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Strong and weak convergence rates for fully coupled multiscale stochastic differential equations driven by $α$-stable processes

Kun Yin

Abstract

We first establish strong convergence rates for multiscale systems driven by $α$-stable processes, with analyses constructed in two distinct scaling regimes. When addressing weak convergence rates of this system, we derive four averaged equations with respect to four scaling regimes. Notably, under sufficient Hölder regularity conditions on the time-dependent drifts of slow process, the strong convergence orders are related to the known optimal strong convergence order $1-\frac{1}α$, and the weak convergence orders are 1. Our primary approach involves employing nonlocal Poisson equations to construct ``corrector equations" that effectively eliminate inhomogeneous terms.

Strong and weak convergence rates for fully coupled multiscale stochastic differential equations driven by $α$-stable processes

Abstract

We first establish strong convergence rates for multiscale systems driven by -stable processes, with analyses constructed in two distinct scaling regimes. When addressing weak convergence rates of this system, we derive four averaged equations with respect to four scaling regimes. Notably, under sufficient Hölder regularity conditions on the time-dependent drifts of slow process, the strong convergence orders are related to the known optimal strong convergence order , and the weak convergence orders are 1. Our primary approach involves employing nonlocal Poisson equations to construct ``corrector equations" that effectively eliminate inhomogeneous terms.
Paper Structure (23 sections, 252 equations)

This paper contains 23 sections, 252 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Sketch of this paper
  4. Organization of this paper
  5. Some settings and main results
  6. Notations and assumptions
  7. Main results
  8. Well-posedness and some moment estimates of $(X_{t}^{\varepsilon},Y_{t}^{\varepsilon})$
  9. The frozen equation for \ref{['1.1']}
  10. Invariant measure of \ref{['4.1']}
  11. Moment estimates of $Y_{t}^{ x,y}$
  12. Strong convergence estimates for \ref{['1.1']}
  13. Regularity estimates of nonlocal Poisson equation
  14. LLN type estimate for $b(t,x,y)$
  15. CLT type estimate for $\frac{1}{\gamma_{\varepsilon}}H(t,x,y)$
  16. ...and 8 more sections

Theorems & Definitions (16)

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  • ...and 6 more