| 1. |
- Andersson, Patrik, 1981-
(author)
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Four applications of stochastic processes : Contagious disease, credit risk, gambling and bond portfolios
- 2011
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Doctoral thesis (other academic/artistic)abstract
- 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.
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| 2. |
- Andersson, Patrik, et al.
(author)
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Optimal bond portfolios with fixed time to maturity
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Journal article (peer-reviewed)abstract
- 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.
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| 3. |
- Jagers, Peter, 1941, et al.
(author)
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General branching processes conditioned on extinction are still branching processes
- 2008
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In: Electronic Communications in Probability. - 1083-589X. ; 13, s. 540-547
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Journal article (peer-reviewed)abstract
- It is well known that a simple, supercritical Bienaymé-Galton-Watson process turns into a subcritical such process, if conditioned to die out. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction.
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| 4. |
- Lagerås, Andreas Nordvall
(author)
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A population model for $\Lambda$-coalescents with neutral mutations
- 2007
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In: Electronic Communications in Probability. - 1083-589X. ; 12, s. 9-20
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Journal article (peer-reviewed)abstract
- Bertoin and Le Gall (2003) introduced a certain probability measure valued Markov process that describes the evolution of a population, such that a sample from this population would exhibit a genealogy given by the so-called $\Lambda$-coalescent, or coalescent with multiple collisions, introduced independently by Pitman (1999) and Sagitov (1999). We show how this process can be extended to the case where lineages can experience mutations. Regenerative compositions enter naturally into this model, which is somewhat surprising, considering a negative result by Möhle (2007).
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| 5. |
- Nordvall Lagerås, Andreas
(author)
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Copulas for Markovian dependence
- 2008
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Reports (other academic/artistic)abstract
- Copulas have been popular to model dependence for multivariate distributions, but have not been used much in modelling temporal dependence of univariate time series. This paper shows some difficulties with using copulas even for Markov processes: some tractable copulas such as mixtures between copulas of complete co- and countermonotonicity and independence (Fréchet copulas) are shown to imply quite a restricted type of Markov process, and Archimedean copulas are shown to be incompatible with Markov chains. We also investigate Markov chains that are spreadable, or equivalently, conditionally i.i.d.
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| 6. |
- Nordvall Lagerås, Andreas, et al.
(author)
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How do Interactions Influence Formation of Social Networks? : A General Microfounded Explanation
- 2009
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Reports (other academic/artistic)abstract
- This paper investigates the strategic interaction effects that precede network formation. We find that for a general class of payoff functions which, among other things, feature strict supermodularity, the degree of a node is a sufficient statistic for the action it undertakes. Dynamically, we construct a general model where each period consists of two stages: first, a game on the given network is played and second, a link is either created or severed. It turns out that the payoff functions we consider give absolute convergence to the absorbing class of networks called nested split graphs. These networks do not only possess mathematically tractable characteristics, but we can also interpret real-world networks as perturbed nested split graphs. The general framework provided here can be applied to more or less complex models of network formation.
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| 7. |
- Nordvall Lagerås, Andreas, 1981-
(author)
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Markov Chains, Renewal, Branching and Coalescent Processes : Four Topics in Probability Theory
- 2007
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Doctoral thesis (other academic/artistic)abstract
- This thesis consists of four papers.In paper 1, we prove central limit theorems for Markov chains under (local) contraction conditions. As a corollary we obtain a central limit theorem for Markov chains associated with iterated function systems with contractive maps and place-dependent Dini-continuous probabilities.In paper 2, properties of inverse subordinators are investigated, in particular similarities with renewal processes. The main tool is a theorem on processes that are both renewal and Cox processes.In paper 3, distributional properties of supercritical and especially immortal branching processes are derived. The marginal distributions of immortal branching processes are found to be compound geometric.In paper 4, a description of a dynamic population model is presented, such that samples from the population have genealogies as given by a Lambda-coalescent with mutations. Depending on whether the sample is grouped according to litters or families, the sampling distribution is either regenerative or non-regenerative.
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| 8. |
- Nordvall Lagerås, Andreas, et al.
(author)
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Reduced branching processes with very heavy tails
- 2008
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In: Journal of Applied Probability. - 0021-9002. ; 45:1, s. 190-200
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Journal article (peer-reviewed)abstract
- The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter α∈(0,1]. We turn to the case of very heavy tailed reproduction distribution α=0 assuming Zubkov’s regularitycondition with parameter β∈(0,∞). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on non-extinction during a long time interval.
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