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Boundary-preserving numerical schemes for stochastic ordinary and partial differential equations

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Názov: Boundary-preserving numerical schemes for stochastic ordinary and partial differential equations
Autori: Ulander, Johan, 1995
Predmety: strong convergence, stochastic partial differential equations, Stochastic ordinary differential equations, geometric numerical integration, positivity-preserving, Lie--Trotter time splitting, boundary-preserving, weak convergence.
Popis: This thesis contributes to the development of boundary-preserving numerical schemes for the strong and weak approximation for stochastic ordinary and partial differential equations (SDEs and SPDEs, respectively). Several of the considered equations model a physical quantity with an inherently restricted range, such as temperature (positive values), stock prices (positive values) or fractions (values in [0,1]), referred to as the invariant domain of the equation. A numerical scheme is said to be boundary-preserving if its numerical approximations are guaranteed to remain within this domain. Boundary preservation is important for the physical interpretability and stability of the numerical approximations. Some established approaches to constructing boundary-preserving schemes are surveyed in the first part of the thesis, and the appended papers explore and develop new methods to guarantee this property. Paper I combines the Lamperti transform with a Lie--Trotter time splitting to construct a family of boundary-preserving numerical schemes for some scalar SDEs achieving strong convergence of order 1. Paper II constructs boundary-preserving numerical schemes for scalar SDEs by introducing auxiliary stochastic processes to convert the considered SDE into an associated reflected SDE. Paper III constructs a positivity-preserving temporal numerical scheme for some semilinear stochastic heat equations perturbed by temporal white noise. The proposed scheme employs a Lie-–Trotter time splitting method, allowing the deterministic and stochastic parts of the equation to be treated independently. Paper IV combines the ideas from Paper III with a finite difference spatial discretisation to obtain the first positivity-preserving numerical scheme for some semilinear stochastic heat equations perturbed by space-time white noise. Paper V combines the ideas from Paper IV with exact simulation for SDEs to obtain the first boundary-preserving numerical scheme for some semilinear SPDEs perturbed by space-time white noise with bounded invariant domain.
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Prístupová URL adresa: https://research.chalmers.se/publication/548883
https://research.chalmers.se/publication/548883/file/548883_Fulltext.pdf
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  Data: Boundary-preserving numerical schemes for stochastic ordinary and partial differential equations
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Ulander%2C+Johan%22">Ulander, Johan</searchLink>, 1995
– Name: Subject
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  Group: Su
  Data: <searchLink fieldCode="DE" term="%22strong+convergence%22">strong convergence</searchLink><br /><searchLink fieldCode="DE" term="%22stochastic+partial+differential+equations%22">stochastic partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Stochastic+ordinary+differential+equations%22">Stochastic ordinary differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22geometric+numerical+integration%22">geometric numerical integration</searchLink><br /><searchLink fieldCode="DE" term="%22positivity-preserving%22">positivity-preserving</searchLink><br /><searchLink fieldCode="DE" term="%22Lie--Trotter+time+splitting%22">Lie--Trotter time splitting</searchLink><br /><searchLink fieldCode="DE" term="%22boundary-preserving%22">boundary-preserving</searchLink><br /><searchLink fieldCode="DE" term="%22weak+convergence%2E%22">weak convergence.</searchLink>
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  Label: Description
  Group: Ab
  Data: This thesis contributes to the development of boundary-preserving numerical schemes for the strong and weak approximation for stochastic ordinary and partial differential equations (SDEs and SPDEs, respectively). Several of the considered equations model a physical quantity with an inherently restricted range, such as temperature (positive values), stock prices (positive values) or fractions (values in [0,1]), referred to as the invariant domain of the equation. A numerical scheme is said to be boundary-preserving if its numerical approximations are guaranteed to remain within this domain. Boundary preservation is important for the physical interpretability and stability of the numerical approximations. Some established approaches to constructing boundary-preserving schemes are surveyed in the first part of the thesis, and the appended papers explore and develop new methods to guarantee this property. Paper I combines the Lamperti transform with a Lie--Trotter time splitting to construct a family of boundary-preserving numerical schemes for some scalar SDEs achieving strong convergence of order 1. Paper II constructs boundary-preserving numerical schemes for scalar SDEs by introducing auxiliary stochastic processes to convert the considered SDE into an associated reflected SDE. Paper III constructs a positivity-preserving temporal numerical scheme for some semilinear stochastic heat equations perturbed by temporal white noise. The proposed scheme employs a Lie-–Trotter time splitting method, allowing the deterministic and stochastic parts of the equation to be treated independently. Paper IV combines the ideas from Paper III with a finite difference spatial discretisation to obtain the first positivity-preserving numerical scheme for some semilinear stochastic heat equations perturbed by space-time white noise. Paper V combines the ideas from Paper IV with exact simulation for SDEs to obtain the first boundary-preserving numerical scheme for some semilinear SPDEs perturbed by space-time white noise with bounded invariant domain.
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    Identifiers:
      – Type: doi
        Value: 10.63959/chalmers.dt/5767
    Languages:
      – Text: English
    Subjects:
      – SubjectFull: strong convergence
        Type: general
      – SubjectFull: stochastic partial differential equations
        Type: general
      – SubjectFull: Stochastic ordinary differential equations
        Type: general
      – SubjectFull: geometric numerical integration
        Type: general
      – SubjectFull: positivity-preserving
        Type: general
      – SubjectFull: Lie--Trotter time splitting
        Type: general
      – SubjectFull: boundary-preserving
        Type: general
      – SubjectFull: weak convergence.
        Type: general
    Titles:
      – TitleFull: Boundary-preserving numerical schemes for stochastic ordinary and partial differential equations
        Type: main
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          Name:
            NameFull: Ulander, Johan
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          Dates:
            – D: 01
              M: 01
              Type: published
              Y: 2025
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