Browsing by Subject "Asymptotics"

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.

In this thesis we prove a short time asymptotic formula for a path integral solution to the Fokker Planck heat equation on a Riemannian manifold. The result is inspired by multiple developments regarding the theory of stochastic differential equations on a Riemannian manifold. Most notably the papers by Itô (1962) which describes the stochastic differential equation, Graham (1985) which describes the probabilistic time development of the stochastic differential equation via a path integral and Anderson and Driver (1999) which proves that Graham's path integral converges to the correct notion of probability. The starting point of the thesis is a paper by R. Graham (1985) where a path integral formula for the solution of the heat equation on a Riemannian manifold is given in terms of a stochastic differential equation in Itô sense. The path integral formula contains an integrand of the exponential of an action function. The action function is defined by the given stochastic differential equation and additional integration variables denoted as the momenta of the paths appearing in the integral. The path integral is defined as the time continuum limit of a product of integrals on a discrete time lattice. The result obtained in this thesis is proven by considering the saddle point approximation of the action appearing in the finite version of Graham's path integral formula. The saddle point approximation gives a power series approximation of the action up to the second order by taking the first and second variations of the action and setting the first variation as zero. We say that the saddle point approximation is evaluated along the critical path of the action which is defined by taking the first variation as zero. The second variation of the action is called the Hessian matrix. With the saddle point approximation of the action, we obtain an asymptotic formula of the path integral which contains the exponential of the action evaluated along the critical path and the determinant of the Hessian. The main part of the proof is the evaluation of the determinant of the Hessian in the continuum limit. To this end we prove a finite dimensional version of a theorem due to R. Forman (1987), called Forman's theorem, which allows us to calculate the ratio of determinants of the Hessian parametrized by two different boundary conditions as a ratio of finite dimensional determinants. We then show that in the continuum limit the ratio of determinants of the Hessian can be written in terms of the Jacobi ow. With the Forman's theorem we then get the short time asymptotic formula by evaluating the determinants on a short time interval.