Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions

The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory.Comment: 22 pages, 4 figures; deleted a section; shortened the presentation of results; corrected typo

Evolution of continental crust in the Proterozoic : growth and reworking in orogenic systems

To understand the growth of continental crust, the balance between juvenile mantle derived extraction, infracrustal reworking and crustal recycling, needs to be estimated. Since the beginning of the century, the use of coupled in situ zircon U–Pb, Lu–Hf and O isotope analyses as a tool to address these questions have increased exponentially. Numerous compilations of ever growing datasets have been presented, leading to new, and sometimes contrasting models of continental growth. Many of theses models, however, suffer from a number of assumptions, including a mantle reservoir that has been homogeneously and linearly depleted since the Hadean. Further, the use of (mainly) detrital zircon, taken out of their geological context, and the application of their depleted model-ages clearly hamper the validity of these models. To accurately address the question regarding continental crustal growth using combined zircon U–Pb-Lu–Hf(-O) isotope data, one needs to have contextual control and minimise the uncertainties of the applied models. In papers included in this thesis such an approach has been used on three different Palaeo- to Meso-Proterozoic orogenic belts; in Fennoscandia, in North American Grenville and in the Birimian terrane of the West African craton. The eastern part of the Sveconorwegian Province, located in the southwestern part of the Fennoscandian Shield, is made up of granitiod rocks that were emplaced through sequential tapping of a reservoir that formed through mixing between a 2.1–1.9 Ga juvenile component and Archaean crust. Between 1.7 and 1.4 Ga the continental crust of the Eastern Segment was reworked with little or no generation of new crust. Further to the west, in the Idefjorden terrane of the Sveconorwegian Province, 1.65 to 1.33 Ga rocks have isotopic signatures that indicate reworking of older continental crust, including sediments. However, overall the isotopic signatures in the Idefjorden terrane indicate an increase in juvenile material with time, consistent with development of an extensional back-arc rift geotectonic setting, accommodating deposition of the local metasedimentary basin, Stora Le-Marstrand. Isotope data from rocks within the Grenville orogen in subsurface Ohio suggest a common c. 1.65 Ga juvenile source to a majority of the sampled bedrock. Emplacement of this juvenile crustal contribution was followed by sequential reworking of that reservoir with little or no additional contribution to the source. The c. 2.31–2.06 Ga Birimian terrane in Ghana, West African craton, is a commonly cited example of plume initiated crustal growth, that is known to have largely juvenile signatures. However, we can show that reworked Archaean crust contribute in a much larger extent than previously known, once again highlighting the importance of infracrustal reworking during emplacement of continental crust. Further, the emplacement of felsic rocks during the Eoeburnean pre-dates suggested plume related rocks, contradicting a suggested plume initiated crustal growth. Collectively, these studies highlight the importance of infracrustal reworking in Palaeo- to Meso-Proterozoic accretionary orogens. These studies also provide good examples of combined zircon U–Pb-Lu–Hf-(O) isotope analyses on rocks and rock suites with known affinity where the validity of chosen models can be justified

Approximation of SPDE covariance operators by finite elements: A semigroup approach

The problem of approximating the covariance operator of the mild solution to a linear stochastic partial differential equation is considered. An integral equation involving the semigroup of the mild solution is derived and a general error decomposition is proven. This formula is applied to approximations of the covariance operator of a stochastic advection-diffusion equation and a stochastic wave equation, both on bounded domains. The approximations are based on finite element discretizations in space and rational approximations of the exponential function in time. Convergence rates are derived in the trace class and Hilbert--Schmidt norms with numerical simulations illustrating the results.Comment: 31 pages, 8 figures; added derivation of trace error formula; to appear in IMA Journal of Numerical Analysi

Hilbert--Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations

Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert--Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates, which couple the regularity of the driving noise with the properties of the differential operator, have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert--Schmidt embeddings of Sobolev spaces. Both non-homogeneous and homogeneous kernels are considered. In the latter case, results in a general Schatten class norm are also provided. Important examples of homogeneous kernels covered by the results of the paper include the class of Mat\'ern kernels.Comment: 26 pages; paper amended to include Schatten class estimates etc.; to appear in "Stochastic Analysis and Applications

Överensstämmelse mellan verksamhet och informationssystem. - En fallstudie på handelskoncernen Haendig.

Den tekniska utvecklingen har gått fort det senaste årtiondet och informationshantering har fått en helt förändrad innebörd i dagens moderna organisationer. De organisationer som inte är beredda att följa denna utveckling löper stor risk att hamna efter eller förlora konkurrenskraft till andra aktörer på marknaden. Genom investeringar i nya informationssystem och ny informationsteknologi skaffar sig organisationen verktyg att använda till att effektivisera och rationalisera sin verksamhet. Sådana investeringar är dock förenade med höga risker och raden av företag som genomfört misslyckade implementeringsförsök är lång. Flera faktorer avgör hur ett företag lyckas tillämpa informationssystem och informationsteknologi som resurser och utnyttja dessa på ett lönsamt sätt. Syftet med vår studie är att beskriva och analysera överensstämmelsen mellan en koncerns informationssystem och dess verksamhet. En fallstudie genomfördes på handelskoncernen Haendig AB. Det empiriska materialet analyserades med hjälp av teoretiska referensramar såsom affärsmodellen och Evgenious posistioneringsmatris. Behovet av överskådlighet och flexibilitet användes som parametrar som styr hur koncernens verksamhet ser ut. En rad kritiska faktorer kunde pekas ut som avgör överensstämmelsen mellan Haendigs verksamhet och dess informationssystem. I en koncern som strävar efter hög överskådlighet och hög flexibilitet i organisationen kan en lönsam informationssystemmiljö utgöras av EAI-teknologi. Men en sådan implementering bör genomföras stegvis och under strukturerade former med tydliga och långsiktiga mål eftersom de kortsiktiga effekterna av en sådan investering är svåra att mäta. Essensen av studien är att företaget bör utgå både från verksamheten och användandet när de analyserar sina informationssystem. Det är viktigt att man har en verksamhet som överensstämmer med den systemmiljö som tillämpas (Evgenious matris). Det är dock lika viktigt att man använder sina informationssystem på ett effektivt sätt för att de ska innebära varaktiga framgångsfaktorer (affärsmodellen). Har man valt en lämplig systemmiljö för sin verksamhet, och dessutom använder sina informationsresurser på ett effektivt sätt, har man goda chanser att kunna använda denna som ett verktyg för att uppnå varaktiga konkurrensfördelar

Svenska livsmedel - varför?

The background of this research is that Swedish food products have hard to compete at the home market, depending on imported food products that increase more and more for every year that pass by. The ethic values LRF associate with Swedish food products, many customers have hard to put in relation with condition of foreign agriculture. Most customers today inquire high quality, safety and convenience in the food which they are eating. The source of their food seems to have less meaning. Many restaurants and bigger households appreciate Swedish food products and wants to buy them but price differences against foreign wares are to high. We have done oral interviews with wholesale dealers, restaurants, public purchasers and with different market organisations, there we have discussed buying strategies, the way they see the primary producer, the conscious of guests, choice and attitudes to Swedish and foreign products, distribution of power in the food market chain and to what position and suspension, they want the ranks of suppliers to have. The result shows the share of imported wares within restaurants and bigger households are at a high level and will continuing increasing. Within these sectors the final consumer in many cases put less effort in the origin of the raw source of food. What the company and the branch really appreciate from primary producers are high quality, high level of service with frequent deliveries, convenience and competitive prices on the products. To be able to put on high price values on raw food the guest must feel a immediate association with these values in for example the local environment, to do this group of costumers interested in paying a higher price. A possibility for primary producers to increase their focus on the costumers is to create and build up a network together with other suppliers to offer a wider product mix of local produced wares. There must be an entrepreneur who is head of this network, that joins all the suppliers together and at the same time can offer a comfortable concept for restaurants and bigger households, that create interest and conscious for domestic products.Bakgrunden till arbetet är de svenska livsmedelsprodukternas allt sämre förmåga att konkurrera på den inhemska marknaden samtidigt som importen av livsmedel fortsätter att öka. De värden LRF förknippar med svenska produkter har många konsumenter svårt att sätta i relation med förutsättningar i det utländska jordbruket. Dagens konsument efterfrågar hög kvalitet, säkerhet och bekvämlighet i den mat de äter och ursprunget verkar vara av mindre betydelse. Många restauranger och storhushåll värdesätter och vill köpa svenska produkter men prisskillnaderna är för stora. Vi har genomfört muntliga intervjuer med grossister, restauranger, offentliga inköpare och olika marknadsorganisationer, där vi diskuterat inköpsstrategier, syn på leverantörer, gästens medvetenhet, valet och inställningen till svenska och utländska produkter, maktfördelningen i livsmedelskedjan samt vilken roll och förhållningssätt de tycker producentledet ska inta. Resultatet visar att andelen importerade varor inom restaurang och storhushåll är betydande och fortsätter att stiga. Inom dessa sektorer lägger slutkonsumenten i många fall mindre vikt vid råvarornas ursprung. Det företagen och branschen framförallt värdesätter av leverantörerna är hög kvalite, hög servicegrad med täta leveranser, bekvämlighet och ett konkurrenskraftigt pris på varan. För att man ska kunna ålägga råvarorna med mervärden måste gästen direkt känna en samhörighet med dessa t.ex. i närmiljön, för att vissa konsumentgrupper ska vara beredda att betala ett högre pris. En möjlighet för producenten att öka sin kundfokus är att tillsammans med andra producenter bygga upp nätverk som kan erbjuda ett brett sortiment av närproducerade råvaror. I detta nätverk behövs en verksam entreprenör som knyter samman producenterna, samtidigt som han kan erbjuda ett bekvämt koncept för restauranger och storhushåll som skapar intresse och medvetenhet för de inhemska produkterna

The heat modulated infinite dimensional Heston model and its numerical approximation

The HEat modulated Infinite DImensional Heston (HEIDIH) model and its numerical approximation are introduced and analyzed. This model falls into the general framework of infinite dimensional Heston stochastic volatility models of (F.E. Benth, I.C. Simonsen '18), introduced for the pricing of forward contracts. The HEIDIH model consists of a one-dimensional stochastic advection equation coupled with a stochastic volatility process, defined as a Cholesky-type decomposition of the tensor product of a Hilbert-space valued Ornstein-Uhlenbeck process, the mild solution to the stochastic heat equation on the real half-line. The advection and heat equations are driven by independent space-time Gaussian processes which are white in time and colored in space, with the latter covariance structure expressed by two different kernels. First, a class of weight-stationary kernels are given, under which regularity results for the HEIDIH model in fractional Sobolev spaces are formulated. In particular, the class includes weighted Mat\'ern kernels. Second, numerical approximation of the model is considered. An error decomposition formula, pointwise in space and time, for a finite-difference scheme is proven. For a special case, essentially sharp convergence rates are obtained when this is combined with a fully discrete finite element approximation of the stochastic heat equation. The analysis takes into account a localization error, a pointwise-in-space finite element discretization error and an error stemming from the noise being sampled pointwise in space. The rates obtained in the analysis are higher than what would be obtained using a standard Sobolev embedding technique. Numerical simulations illustrate the results.Comment: 35 pages, 7 figure

Потенциал довузовского образования в развитии предпринимательских навыков слушателей

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