| 1. |
- Engström, Christopher, et al.
(författare)
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Generalisation of the Damping Factor in PageRank for Weighted Networks
- 2014
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Ingår i: Modern Problems in Insurance Mathematics. - Cham : Springer International Publishing. - 9783319066523 - 9783319066530 ; , s. 313-333
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Bokkapitel (refereegranskat)abstract
- In this article we will look at the PageRank algorithm used to rank nodes in a network. While the method was originally used by Brin and Page to rank home pages in order of “importance”, since then many similar methods have been used for other networks such as financial or P2P networks. We will work with a non-normalised version of the usual PageRank definition which we will then generalise to enable better options, such as adapting the method or allowing more types of data. We will show what kind of effects the new options creates using examples as well as giving some thoughts on what it can be used for. We will also take a brief look at how adding new connections between otherwise unconnected networks can change the ranking.
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| 2. |
- Lundengård, Karl, 1987-, et al.
(författare)
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Asian Options, Jump-Diffusion Processes on a Lattice, and Vandermonde Matrices
- 2014
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Ingår i: Modern Problems in Insurance Mathematics. - Cham : Springer International Publishing. - 9783319066523 - 9783319066530 ; , s. 335-363
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Bokkapitel (refereegranskat)abstract
- Asian options are options whose value depends on the average asset price during its lifetime. They are useful because they are less subject to price manipulations. We consider Asian option pricing on a lattice where the underlying asset follows the Merton–Bates jump-diffusion model. We describe the construction of the lattice using the moment matching technique which results in an equation system described by a Vandermonde matrix. Using some properties of Vandermonde matrices we calculate the jump probabilities of the resulting system. Some conditions on the possible jump sizes in the lattice are also given.
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| 3. |
- Malmberg, Hannes, et al.
(författare)
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Probabilistic choice with an infinite set of options : An Approach Based on Random Sup Measures
- 2014
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Ingår i: Modern Problems in Insurance Mathematics. - London : Springer. - 9783319066523 - 9783319066530 ; , s. 291-312
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Bokkapitel (refereegranskat)abstract
- This chapter deals with probabilistic choice when the number of options is infinite. The choice space is a compact set S⊆R k and we model choice over S as a limit of choices over triangular sequences {x n1 ,…,x nn }⊆S as n→∞ . We employ the theory of random sup measures and show that in the limit when n→∞ , people behave as though they are maximising over a random sup measure. Thus, our results complement Resnick and Roy’s [18] theory of probabilistic choice over infinite sets. They define choice as a maximisation over a stochastic process on S with upper semi-continuous (usc) paths. This connects to our model as their random usc function can be defined as a sup-derivative of a random sup measure, and their maximisation problem can be transformed into a maximisation problem over this random sup measure. One difference remains though: with our model the limiting random sup measures are independently scattered, without usc paths. A benefit of our model is that we provide a way of connecting the stochastic process in their model with finite case distributional assumptions, which are easier to interpret. In particular, when choices are valued additively with one deterministic and one random part, we explore the importance of the tail behaviour of the random part, and show that the exponential distribution is an important boundary case between heavy-tailed and light-tailed distributions.
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| 4. |
- Ni, Ying
(författare)
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Exponential asymptotical expansions for ruin probability in a classical risk process with non-polynomial perturbations
- 2014
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Ingår i: Modern Problems in Insurance Mathematics. - Cham : Springer. - 9783319066523 - 9783319066530 ; , s. 69-94
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Bokkapitel (refereegranskat)abstract
- In this paper we investigate the asymptotical behaviour of ruin probability in a classical compound Poisson risk process associated with perturbations in the claim size distributions and/or other parameters of the risk process. The novelty of this study is that we consider non-polynomial perturbations which include the polynomial perturbations as particular cases. The aim of the study is to develop exponential asymptotical expansions for the ruin probability as the initial capital goesto infinity and the perturbation parameter goes to zero, simultaneously but in a balanced manner. Numerical examples of risk processes with such type of perturbations are also given for illustrative purposes.
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| 5. |
- Petersson, Mikael, 1984-
(författare)
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Asymptotics of Ruin Probabilities for Perturbed Discrete Time Risk Processes
- 2014
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Ingår i: Modern Problems in Insurance Mathematics. - Cham : Springer. - 9783319066523 - 9783319066530 ; , s. 95-112
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Bokkapitel (refereegranskat)abstract
- We consider the problem of approximating the infinite time horizon ruin probabilities for discrete time risk processes. The approach is based on asymptotic results for non-linearly perturbed discrete time renewal equations. Under some moment conditions on the claim distributions, the approximations take the form of exponential asymptotic expansions with respect to the perturbation parameter. We show explicitly how the coefficients of these expansions can be computed as functions of the coefficients of the expansions of local characteristics for perturbed risk processes.
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| 6. |
- Silvestrov, Dmitrii, 1947-
(författare)
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Improved Asymptotics for Ruin Probabilities
- 2014
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Ingår i: Modern Problems in Insurance Mathematics. - Cham : Springer. - 9783319066523 - 9783319066530 ; , s. 37-68
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Bokkapitel (refereegranskat)abstract
- This paper presents a survey of results on improved asymptotics for ruin probabilities in the Cramér-Lundberg, diffusion, and stable approximations of ruin probabilities for perturbed risk processes, obtained by the author and his collaborators. These results are: exponential asymptotic expansions for ruin probabilities in the Cramér-Lundberg anddiffusion approximations of ruin probabilities; necessary and sufficient conditions for convergence of ruin probabilities in the model of diffusion and stable approximations; and explicit exponential rates of convergence in the Cramér-Lundberg approximation for ruin probabilities for reinsurance risk processes.
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| 7. |
- Ekheden, Erland, 1982-, et al.
(författare)
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Analysis of the Stochasticity of Mortality Using Variance Decomposition
- 2014
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Ingår i: Modern Problems in Insurance Mathematics. - Cham : Springer Publishing Company. - 9783319066523 ; , s. 199-222
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Bokkapitel (refereegranskat)abstract
- We analyse the stochasticity in mortality data from the USA, the UK and Sweden, and in particular to which extent mortality rates are explained by systematic variation, due to various risk factors, and random noise. We formalise this in terms of a mixed regression model with a logistic link function, and decomposethe variance of the observations into three parts: binomial risk, the variance due to random mortality variation in a finite population, systematic risk explained by the covariates and unexplained systematic risk, variance that comes from real changes in mortality rates, not captured by the covariates. The fraction of unexplained variance caused by binomial risk provides a limit in terms of the resolution that can be achieved by a model. This can be used as a model selection tool for selecting the number of covariates and regression parameters of the deterministic part of the regression function, and for testing whether unexplained systematic variation should be explicitly modelled or not. We use a two-factor model with ageand calendar year as covariates, and perform the variance decomposition for a simple model with a linear time trend on the logit scale. The population size turns out to be crucial, and for Swedish data, the simple model works surprisingly well, leaving only a small fraction of unexplained systematic risk, whereas for the UK and the USA, the amount of unexplained systematic risk is larger, so that more elaborate models might work better.
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