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Volume 28, Issue 5
Weak Approximation of Obliquely Reflected Diffusions on Time-Dependent Domains

Kaj Nyström & Thomas Önskog

J. Comp. Math., 28 (2010), pp. 579-605.

Published online: 2010-10

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  • Abstract

In an earlier paper, we proved the existence of solutions to the Skorohod problem with oblique reflection in time-dependent domains and, subsequently, applied this result to the problem of constructing solutions, in time-dependent domains, to stochastic differential equations with oblique reflection. In this paper we use these results to construct weak approximations of solutions to stochastic differential equations with oblique reflection, in time-dependent domains in $\mathbb{R}^{d}$, by means of a projected Euler scheme. We prove that the constructed method has, as is the case for normal reflection and time-independent domains, an order of convergence equal to $1/2$ and we evaluate the method empirically by means of two numerical examples. Furthermore, using a well-known extension of the Feynman-Kac formula, to stochastic differential equations with reflection, our method gives, in addition, a Monte Carlo method for solving second order parabolic partial differential equations with Robin boundary conditions in time-dependent domains.

  • AMS Subject Headings

65MXX, 35K20, 65CXX, 60J50, 60J60

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COPYRIGHT: © Global Science Press

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@Article{JCM-28-579, author = {}, title = {Weak Approximation of Obliquely Reflected Diffusions on Time-Dependent Domains}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {5}, pages = {579--605}, abstract = {

In an earlier paper, we proved the existence of solutions to the Skorohod problem with oblique reflection in time-dependent domains and, subsequently, applied this result to the problem of constructing solutions, in time-dependent domains, to stochastic differential equations with oblique reflection. In this paper we use these results to construct weak approximations of solutions to stochastic differential equations with oblique reflection, in time-dependent domains in $\mathbb{R}^{d}$, by means of a projected Euler scheme. We prove that the constructed method has, as is the case for normal reflection and time-independent domains, an order of convergence equal to $1/2$ and we evaluate the method empirically by means of two numerical examples. Furthermore, using a well-known extension of the Feynman-Kac formula, to stochastic differential equations with reflection, our method gives, in addition, a Monte Carlo method for solving second order parabolic partial differential equations with Robin boundary conditions in time-dependent domains.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1003-m2957}, url = {http://global-sci.org/intro/article_detail/jcm/8539.html} }
TY - JOUR T1 - Weak Approximation of Obliquely Reflected Diffusions on Time-Dependent Domains JO - Journal of Computational Mathematics VL - 5 SP - 579 EP - 605 PY - 2010 DA - 2010/10 SN - 28 DO - http://doi.org/10.4208/jcm.1003-m2957 UR - https://global-sci.org/intro/article_detail/jcm/8539.html KW - Stochastic differential equations, Oblique reflection, Robin boundary conditions, Skorohod problem, Time-dependent domain, Weak approximation, Monte Carlo method, Parabolic partial differential equations, Projected Euler scheme. AB -

In an earlier paper, we proved the existence of solutions to the Skorohod problem with oblique reflection in time-dependent domains and, subsequently, applied this result to the problem of constructing solutions, in time-dependent domains, to stochastic differential equations with oblique reflection. In this paper we use these results to construct weak approximations of solutions to stochastic differential equations with oblique reflection, in time-dependent domains in $\mathbb{R}^{d}$, by means of a projected Euler scheme. We prove that the constructed method has, as is the case for normal reflection and time-independent domains, an order of convergence equal to $1/2$ and we evaluate the method empirically by means of two numerical examples. Furthermore, using a well-known extension of the Feynman-Kac formula, to stochastic differential equations with reflection, our method gives, in addition, a Monte Carlo method for solving second order parabolic partial differential equations with Robin boundary conditions in time-dependent domains.

Kaj Nyström & Thomas Önskog. (1970). Weak Approximation of Obliquely Reflected Diffusions on Time-Dependent Domains. Journal of Computational Mathematics. 28 (5). 579-605. doi:10.4208/jcm.1003-m2957
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