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Existence, uniqueness and regularity of solutions to systems of nonlocal obstacle problems related to optimal switching
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0003-2716-3195
2019 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813Article in journal (Refereed) Published
Abstract [en]

We study viscosity solutions to a system of nonlinear degenerate parabolic partial integro-differential equations with interconnected obstacles. This type of problem occurs in the context of optimal switching problems when the dynamics of the underlying state variableis described by an n-dimensional Lévy 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 Hölder 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) [BJK10] and of Lundström, Nyström and Olofsson (2014) [LNO14, LNO14b] to hold for more general systems of equations.

Place, publisher, year, edition, pages
2019.
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
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-249063DOI: 10.1016/j.jmaa.2018.11.003ISI: 000464490800002Scopus ID: 2-s2.0-85061800225OAI: oai:DiVA.org:kth-249063DiVA, id: diva2:1303764
Note

QC 20190507

Available from: 2019-04-10 Created: 2019-04-10 Last updated: 2019-05-14Bibliographically approved

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Önskog, Thomas

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