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Existence, uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2019 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 475, no 1, p. 13-31Article in journal (Refereed) Published
Abstract [en]

We study viscosity solutions to a system of nonlinear degenerate parabolic partial integrodifferential equations with interconnected obstacles. This type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variable is described by an n-dimensional L´evy process. We first establish a continuous dependence estimate for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the partial integro-differential operator and on the obstacles and terminal conditions. Using the continuous dependence estimate, we obtain the comparison principle and uniqueness of viscosity solutions as well as Lipschitz regularity in the spatial variables. Our main contribution is construction of suitable families of viscosity sub- and supersolutions which we use as “barrier functions” to prove Ho¨lder continuity in the time variable, and, through Perron’s method, existence of a unique viscosity solution. This paper generalizes parts of the results of Biswas, Jakobsen and Karlsen (2010) and of Lundström, Nyström and Olofsson (2014) to hold for more general systems of equations.

Place, publisher, year, edition, pages
Elsevier, 2019. Vol. 475, no 1, p. 13-31
Keywords [en]
variational inequality, existence, viscosity solution, nonlocal operator, partial integro-differential operator, Lévy process, jump diffusion, optimal switching problem, regularity, continuous dependence, well posed
National Category
Mathematics Mathematical Analysis
Identifiers
URN: urn:nbn:se:umu:diva-153270DOI: 10.1016/j.jmaa.2018.11.003ISI: 000464490800002OAI: oai:DiVA.org:umu-153270DiVA, id: diva2:1263010
Available from: 2018-11-14 Created: 2018-11-14 Last updated: 2019-06-12Bibliographically approved

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Lundström, Niklas L.P.

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CiteExportLink to record
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Citation style
  • apa
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  • vancouver
  • Other style
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  • de-DE
  • en-GB
  • en-US
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  • nn-NO
  • nn-NB
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  • Other locale
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