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Wong-Zakai approximations and support theorems for SDEs under Lyapunov conditions

Qi Li, Jianliang Zhai, Tusheng Zhang

Abstract

In this paper, we establish the Stroock-Varadhan type support theorems for stochastic differential equations (SDEs) under Lyapunov conditions, which significantly improve the existing results in the literature where the coefficients of the SDEs are required to be globally Lipschitz and of linear growth. Our conditions are very mild to include many important models, e.g. Threshold Ornstein-Ulenbeck process, Stochastic SIR model, Stochastic Lotka-Volterra systems, Stochastic Duffing-van der Pol oscillator model, which have polynomial the coefficients. To obtain the support theorem, we prove a new Wong-Zakai approximation problem, which is of independent interest.

Wong-Zakai approximations and support theorems for SDEs under Lyapunov conditions

Abstract

In this paper, we establish the Stroock-Varadhan type support theorems for stochastic differential equations (SDEs) under Lyapunov conditions, which significantly improve the existing results in the literature where the coefficients of the SDEs are required to be globally Lipschitz and of linear growth. Our conditions are very mild to include many important models, e.g. Threshold Ornstein-Ulenbeck process, Stochastic SIR model, Stochastic Lotka-Volterra systems, Stochastic Duffing-van der Pol oscillator model, which have polynomial the coefficients. To obtain the support theorem, we prove a new Wong-Zakai approximation problem, which is of independent interest.
Paper Structure (5 sections, 8 theorems, 54 equations)

This paper contains 5 sections, 8 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Wong-Zakai approxiamtions
  3. Support theorems
  4. Examples
  5. Acknowledgement

Key Result

Theorem 2.1

Let $Y^n$, $Z$ be solutions of SDEs (Yn) and (Z), respectively. Suppose Assumption assump1 is in place. Then, for $\delta>0$

Theorems & Definitions (11)

  • Theorem 2.1
  • proof
  • Remark 3.1
  • Proposition 3.1
  • Theorem 3.1
  • proof
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • ...and 1 more