Krylov integrators for Hamiltonian systems

We consider Arnoldi-like processes to obtain symplectic subspaces for Hamiltonian systems. Large dimensional systems are locally approximated by ones living in low dimensional subspaces, and we especially consider Krylov subspaces and some of their extensions. These subspaces can be utilized in two ways: by solving numerically local small dimensional systems and then mapping back to the large dimension, or by using them for the approximation of necessary functions in exponential integrators applied to large dimensional systems. In the former case one can expect an excellent energy preservation and in the latter this is so for linear systems. We consider second order exponential integrators which solve linear systems exactly and for which these two approaches are in a certain sense equivalent. We also consider the time symmetry preservation properties of the integrators. In numerical experiments these methods combined with symplectic subspaces show promising behavior also when applied to nonlinear Hamiltonian problems.Peer reviewe

On the stability of two-chunk file-sharing systems

We consider five different peer-to-peer file sharing systems with two chunks, with the aim of finding chunk selection algorithms that have provably stable performance with any input rate and assuming non-altruistic peers who leave the system immediately after downloading the second chunk. We show that many algorithms that first looked promising lead to unstable or oscillating behavior. However, we end up with a system with desirable properties. Most of our rigorous results concern the corresponding deterministic large system limits, but in two simplest cases we provide proofs for the stochastic systems also.Comment: 19 pages, 7 figure

Quadratic backward stochastic differential equations

Tässä tutkielmassa analysoimme takaperoisia stokastisia differentiaaliyhtälöitä. Aloitamme esittelemällä stokastiset prosessit, Brownin liikkeen, stokastiset integraalit ja Itôn kaavan. Tämän jälkeen siirrymme tarkastelemaan stokastisia differentiaaliyhtälöitä ja lopulta takaperoisia stokastisia differentiaaliyhtälöitä. Tämän tutkielman pääaiheena on takaperoiset stokastiset differentiaaliyhtälöt kvadraattisilla oletuksilla. Näillä oletuksilla todistamme olemassaoloteoreeman ja tietyt säännöllisyysehdot takaperoisen stokastisen differentiaaliyhtälön ratkaisulle.In this thesis, we analyze backward stochastic differential equations. We begin by introducing stochastic processes, Brownian motion, stochastic integrals, and Itô's formula. After that, we move on to consider stochastic differential equations and finally backward stochastic differential equations. The main topic of this thesis are backward stochastic differential equations under quadratic assumptions. Under these assumptions we prove an existence theorem and certain regularity conditions for the solution of the backward stochastic differential equation

Substituutiojouston estimointi : simulointihavaintoja oletusten vaikutuksista tuloksiin

Tässä työssä tutkitaan substituutiojouston estimointia. Substituutiojousto on tunnuslu-ku, joka kuvaa tuotantopanoksen korvaamisen helppoutta jollakin toisella tuotanto-panoksella. Substituutiojousto voidaan estimoida jonkin tuotantofunktion parametrien avulla. Tässä työssä tarkastellaan lähemmin CES- ja Translog-tuotantofunktioita. Substituutiojousto on määritelty alun perin kahden tuotantopanoksen tuotantofunk-tioille. Määritelmä voidaan kuitenkin yleistää usean tuotantopanoksen tuotantofunkti-oille. Esimerkkejä substituutiojouston yleistyksistä usean tuotantopanoksen tilanteessa ovat Allen-substituutiojousto, Morishima-substituutiojousto ja varjosubstituutiojousto. Sekä substituutiojouston että sen yleistysten estimointi vaatii useita taustaoletuksia. Tässä työssä hahmotellaan erästä oletuskehikkoa, jossa yritysten tuotantofunktiot ovat CES-muotoisia. Oletuksia tarkastellaan myös esittämällä simulointiesimerkkejä, joissa oletukset eivät toteudu kaikilta osin. Simuloinnin avulla on mahdollista tarkastella, millä tavalla oletusten vapauttaminen vaikuttaa estimoinnin onnistumiseen

Detection Of Generic One Parameter Bifurcations Of Hamiltonian Equilibria

. In this paper several possibilities are considered for constructing test functions which reveal generic one parameter bifurcations of equilibria of Hamiltonian systems. These are alternatives to computing all the eigenvalues of the linearized system. Because of the Hamiltonian structure, most test functions that are used for general systems do not work here. Typically a test function that should change sign at a bifurcation point has a multiple zero and no sign change there for Hamiltonian systems. Methods based on characteristic polynomials and bialternate products of matrices are discussed in detail. 1. Introduction Consider a Hamiltonian system depending on a real parameter ff and defined by a smooth function h : R 2n \Theta R ! R x = Jh x (x; ff) ; where J = \Theta 0 I \GammaI 0 : Let x ff be an equilibrium depending on ff h x (x ff ; ff) = 0 and denote by H(x ff ; ff) = Jh xx (x ff ; ff) the matrix of the linearized system and by oe(H(x ff ; ff)) its spectrum. A lo..

Risky debt, bad bank and government

The purpose of this paper is to put forward a valuation framework for interest rate sensitive claims. We concentrate on secured loans. The value of the secured loan depends upon the coupon rate, the maturity, the term structure of interest rates and the value of the collateral as well as the probability of default. We follow Schwartz and Torous (1992) and assume that borrower's conditional probability of default is given by a hazards function. Furthermore, we value guarantees, junior secured debt and unemployment insurance