uu.seUppsala University Publications

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Optimal Switching Problems and Related EquationsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2015 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Uppsala: Department of Mathematics, 2015. , 37 p.
##### Series

Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 87
##### Keyword [en]

optimal switching, stochastic control, variational inequalities, backward stochastic differential equations, incomplete information, stochastic filtering
##### National Category

Mathematical Analysis
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:uu:diva-247298ISBN: 978-91-506-2448-9 (print)OAI: oai:DiVA.org:uu-247298DiVA: diva2:796336
##### Public defence

2015-05-08, Polhemsalen, Lägerhyddsvägen 1, Uppsala, 10:15 (English)
##### Opponent

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##### Supervisors

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#####

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Available from: 2015-04-17 Created: 2015-03-17 Last updated: 2015-04-17
##### List of papers

This thesis consists of five scientific papers dealing with equations related to the optimal switching problem, mainly backward stochastic differential equations and variational inequalities. Besides the scientific papers, the thesis contains an introduction to the optimal switching problem and a brief outline of possible topics for future research.

Paper I concerns systems of variational inequalities with operators of Kolmogorov type. We prove a comparison principle for sub- and supersolutions and prove the existence of a solution as the limit of solutions to iteratively defined interconnected obstacle problems. Furthermore, we use regularity results for a related obstacle problem to prove Hölder continuity of this solution.

Paper II deals with systems of variational inequalities in which the operator is of non-local type. By using a maximum principle adapted to this non-local setting we prove a comparison principle for sub- and supersolutions. Existence of a solution is proved using this comparison principle and Perron's method.

In Paper III we study backward stochastic differential equations in which the solutions are reflected to stay inside a time-dependent domain. The driving process is of Wiener-Poisson type, allowing for jumps. By a penalization technique we prove existence of a solution when the bounding domain has convex and non-increasing time slices. Uniqueness is proved by an argument based on Ito's formula.

Paper IV and Paper V concern optimal switching problems under incomplete information. In Paper IV, we construct an entirely simulation based numerical scheme to calculate the value function of such problems. We prove the convergence of this scheme when the underlying processes fit into the framework of Kalman-Bucy filtering. Paper V contains a deterministic approach to incomplete information optimal switching problems. We study a simplistic setting and show that the problem can be reduced to a full information optimal switching problem. Furthermore, we prove that the value of information is positive and that the value function under incomplete information converges to that under full information when the noise in the observation vanishes.

1. Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type$(function(){PrimeFaces.cw("OverlayPanel","overlay600567",{id:"formSmash:j_idt781:0:j_idt786",widgetVar:"overlay600567",target:"formSmash:j_idt781:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness$(function(){PrimeFaces.cw("OverlayPanel","overlay640000",{id:"formSmash:j_idt781:1:j_idt786",widgetVar:"overlay640000",target:"formSmash:j_idt781:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Reflected BSDE of Wiener-Poisson type in time-dependent domains$(function(){PrimeFaces.cw("OverlayPanel","overlay716031",{id:"formSmash:j_idt781:2:j_idt786",widgetVar:"overlay716031",target:"formSmash:j_idt781:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Optimal switching problems under partial information$(function(){PrimeFaces.cw("OverlayPanel","overlay703457",{id:"formSmash:j_idt781:3:j_idt786",widgetVar:"overlay703457",target:"formSmash:j_idt781:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. A Brownian optimal switching problem under incomplete information$(function(){PrimeFaces.cw("OverlayPanel","overlay783913",{id:"formSmash:j_idt781:4:j_idt786",widgetVar:"overlay783913",target:"formSmash:j_idt781:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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