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<p>This work explores the use of Quantized State Systems (QSS) methods for the simulation of Stochastic Differential Equations (SDEs). To that purpose, an extension of these algorithms is proposed wherein the governing Wiener process is sampled at regular intervals, while the states are updated asynchronously when they satisfy the threshold conditions corresponding to the respective QSS method. We show that the resulting schemes produce trajectories that converge to the actual solutions of the SDEs as the sampling interval <i>h</i> and the quantum <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Delta {\textbf{Q}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi mathvariant="bold">Q</mi> </mrow> </math></EquationSource> </InlineEquation> approach zero. Moreover, we prove that, in stable linear time-invariant cases, the expected norm of the error is globally bounded by a linear function of the quantum <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Delta {\textbf{Q}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi mathvariant="bold">Q</mi> </mrow> </math></EquationSource> </InlineEquation> and the sampling interval <i>h</i>, demonstrating that the proposed scheme preserves practical stability regardless of the choice of these parameters. We also present simulation experiments that illustrate some potential advantages of the scheme.</p>

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Quantized state system methods for stochastic differential equations

  • 期刊论文

  • 发表日期：

    2025年12月18日

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  • Noelia Pizzi,
  • Mariana Bergonzi,
  • Joaquín Fernandez,
  • Damián Marelli,
  • Ernesto Kofman

摘要

This work explores the use of Quantized State Systems (QSS) methods for the simulation of Stochastic Differential Equations (SDEs). To that purpose, an extension of these algorithms is proposed wherein the governing Wiener process is sampled at regular intervals, while the states are updated asynchronously when they satisfy the threshold conditions corresponding to the respective QSS method. We show that the resulting schemes produce trajectories that converge to the actual solutions of the SDEs as the sampling interval h and the quantum \(\Delta {\textbf{Q}}\) Δ Q approach zero. Moreover, we prove that, in stable linear time-invariant cases, the expected norm of the error is globally bounded by a linear function of the quantum \(\Delta {\textbf{Q}}\) Δ Q and the sampling interval h, demonstrating that the proposed scheme preserves practical stability regardless of the choice of these parameters. We also present simulation experiments that illustrate some potential advantages of the scheme.