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Sharp estimates for Lyapunov exponents of Milstein approximation of stochastic differential systems

Vu Thi Hue

TL;DR

The paper tackles the problem of preserving exponential stability and exponential blow-up from a continuous 2\times 2 linear SDE to its Milstein discretization. It introduces sharp, both upper and lower, bounds for the discrete Lyapunov exponent, enabling precise characterization of mean-square and almost-sure stability for small step sizes $\Delta t$ and for both the Milstein and $\theta$-Milstein schemes. The results are first established for linear SDEs and then extended to the $\theta$-Milstein variant, including explicit expressions for the per-step multipliers and their impact on the discrete exponents. Numerical simulations corroborate the theoretical bounds and illustrate the dependence on step size, validating the equivalence between discrete and continuum stability regimes in practical settings.

Abstract

The Milstein approximation with step size $Δt>0$ of the solution $(X, Y)$ to a two-by-two system of linear stochastic differential equations is considered. It is proved that when the solution of the underlying model is exponentially stable or exponentially blowing up at infinite time, these behaviours are preserved at the level of the Milstein approximate solution $\{(X_n, Y_n)\}$ in both the mean-square and almost-sure senses, provided sufficiently small step size $Δt$. This result is based on sharp estimates, from both above and below, of the discrete Lyapunov exponent. This type of sharp estimate for approximate solutions to stochastic differential equations seems to be first studied in this work. In particular, the proposed method covers the setting for linear stochastic differential equations as well as the $θ$-Milstein scheme's setting.

Sharp estimates for Lyapunov exponents of Milstein approximation of stochastic differential systems

TL;DR

The paper tackles the problem of preserving exponential stability and exponential blow-up from a continuous 2\times 2 linear SDE to its Milstein discretization. It introduces sharp, both upper and lower, bounds for the discrete Lyapunov exponent, enabling precise characterization of mean-square and almost-sure stability for small step sizes and for both the Milstein and -Milstein schemes. The results are first established for linear SDEs and then extended to the -Milstein variant, including explicit expressions for the per-step multipliers and their impact on the discrete exponents. Numerical simulations corroborate the theoretical bounds and illustrate the dependence on step size, validating the equivalence between discrete and continuum stability regimes in practical settings.

Abstract

The Milstein approximation with step size of the solution to a two-by-two system of linear stochastic differential equations is considered. It is proved that when the solution of the underlying model is exponentially stable or exponentially blowing up at infinite time, these behaviours are preserved at the level of the Milstein approximate solution in both the mean-square and almost-sure senses, provided sufficiently small step size . This result is based on sharp estimates, from both above and below, of the discrete Lyapunov exponent. This type of sharp estimate for approximate solutions to stochastic differential equations seems to be first studied in this work. In particular, the proposed method covers the setting for linear stochastic differential equations as well as the -Milstein scheme's setting.

Paper Structure

This paper contains 10 sections, 6 theorems, 84 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Exponential stability versus exponential blow-up
  3. Mean-square exponential stability
  4. Almost-sure exponential stability
  5. Linear SDEs and theta-Milstein scheme
  6. Milstein scheme of linear SDE
  7. The $\theta$ -Milstein scheme
  8. Further discussion and numerical simulation
  9. Further discussion with different noisy intensities
  10. Numerical simulations

Key Result

Proposition 2.1

Let $\{Z_n\}$ be the solution to the Milstein scheme FinalLable with a step size $0<\Delta t<1$. Then, it holds where the remainder $\mathbf{R}_{\lambda,\varepsilon,\sigma}^{\mathsf{ms}}(\Delta t)$ is formulated as Moreover, there exists $\Delta t_\dagger>0$ sufficiently small such that, for all $0<\Delta t<\Delta t_\dagger$,

Figures (4)

  • Figure 1: Stable and blowing-up regions of noise parameters $\sigma,\varepsilon$ given drift coefficient $\lambda$.
  • Figure 2: Almost-sure exponential blow-up of $| Z_n |$ via the graph of $\log|Z_n|$.
  • Figure 3: Almost-sure exponential stability of $| Z_n |$ via the graph of $\log|Z_n|$.
  • Figure 4: Continuum and discrete Lyapunov exponents as the time step $\Delta t$ varies.

Theorems & Definitions (14)

  • Remark 1.1
  • Proposition 2.1: Exponential stability versus exponential blow-up in the mean-square sense
  • proof
  • Remark 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.2: Exponential stability versus exponential blow-up in the almost-sure sense
  • proof
  • Remark 2.2
  • ...and 4 more