Browsing by Subject "Asymptotics"

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  • Pyrylä, Atte (Helsingin yliopisto, 2020)
    In this thesis we will look at the asymptotic approach to modeling randomly weighted heavy-tailed random variables and their sums. The heavy-tailed distributions, named after the defining property of having more probability mass in the tail than any exponential distribution and thereby being heavy, are essentially a way to have a large tail risk present in a model in a realistic manner. The weighted sums of random variables are a versatile basic structure that can be adapted to model anything from claims over time to the returns of a portfolio, while giving the primary random variables heavy-tails is a great way to integrate extremal events into the models. The methodology introduced in this thesis offers an alternative to some of the prevailing and traditional approaches in risk modeling. Our main result that we will cover in detail, originates from "Randomly weighted sums of subexponential random variables" by Tang and Yuan (2014), it draws an asymptotic connection between the tails of randomly weighted heavy-tailed random variables and the tails of their sums, explicitly stating how the various tail probabilities relate to each other, in effect extending the idea that for the sums of heavy-tailed random variables large total claims originate from a single source instead of being accumulated from a bunch of smaller claims. A great merit of these results is how the random weights are allowed for the most part lack an upper bound, as well as, be arbitrarily dependent on each other. As for the applications we will first look at an explicit estimation method for computing extreme quantiles of a loss distributions yielding values for a common risk measure known as Value-at-Risk. The methodology used is something that can easily be adapted to a setting with similar preexisting knowledge, thereby demonstrating a straightforward way of applying the results. We then move on to examine the ruin problem of an insurance company, developing a setting and some conditions that can be imposed on the structures to permit an application of our main results to yield an asymptotic estimate for the ruin probability. Additionally, to be more realistic, we introduce the approach of crude asymptotics that requires little less to be known of the primary random variables, we formulate a result similar in fashion to our main result, and proceed to prove it.
  • Bakharev, F. L.; Cardone, G.; Nazarov, S. A.; Taskinen, J. (2017)
    We give an example of a scalar second order differential operator in the plane with double periodic coefficients and describe its modification, which causes an additional spectral band in the essential spectrum. The modified operator is obtained by applying to the coefficients a mirror reflection with respect to a vertical or horizontal line. This change gives rise to Rayleigh type waves localized near the line. The results are proven using asymptotic analysis, and they are based on high contrast of the coefficient functions.
  • Tolvanen, Tuuli (Helsingin yliopisto, 2020)
    The objective of this thesis is to introduce the concept of compound variables and explain their use in one application specifically, as the total claim amount of an insurance company can be viewed as a compound variable. We study both the average behaviour as well as the tail behaviour of compound variables. Before delving into the results concerning the tails of compound variables, we aim to present an overview about the general theory and treat the average behaviour of compound variables first. We familiarize the reader with rudimentary concepts such as moment and cumulant generating functions. Along the way, the reader will also gain an understanding of both mixed variables as well as compound mixed variables. We state and prove some fundamental results concerning the expectation, variance and moment generating functions of compound variables. When the concept of compound variable is used to interpret the total claim amount, we also find the number of claims to be of interest. Since it is a random variable, we wish to be able to model it somehow. In the case of a general compound variable, the number of claims simply corresponds to the number of summands in the variable. We consider compound Poisson variables as a special case of compound variables. The reason for this is that if the counting variable or the number of claims variable is Poisson distributed, then the compound variable is a compound Poisson random variable. We also enhance the modelling of the number of claims by presenting mixing variables into the model. As a more general version for determining the expectation of a random sum we prove Wald's identity. It does not assume the independence of the counting variable and the increments in the same way we do in the definition of a compound variable. Towards the end, we shift the focus from general theory and average behaviour to tail behaviour of compound variables. We introduce the reader to the necessary classes of heavy-tailed and subexponential distributions to be able to formulate a few results that give an asymptotically equivalent approximation for the tail function of the compound variable. We prove the result for the case of the negative expectation of the increments (summands). We also present results for the case of non-negative expectation of the increments. Such a situation would be of interest in particular for total claim amounts, if we assume the claims being non-negative random variables.