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High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

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Abstract

An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.

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Authors and Affiliations

  1. Universidad de Ciencias Informáticas, La Habana, Cuba

    H. De la Cruz Cancino

  2. Instituto de Cibernética, Matemática y Física, La Habana, Cuba

    R. J. Biscay & J. C. Jimenez

  3. Montreal Neurological Institute, McGill University, Montreal, Canada

    F. Carbonell

  4. Institute of Statistical Mathematics, Tokyo, Japan

    T. Ozaki

Authors
  1. H. De la Cruz Cancino
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  2. R. J. Biscay
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  3. J. C. Jimenez
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  4. F. Carbonell
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  5. T. Ozaki
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Correspondence to H. De la Cruz Cancino.

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Communicated by Anders Szepessy.

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De la Cruz Cancino, H., Biscay, R.J., Jimenez, J.C. et al. High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise. Bit Numer Math 50, 509–539 (2010). https://doi.org/10.1007/s10543-010-0272-6

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