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LevyArea

Iterated stochastic integrals in Julia.

https://github.com/stochastics-uni-luebeck/levyarea.jl

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Iterated stochastic integrals in Julia.

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Created over 7 years ago · Last pushed about 4 years ago
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README.md

LevyArea.jl

Iterated Stochastic Integrals in Julia

Stable Dev Build Status DOI <!-- Concept DOI --> arXiv

This package implements state-of-the-art methods for the simulation of iterated stochastic integrals. These appear e.g. in higher order algorithms for the solution of stochastic (partial) differential equations.

Installation

This package can be installed from the Julia package manager (type ]) julia pkg> add LevyArea

Usage Example

Load the package and generate a Wiener increment: julia-repl julia> using LevyArea julia> m = 5; # dimension of Wiener process julia> h = 0.01; # step size or length of time interval julia> err = 0.05; # error bound julia> W = sqrt(h) * randn(m); # increment of Wiener process Here, $W$ is the $m$-dimensional vector of increments of the driving Wiener process on some time interval of length $h$.

The default call uses h^(3/2) as the precision and chooses the best algorithm automatically: julia-repl julia> II = iterated_integrals(W,h) If not stated otherwise, the default error criterion is the $\max,L^2$-error and the function returns the $m \times m$ matrix II containing a realisation of the approximate iterated stochastic integrals that correspond to the given increment $W$.

The desired precision can be optionally provided using a third positional argument: julia-repl julia> II = iterated_integrals(W,h,err) Again, the software package automatically chooses the optimal algorithm.

To determine which algorithm is chosen by the package without simulating any iterated stochastic integrals yet, the function optimal_algorithm can be used. The arguments to this function are the dimension of the Wiener process, the step size and the desired precision: julia-repl julia> alg = optimal_algorithm(m,h,err); # output: Fourier()

It is also possible to choose the algorithm directly using the keyword argument alg. The value can be one of Fourier(), Milstein(), Wiktorsson() and MronRoe(): julia-repl julia> II = iterated_integrals(W,h; alg=Milstein())

As the norm for the considered error, e.g., the $\max,L^2$- and $\mathrm{F},L^2$-norm can be selected using a keyword argument. The corresponding values are MaxL2() and FrobeniusL2(): julia-repl julia> II = iterated_integrals(W,h,err; error_norm=FrobeniusL2())

If iterated stochastic integrals for some $Q$-Wiener process need to be simulated, like for the numerical simulation of solutions to SPDEs, then the increment of the $Q$-Wiener process together with the square roots of the eigenvalues of the associated covariance operator have to be provided: julia-repl julia> q = [1/k^2 for k=1:m]; # eigenvalues of cov. operator julia> QW = sqrt(h) * sqrt.(q) .* randn(m); # Q-Wiener increment julia> IIQ = iterated_integrals(QW,sqrt.(q),h,err) In this case, the function iterated_integrals utilizes a scaling of the iterated stochastic integrals and also adjusts the error estimates appropriately such that the error bound holds w.r.t.\ the iterated stochastic integrals $\mathcal{I}^{Q}(h)$ based on the $Q$-Wiener process. Here the error norm defaults to the $\mathrm{F},L^2$-error.

Note that all discussed keyword arguments are optional and can be combined as needed.

Additional information can be found, e.g., using the Julia help mode: julia-repl julia> ?iterated_integrals julia> ?optimal_algorithm or by reading the documentation.

Citing

Please cite this package and/or the accompanying paper if you found this package useful. Example BibLaTeX code can be found in the CITATION.bib file.

Related Packages

A Matlab version of this package is also available under LevyArea.m.

Citation (CITATION.bib) 

@Article{Kastner2022,
  author      = {Kastner, Felix and Rößler, Andreas},
  date        = {2022-01-20},
  title       = {An Analysis of Approximation Algorithms for Iterated Stochastic Integrals and a Julia and MATLAB Simulation Toolbox},
  eprint      = {2201.08424},
  eprintclass = {math.NA},
  eprinttype  = {arXiv},
  keywords    = {Levy Area, Iterated Integrals, Stochastic Differential Equations, Julia},
}

@Software{Kastner2022a,
  author    = {Kastner, Felix and Rößler, Andreas},
  date      = {2022},
  title     = {LevyArea.jl},
  doi       = {10.5281/ZENODO.5883748},
  url       = {https://github.com/stochastics-uni-luebeck/LevyArea.jl},
  copyright = {MIT License},
  keywords  = {Levy Area, Iterated Integrals, Stochastic Differential Equations, Julia},
  publisher = {Zenodo},
}

@Comment{jabref-meta: databaseType:biblatex;}

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juliahub.com: LevyArea

Iterated stochastic integrals in Julia.

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