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Rough differential equations containing path-dependent bounded variation terms

Shigeki Aida

Abstract

We consider rough differential equations whose coefficients contain path-dependent bounded variation terms and prove the existence and a priori estimate of solutions. These equations include classical path-dependent SDEs containing running maximum processes and normal reflection terms. We apply these results to determine the topological support of the solution processes.

Rough differential equations containing path-dependent bounded variation terms

Abstract

We consider rough differential equations whose coefficients contain path-dependent bounded variation terms and prove the existence and a priori estimate of solutions. These equations include classical path-dependent SDEs containing running maximum processes and normal reflection terms. We apply these results to determine the topological support of the solution processes.

Paper Structure

This paper contains 13 sections, 36 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Preliminary and Main Theorem
  3. Proof of Theorem \ref{['main theorem']}
  4. A continuity property of the solution mapping
  5. Examples
  6. Reflected rough differential equations
  7. Perturbed reflected SDEs: a short review
  8. Perturbed reflected rough differential equations
  9. Related path-dependent RDEs
  10. Continuity property and Support theorem
  11. Appendix
  12. Uniqueness of the Gubinelli derivative
  13. Path-dependent RDE with drift

Key Result

Lemma 2.1

Theorems & Definitions (79)

  • Lemma 2.1
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8
  • ...and 69 more