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Optimal Switching Problems and Related Equations
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

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.

##### Place, publisher, year, edition, pages
Uppsala: Department of Mathematics, 2015. , p. 37
##### 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
Mathematics
##### Identifiers
ISBN: 978-91-506-2448-9 (print)OAI: oai:DiVA.org:uu-247298DiVA, id: diva2:796336
##### Public defence
2015-05-08, Polhemsalen, Lägerhyddsvägen 1, Uppsala, 10:15 (English)
##### Supervisors
Available from: 2015-04-17 Created: 2015-03-17 Last updated: 2015-04-17
##### List of papers
1. Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type
Open this publication in new window or tab >>Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type
2014 (English)In: Annali di Matematica Pura ed Applicata, ISSN 0373-3114, E-ISSN 1618-1891, Vol. 193, no 4, p. 1213-1247Article in journal (Refereed) Published
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-192711 (URN)10.1007/s10231-013-0325-y (DOI)000339962000017 ()
Available from: 2013-01-25 Created: 2013-01-24 Last updated: 2017-12-06Bibliographically approved
2. Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness
Open this publication in new window or tab >>Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness
2014 (English)In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 145, no 3-4, p. 407-432Article 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 $t$.

Mathematics
##### Identifiers
urn:nbn:se:uu:diva-204876 (URN)10.1007/s00229-014-0683-9 (DOI)000343881600008 ()
Available from: 2013-08-12 Created: 2013-08-12 Last updated: 2017-12-06Bibliographically approved
3. Reflected BSDE of Wiener-Poisson type in time-dependent domains
Open this publication in new window or tab >>Reflected BSDE of Wiener-Poisson type in time-dependent domains
2016 (English)In: Stochastic Models, ISSN 1532-6349, E-ISSN 1532-4214, Vol. 32, no 2, p. 275-300Article in journal (Refereed) Published
##### Abstract [en]

In the paper we study multi-dimensional reflected backward stochasticdifferential equations driven by Wiener-Poisson type processes. We prove existence and uniqueness of solutions, with reflection in the inward spatial normal direction, in the setting of certain time-dependent domains.

##### Keyword
Backward stochastic differential equation, convex domain, reflected backward stochastic differential equation, time-dependent domain, 60H10, 60H20
##### National Category
Mathematical Analysis Probability Theory and Statistics
##### Identifiers
urn:nbn:se:uu:diva-224258 (URN)10.1080/15326349.2015.1116011 (DOI)000377141900005 ()
Available from: 2014-05-07 Created: 2014-05-07 Last updated: 2017-12-05Bibliographically approved
4. Optimal switching problems under partial information
Open this publication in new window or tab >>Optimal switching problems under partial information
2015 (English)In: Monte Carlo Methods and Applications, ISSN 1569-3961, Vol. 21, no 2, p. 91-120Article in journal (Refereed) Published
##### Abstract [en]

In this paper we formulate and study an optimal switching problem under partial information. In our model the agent/manager/investor attempts to maximize the expected reward by switching between different states/investments. However, he is not fully aware of his environment and only an observation process, which contains partial information about the environment/underlying, is accessible. It is based on the partial information carried by this observation process that all decisions must be made. We propose a probabilistic numerical algorithm based on dynamic programming, regression Monte Carlo methods, and stochastic filtering theory to compute the value function. In this paper, the approximation of the value function and the corresponding convergence result are obtained when the underlying and observation processes satisfy the linear Kalman-Bucy setting. A numerical example is included to show some specifc features of partial information.

Mathematics
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-219911 (URN)10.1515/mcma-2014-0013 (DOI)
Available from: 2014-03-06 Created: 2014-03-06 Last updated: 2016-04-13Bibliographically approved
5. A Brownian optimal switching problem under incomplete information
Open this publication in new window or tab >>A Brownian optimal switching problem under incomplete information
##### Abstract [en]

In this paper we consider an incomplete information optimal switching problem in which the manager can only make use of noisy observations of the underlying Brownian motion $\{W_t\}_{t \geq 0}$. The manager can, at a fixed cost, switch between having the production facility open or closed and must find the optimal management strategy using only the noisy observations. Using the theory of linear stochastic filtering, we reduce the problem to a full information setting, show that the value function is non-decreasing with the amount of information available, and that the value function of the incomplete information problem converges to the value function of the corresponding full information problem as the noise in the observed process tends to $0$.

##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-242556 (URN)
Available from: 2015-01-27 Created: 2015-01-27 Last updated: 2015-03-17

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Cite
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