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Long-time behaviors of some stochastic differential equations driven by Lévy noise

I. Orlovskyi, F. Proske, O. Tymoshenko

Abstract

Using key tools such as Itô formula for general semi-martingales, moments estimates for Lévy-type stochastic integrals and properties of regular varying functions we find conditions under which solutions of stochastic differential equation with jumps are almost sure asymptotically equivalent nonrandom function with $t\to \infty$.

Long-time behaviors of some stochastic differential equations driven by Lévy noise

Abstract

Using key tools such as Itô formula for general semi-martingales, moments estimates for Lévy-type stochastic integrals and properties of regular varying functions we find conditions under which solutions of stochastic differential equation with jumps are almost sure asymptotically equivalent nonrandom function with .
Paper Structure (8 sections, 9 theorems, 58 equations)

This paper contains 8 sections, 9 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Itô formula
  4. Almost-sure asymptotic equivalence solution of SDEs driven by Lévy noise and solution ODE
  5. Auxiliary assertions
  6. The Proofs of Theorems 4.1 and 4.2
  7. Unboudedness of the solution of non-homogeneous SDEs driven by Lévy noise
  8. Conclusion

Key Result

Theorem 3.1

Suppose Then we have

Theorems & Definitions (23)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Remark 2.5
  • Theorem 3.1
  • Theorem 3.2: Dynkin formula
  • proof
  • Theorem 4.1
  • Theorem 4.2
  • ...and 13 more