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Duals and inverse flows of generalized Ornstein-Uhlenbeck processes

Anita Behme, Henriette E. Heinrich, Alexander Lindner

Abstract

We derive explicit representations for the (Siegmund) dual and the inverse flow of generalized Ornstein-Uhlenbeck processes whenever these exist. It turns out that the dual and the process corresponding to the inverse stochastic flow are again generalized Ornstein-Uhlenbeck processes. Further, we observe that the stationary distribution of the dual process provides information about the hitting time of zero of the original process.

Duals and inverse flows of generalized Ornstein-Uhlenbeck processes

Abstract

We derive explicit representations for the (Siegmund) dual and the inverse flow of generalized Ornstein-Uhlenbeck processes whenever these exist. It turns out that the dual and the process corresponding to the inverse stochastic flow are again generalized Ornstein-Uhlenbeck processes. Further, we observe that the stationary distribution of the dual process provides information about the hitting time of zero of the original process.

Paper Structure

This paper contains 8 sections, 13 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalized Ornstein-Uhlenbeck processes
  4. Hitting probabilities
  5. The dual of the generalized Ornstein-Uhlenbeck process
  6. Siegmund duality on $\mathbb{R}$
  7. Non-negative Siegmund dual processes
  8. The inverse flow of the generalized Ornstein-Uhlenbeck process

Key Result

Lemma 2.1

Under Assumption eq:assU(A), the solution $(V_t^x)_{t\geq 0}$ of eq:GOUSDE, called the generalized Ornstein-Uhlenbeck (GOU) process, is unique and given explicitly by Hereby, $(\mathcal{E}(U)_t)_{t\geq 0}$ is the stochastic exponential of $(U_t)_{t\geq 0}$ and $\eta=(\eta_t)_{t\geq 0}$ is the Lévy process given by

Theorems & Definitions (29)

  • Lemma 2.1: BLM2011
  • Lemma 2.2: BLM2011
  • Lemma 2.3: BLM2011 or Karandikar1991
  • Lemma 2.4: BLM2011 and its proof
  • Lemma 2.5
  • proof
  • Example 2.6
  • Proposition 2.7: BankSly
  • Definition 3.1
  • Example 3.2
  • ...and 19 more