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Four applications of stochastic processes: Contagious disease, credit risk, gambling and bond portfolios
Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
2011 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
Abstract [en]

This thesis consists of four papers on applications of stochastic processes.

In Paper I we study an open population SIS (Susceptible - Infective - Susceptible) stochastic epidemic model from the time of introduction of the disease, through a possible outbreak and to extinction. The analysis uses coupling arguments and diffusion approximations.

In Paper II we propose a model describing an economy where companies may default due to contagion. The features of the model are analyzed using diffusion approximations. We show that the model can reproduce oscillations in the default rates similar to what has been observed empirically.

In Paper III we consider the problem of finding an optimal betting strategy for a house-banked casino card game that is played for several coups before reshuffling. A limit result for the return process is found and the optimal card counting strategy is derived. This continuous time strategy is shown to be a natural generalization of the discrete time strategy where the so called effects of removals are replaced by the infinitesimal generator of the card process.

In Paper IV we study interest rate models where the term structure is given by an affine relation and in particular where the driving stochastic processes are so-called generalised Ornstein-Uhlenbeck processes. We show that the return and variance of a portfolio of bonds which are continuously rolled over, also called rolling horizon bonds, can be expressed using the cumulant generating functions of the background driving Lévy processes associated with the OU processes. We also show that if the short rate, in a risk-neutral setting, is given by a linear combination of generalised OU processes, the implied term structure can be expressed in terms of the cumulant generating functions.

sted, utgiver, år, opplag, sider
Stockholm: Department of Mathematics, Stockholm University , 2011. , s. 27
Emneord [en]
Stochastic processes
HSV kategori
Forskningsprogram
matematisk statistik
Identifikatorer
URN: urn:nbn:se:su:diva-57949ISBN: 978-91-7447-318-6 (tryckt)OAI: oai:DiVA.org:su-57949DiVA, id: diva2:418801
Disputas
2011-08-26, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (engelsk)
Opponent
Veileder
Merknad
At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Submitted. Paper 3: Submitted. Paper 4: Manuscript.Tilgjengelig fra: 2011-06-06 Laget: 2011-05-24 Sist oppdatert: 2011-06-14bibliografisk kontrollert
Delarbeid
1. A stochastic SIS epidemic with demography: initial stages and time to extinction
Åpne denne publikasjonen i ny fane eller vindu >>A stochastic SIS epidemic with demography: initial stages and time to extinction
2011 (engelsk)Inngår i: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 62, nr 3, s. 333-348Artikkel i tidsskrift (Fagfellevurdert) Published
Abstract [en]

We study an open population stochastic epidemic model from the time of introduction of the disease, through a possible outbreak and to extinction. The model describes an SIS (susceptible–infective–susceptible) epidemic where all individuals, including infectious ones, reproduce at a given rate. An approximate expression for the outbreak probability is derived using a coupling argument. Further, we analyse the behaviour of the model close to quasi-stationarity, and the time to disease extinction, with the aid of a diffusion approximation. In this situation the number of susceptibles and infectives behaves as an Ornstein–Uhlenbeck process, centred around the stationary point, for an exponentially distributed time before going extinct.

Emneord
Stochastic epidemic model, Quasi stationarity, SIS model, Coupling, Ornstein–Uhlenbeck, Diffusion approximation, Outbreak probability
HSV kategori
Forskningsprogram
matematisk statistik
Identifikatorer
urn:nbn:se:su:diva-57948 (URN)10.1007/s00285-010-0336-x (DOI)000287250300002 ()
Tilgjengelig fra: 2011-05-24 Laget: 2011-05-24 Sist oppdatert: 2017-12-11bibliografisk kontrollert
2. Optimal bond portfolios with fixed time to maturity
Åpne denne publikasjonen i ny fane eller vindu >>Optimal bond portfolios with fixed time to maturity
(engelsk)Artikkel i tidsskrift (Fagfellevurdert) Submitted
Abstract [en]

We study interest rate models where the term structure is given by an affine relation and in particular where the driving stochastic processes are so-called generalised Ornstein-Uhlenbeck processes.

For many institutional investors it is natural to consider investment in bonds where the time to maturity of the bonds in the portfolio is kept fixed over time. We show that the return and variance of such a portfolio of bonds which are continuously rolled over, also called rolling horizon bonds, can be expressed using the cumulant generating functions of the background driving L´evy processes associated with the OU processes. This allows us to calculate the efficient mean-variance portfolio. We exemplify the results by a case study on U.S. Treasury bonds.

We also show that if the short rate, in a risk-neutral setting, is given by a linear combination of generalised OU processes, the implied term structure can be expressed in terms of the cumulant generating functions. This makes it possible to quite easily see what kind of term structures can be generated with a particular short rate dynamics.

Emneord
Interest rate models, rolling horizon bonds, generalised Ornstein-Uhlenbeck processes, affine term structure, mean variance portfolio
HSV kategori
Forskningsprogram
matematisk statistik
Identifikatorer
urn:nbn:se:su:diva-57751 (URN)
Tilgjengelig fra: 2011-05-18 Laget: 2011-05-18 Sist oppdatert: 2011-06-14bibliografisk kontrollert
3. Credit default model for a dynamically changing economy
Åpne denne publikasjonen i ny fane eller vindu >>Credit default model for a dynamically changing economy
(engelsk)Artikkel i tidsskrift (Fagfellevurdert) Submitted
HSV kategori
Forskningsprogram
matematisk statistik
Identifikatorer
urn:nbn:se:su:diva-57752 (URN)
Tilgjengelig fra: 2011-05-18 Laget: 2011-05-18 Sist oppdatert: 2011-06-14bibliografisk kontrollert
4. Card counting in continuous time
Åpne denne publikasjonen i ny fane eller vindu >>Card counting in continuous time
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
HSV kategori
Forskningsprogram
matematisk statistik
Identifikatorer
urn:nbn:se:su:diva-57750 (URN)
Tilgjengelig fra: 2011-05-18 Laget: 2011-05-18 Sist oppdatert: 2011-06-14bibliografisk kontrollert

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