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A Multi-level Monte Carlo simulation for invariant distribution of Markovian switching Lévy-driven SDEs with super-linearly growth coefficients

Hoang-Viet Nguyen, Trung-Thuy Kieu, Duc-Trong Luong, Hoang-Long Ngo, Tran Ngoc Khue

TL;DR

An approximation scheme that can be applied to stochastic differential equations with super-linear growth drift and diffusion coefficients with super-linear growth drift and diffusion coefficients is proposed.

Abstract

This paper concerns the numerical approximation for the invariant distribution of Markovian switching Lévy-driven stochastic differential equations. By combining the tamed-adaptive Euler-Maruyama scheme with the Multi-level Monte Carlo method, we propose an approximation scheme that can be applied to stochastic differential equations with super-linear growth drift and diffusion coefficients.

A Multi-level Monte Carlo simulation for invariant distribution of Markovian switching Lévy-driven SDEs with super-linearly growth coefficients

TL;DR

An approximation scheme that can be applied to stochastic differential equations with super-linear growth drift and diffusion coefficients with super-linear growth drift and diffusion coefficients is proposed.

Abstract

This paper concerns the numerical approximation for the invariant distribution of Markovian switching Lévy-driven stochastic differential equations. By combining the tamed-adaptive Euler-Maruyama scheme with the Multi-level Monte Carlo method, we propose an approximation scheme that can be applied to stochastic differential equations with super-linear growth drift and diffusion coefficients.

Paper Structure

This paper contains 12 sections, 11 theorems, 80 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Model assumptions and moment estimates
  3. Adaptive Multi-level Monte Carlo for invariant distribution
  4. Existence and uniqueness of the invariant measure
  5. Approximation algorithm for the invariant measure
  6. TAEM scheme
  7. Construction of the TAEM scheme
  8. Moments of the TAEM scheme
  9. Convergence of the TAEM scheme
  10. Numerical experiments
  11. Path approximation
  12. Invariant measure approximation

Key Result

Proposition 2.2

Assume that $X=(X_t)_{t \geq 0}$ is a solution to equation eqn1, $\sigma$ is bounded on every compact subset of $S \times \mathbb{R}^d$, and C4 holds for $q=2p_0$. Then, for any $p \in (0,p_0]$ and $t \ge 0$, there exists a positive constant $C_p$ which does not depend on $t$ such that

Figures (1)

  • Figure 1: Numerical results for invariant measure approximation.

Theorems & Definitions (23)

  • Remark 2.1
  • Proposition 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 13 more