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<p>Many problems in life sciences require the understanding of averages of a time-continuum evolution of a random phenomena or solutions to a parabolic equation defined on a continuum. The nature of both continua necessitates to implement approximation schemes in both time-space variables. We propose a procedure that identifies <i>exactly</i> these averages as averages of random phenomena defined on a discrete set whose time evolution is perfectly randomized. Consequently, both time and state sets are amenable to precise computation, effectively reducing the overall error to that inherent in simple Monte Carlo or integration methods. Our approach also extends to parabolic problems and enables to identify exact discretization schemes for solving the associated PDEs. These identifications are based on isospectral schemes called <i>gateway relations</i> as they relate two seemingly disconnected worlds, the continuum and the lattice, and their recent refinement, the interweaving relations. We show promising numerical examples leveraging the gateway relationship between the semigroups of squared Bessel and CIR processes, in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {R}}^+\)</EquationSource> </InlineEquation> and in its <i>n</i>-Weyl chamber, demonstrating the benefits of gateway-interweaving inspired simulation compared to traditional methods.</p>

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Isospectral schemes for exact algorithms for stochastic processes and PDE’s

  • 期刊论文

  • 发表日期：

    2026年1月7日

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  • Andrew Chee,
  • X. Y. Han,
  • Yuxhuan Liu,
  • Pierre Patie,
  • Rohan Sarkar,
  • Tonghua Tian

摘要

Many problems in life sciences require the understanding of averages of a time-continuum evolution of a random phenomena or solutions to a parabolic equation defined on a continuum. The nature of both continua necessitates to implement approximation schemes in both time-space variables. We propose a procedure that identifies exactly these averages as averages of random phenomena defined on a discrete set whose time evolution is perfectly randomized. Consequently, both time and state sets are amenable to precise computation, effectively reducing the overall error to that inherent in simple Monte Carlo or integration methods. Our approach also extends to parabolic problems and enables to identify exact discretization schemes for solving the associated PDEs. These identifications are based on isospectral schemes called gateway relations as they relate two seemingly disconnected worlds, the continuum and the lattice, and their recent refinement, the interweaving relations. We show promising numerical examples leveraging the gateway relationship between the semigroups of squared Bessel and CIR processes, in \({\mathbb {R}}^+\) and in its n-Weyl chamber, demonstrating the benefits of gateway-interweaving inspired simulation compared to traditional methods.