(Z,C)$ verifying Equation E:cdt, it holds that $C$ is a $

Theorems & Definitions (18)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 1
  • Proposition 3
  • proof
  • ...and 8 more