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The Skorohod oblique reflection problem in 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.
2010 (English)In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 38, no 6, 2170-2223 p.Article in journal (Refereed) Published
Abstract [en]

The deterministic Skorohod problem plays an important role in the construction and analysis of diffusion processes with reflection. In the form studied here, the multidimensional Skorohod problem was introduced, in time-independent domains, by H. Tanaka and further investigated by P.-L. Lions and A.-S. Sznitman in their celebrated article. Subsequent results of several researchers have resulted in a large literature on the Skorohod problem in time-independent domains. In this article we conduct a thorough study of the multidimensional Skorohod problem in time-dependent domains. In particular, we prove the existence of càdlàg solutions (x, λ) to the Skorohod problem, with oblique reflection, for (D,,w) assuming, in particular, that D is a time-dependent domain (Theorem 1.2). In addition, we prove that if w is continuous, then x is continuous as well (Theorem 1.3). Subsequently, we use the established existence results to construct solutions to stochastic differential equations with oblique reflection (Theorem 1.9) in time-dependent domains. In the process of proving these results we establish a number of estimates for solutions to the Skorohod problem with bounded jumps and, in addition, several results concerning the convergence of sequences of solutions to Skorohod problems in the setting of time-dependent domains.

Place, publisher, year, edition, pages
2010. Vol. 38, no 6, 2170-2223 p.
Keyword [en]
Skorohod problem, oblique reflection, time-dependent domain, stochastic differential equations
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-37418DOI: 10.1214/10-AOP538ISI: 000283010500003OAI: oai:DiVA.org:umu-37418DiVA: diva2:360258
Available from: 2010-11-02 Created: 2010-11-02 Last updated: 2017-12-12Bibliographically 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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Önskog, ThomasNyström, Kaj

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