# Browsing by Subject "Markov chains"

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• (Helsingin yliopisto, 2021)
Bonus-malus systems are used globally to determine insurance premiums of motor liability policy-holders by observing past accident behavior. In these systems, policy-holders move between classes that represent different premiums. The number of accidents is used as an indicator of driving skills or risk. The aim of bonus-malus systems is to assign premiums that correspond to risks by increasing premiums of policy-holders that have reported accidents and awarding discounts to those who have not. Many types of bonus-malus systems are used and there is no consensus about what the optimal system looks like. Different tools can be utilized to measure the optimality, which is defined differently according to each tool. The purpose of this thesis is to examine one of these tools, elasticity. Elasticity aims to evaluate how well a given bonus-malus system achieves its goal of assigning premiums fairly according to the policy-holders’ risks by measuring the response of the premiums to changes in the number of accidents. Bonus-malus systems can be mathematically modeled using stochastic processes called Markov chains, and accident behavior can be modeled using Poisson distributions. These two concepts of probability theory and their properties are introduced and applied to bonus-malus systems in the beginning of this thesis. Two types of elasticities are then discussed. Asymptotic elasticity is defined using Markov chain properties, while transient elasticity is based on a concept called the discounted expectation of payments. It is shown how elasticity can be interpreted as a measure of optimality. We will observe that it is typically impossible to have an optimal bonus-malus system for all policy-holders when optimality is measured using elasticity. Some policy-holders will inevitably subsidize other policy-holders by paying premiums that are unfairly large. More specifically, it will be shown that, for bonus-malus systems with certain elasticity values, lower-risk policy-holders will subsidize the higher-risk ones. Lastly, a method is devised to calculate the elasticity of a given bonus-malus system using programming language R. This method is then used to find the elasticities of five Finnish bonus-malus systems in order to evaluate and compare them.
• (2010)
Department of Computer Science Series of Publications C Report C-2010-39
We consider approximate policy evaluation for finite state and action Markov decision processes (MDP) with the least squares temporal difference algorithm, LSTD(λ), in an explorationenhanced off-policy learning context. We establish for the discounted cost criterion that the off-policy LSTD(λ) converges almost surely under mild, minimal conditions. We also analyze other convergence and boundedness properties of the iterates involved in the algorithm. Our analysis draws on theories of both finite space Markov chains and weak Feller Markov chains on topological spaces. Our results can be applied to other temporal difference algorithms and MDP models. As examples, we give a convergence analysis of an off-policy TD(λ) algorithm and extensions to MDP with compact action and state spaces.
• (2020)
Livestock production in developed countries has undergone profound changes in recent decades and this development seems to continue apace. One consequence is that manure is being - and will be - produced on fewer but larger farms. Data on the bulk of manure nutrients from each country are published by Eurostat, but it is not known how manure is distributed across farms of different sizes. This study 1) puts forward an estimate of the distribution of main manure nutrients between farms of different sizes, 2) estimates how this distribution will change in the near future and 3) discusses the land use effects of this development. Results suggest that by the year 2030 farms housing > 500 livestock units will produce more than two-thirds of all manure phosphorus, whereas the proportion in 2010 was one-third. With the Nitrates Directive limiting the use of organic nitrate of manure, growing farms need to acquire, or conclude contracts for the use of, 4.9 million hectares from exiting farms or the open market in order to comply with manure spreading requirements. This shift will involve 64% of the total spreading area of 2010 and 15% of the total utilized agricultural area of the regions studied. In light of these predictions, international nutrient policies should consider the evolution of farm structure in general and manure phosphorus agglomeration in particular. Also salient is improved co-operation beyond the single farm level to ensure the functionality of crop-livestock systems.
• (2020)
Smooth transition autoregressive models are widely used to capture nonlinearities in univariate and multivariate time series. Existence of stationary solution is typically assumed, implicitly or explicitly. In this paper, we describe conditions for stationarity and ergodicity of vector STAR models. The key condition is that the joint spectral radius of certain matrices is below 1. It is not sufficient to assume that separate spectral radii are below 1. Our result allows to use recently introduced toolboxes from computational mathematics to verify the stationarity and ergodicity of vector STAR models.