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Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
2014 (English)In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 145, no 3-4, 407-432 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we study viscosity solutions to the system  \begin{eqnarray*}&&\min\biggl\{-\mathcal{H}u_i(x,t)-\psi_i(x,t),u_i(x,t)-\max_{j\neq i}(-c_{i,j}(x,t)+u_j(x,t))\biggr\}=0,\notag\\&&u_i(x,T)=g_i(x),\ i\in\{1,\dots,d\},\end{eqnarray*}where $(x,t)\in\mathbb R^{N}\times [0,T]$. Concerning $\mathcal{H}$ we assume that $\mathcal{H}=\mathcal{L}+\mathcal{I}$ where$\mathcal{L}$ is a linear, possibly degenerate, parabolic operator of second order and $\mathcal{I}$ is a non-local integro-partial differential operator. A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems when thedynamics of the underlying state variables is described by $N$-dimensional Levy processes.  We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth andstructural assumptions on the data, i.e., on the operator $\mathcal{H}$ and on continuous functions $\psi_i$, $c_{i,j}$, and$g_i$.   Using the comparison principle we establish the existence of a unique viscosity solution $(u_1,\dots,u_d)$  to the system by using Perron's method. Our contribution, compared to the existing literature, is that we establish existence and uniqueness of viscosity solutions in the setting of Levy processes and non-local operators with no sign assumption on the switching costs $\{c_{i,j}\}$ and allowing $c_{i,j}$  to depend on $x$ as well as $t$.

Place, publisher, year, edition, pages
2014. Vol. 145, no 3-4, 407-432 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-204876DOI: 10.1007/s00229-014-0683-9ISI: 000343881600008OAI: oai:DiVA.org:uu-204876DiVA: diva2:640000
Available from: 2013-08-12 Created: 2013-08-12 Last updated: 2017-12-06Bibliographically approved

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Lundström, NiklasNyström, KajOlofsson, Marcus
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