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On Monte Carlo algorithms applied to Dirichlet problems for parabolic operators in the setting of time-dependent domains
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2009 (English)In: Monte Carlo Methods and Applications, ISSN 1569-3961, Vol. 15, no 1, 11-47 p.Article in journal (Refereed) Published
Abstract [en]

Dirichlet problems for second order parabolic operators in space-time domains Ω⊂ Rn+1  are of paramount importance in analysis, partial differential equations and applied mathematics. These problems can be approached in many different ways using techniques from partial differential equations, potential theory, stochastic differential equations, stopped diffusions and Monte Carlo methods. The performance of any technique depends on the structural assumptions on the operator, the geometry and smoothness properties of the space-time domain Ω, the smoothness of the Dirichlet data and the smoothness of the coefficients of the operator under consideration. In this paper, which mainly is of numerical nature, we attempt to further understand how Monte Carlo methods based on the numerical integration of stochastic differential equations perform when applied to Dirichlet problems for uniformly elliptic second order parabolic operators and how their performance vary as the smoothness of the boundary, Dirichlet data and coefficients change from smooth to non-smooth. Our analysis is set in the genuinely parabolic setting of time-dependent domains, which in itself adds interesting features previously only modestly discussed in the literature. The methods evaluated and discussed include elaborations on the non-adaptive method proposed by Gobet [4] based on approximation by half spaces and exit probabilities and the adaptive method proposed in [3] for weak approximation of stochastic differential equations.

Place, publisher, year, edition, pages
Berlin New York: de Gruyter , 2009. Vol. 15, no 1, 11-47 p.
Keyword [en]
time-dependent domain, non-smooth domain, heat equation, parabolic partial differential equations, Cauchy-Dirichlet problem, stochastic differential equations, stopped diffusion, Euler scheme, adaptive methods
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-21728DOI: 10.1515 /MCMA.2009.002OAI: oai:DiVA.org:umu-21728DiVA: diva2:211758
Available from: 2009-08-06 Created: 2009-04-17 Last updated: 2012-08-16Bibliographically approved
In thesis
1. The Skorohod problem and weak approximation of stochastic differential equations in time-dependent domains
Open this publication in new window or tab >>The Skorohod problem and weak approximation of stochastic differential equations in time-dependent domains
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of a summary and four scientific articles. All four articles consider various aspects of stochastic differential equations and the purpose of the summary is to provide an introduction to this subject and to supply the notions required in order to fully understand the articles.

In the first article we conduct a thorough study of the multi-dimensional Skorohod problem in time-dependent domains. In particular we prove the existence of cádlág solutions to the Skorohod problem with oblique reflection in time-independent domains with corners. We use this existence result to construct weak solutions to stochastic differential equations with oblique reflection in time-dependent domains. In the process of obtaining these results we also establish convergence results for sequences of solutions to the Skorohod problem and a number of estimates for solutions, with bounded jumps, to the Skorohod problem.

The second article considers the problem of determining the sensitivities of a solution to a second order parabolic partial differential equation with respect to perturbations in the parameters of the equation. We derive an approximate representation of the sensitivities and an estimate of the discretization error arising in the sensitivity approximation. We apply these theoretical results to the problem of determining the sensitivities of the price of European swaptions in a LIBOR market model with respect to perturbations in the volatility structure (the so-called ‘Greeks’).

The third article treats stopped diffusions in time-dependent graph domains with low regularity. We compare, numerically, the performance of one adaptive and three non-adaptive numerical methods with respect to order of convergence, efficiency and stability. In particular we investigate if the performance of the algorithms can be improved by a transformation which increases the regularity of the domain but, at the same time, reduces the regularity of the parameters of the diffusion.

In the fourth article we use the existence results obtained in Article I to construct a projected Euler scheme for weak approximation of stochastic differential equations with oblique reflection in time-dependent domains. We prove theoretically that the order of convergence of the proposed algorithm is 1/2 and conduct numerical simulations which support this claim.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2009. 36 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 42
Keyword
Skorohod problem, weak approximation, time-dependent domain, stochastic differential equations, parabolic partial differential equations, oblique reflection, stopped diffusions, Euler scheme, adaptive methods, sensitivity analysis, financial derivatives, 'Greeks'
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-25429 (URN)978-91-7264-823-4 (ISBN)
Distributor:
Institutionen för matematik och matematisk statistik, 90187, Umeå
Public defence
2009-09-18, MA121, MIT-huset, Umeå universitet, Umeå, 13:15 (English)
Opponent
Supervisors
Available from: 2009-08-31 Created: 2009-08-17 Last updated: 2010-11-03Bibliographically approved

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