## About

112

Publications

4,036

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

1,750

Citations

Citations since 2016

Introduction

Additional affiliations

January 2001 - December 2010

**Norges Handelshøyskole**

August 1990 - December 2000

**Stord/Haugesund University College**

January 1983 - August 1990

Education

January 1983 - June 1987

## Publications

Publications (112)

Almost every supplier faces uncertain and time-varying demand. E-commerce and online
shopping have given suppliers unprecedented access to data on customers’ behavior, which sheds light on demand uncertainty. The main purpose of this research project is to provide an analytic tool for decentralized supply channel members to devise optimal long-term...

We extend the single-period decentralized channel optimization problem with stochastic demand into a multi-period, non-autonomous one and propose an efficient general solution algorithm for it. A supply channel composed of two price-setting agents in a bi-level (Stackelberg) framework has to address an uncertain dynamic demand for a perishable good...

The basic, reasonable hypothesis underlying this paper is that many individuals are comfortable with their current combination of job and residential location and have no intention of changing job location or moving from where they live. This causes autocorrelation in the time series of commuting flows and provides a rationale for introducing fixed...

Traditional cost–benefit models of investments in road infrastructure are often based on demand curves assuming a given spatial distribution of jobs and households. We first use numerical experiments based on a spatial general equilibrium model to illustrate how this potentially introduces a serious prediction bias in the willingness-to-pay for the...

This book shows how statistics can be used to gain insights relevant to economics. Combined with my newly developed system for colloborative learning it is ideal for new learning formats like flipped classrooms.

In this chapter we will look at situations that occur when we study two discrete random variables at the same time. One example of that sort could be that X is the price of a stock, while Y is the number of that stock that is sold each day. In many cases there can be more or less strong relations between the two variables, and it is important to be...

In this chapter we will look at very basic statistical concepts. The material is facilitated to make it available to a wide audience and does not require any prerequisites. For some readers this will mean that they are already familiar with the contents and it may be sufficient to browse quickly through the chapter before continuing to the next cha...

In this chapter we will try to find the values of unknown parameters in our models from observations. We know for example that there is a probability p of errors in a shipment of goods, but we do not know the exact value. This requires a line of approach which is different from what we have been using so far. In the previous chapters the distributi...

In this chapter we will study statistical testing of a hypothesis. Hypothesis testing has found widespread applications in many different fields. We can, e.g., ask if a poll confirms that voters have changed their opinion, or if a manager of mutual funds is performing systematically better than another. To test different hypotheses against each aft...

In this chapter we will introduce a general framework we will use to study statistical distributions. A statistical distribution specifies the probability of the different outcomes that can occur. We can, e.g., ask about the probability for exactly x defective items in a sample of 10 items. We let x run from 0 to 10, and compute 11 probabilities. T...

In this short chapter we will go through the basic definitions of probability. To ease the exposition, we will only discuss very simple examples. It is important to notice that the concepts we develop in this chapter are central to all thinking about statistics. Regardless of level and purpose these concepts provide tools that can be used to descri...

In this chapter we will discuss some of the most commonly used probability distributions. We will discuss basic properties of these distributions; in particular, we will provide explicit formulas for the mean and variance. We focus the binomial distribution, the hyper geometric distribution, the Poisson distribution, and the normal distribution. Th...

In this chapter we will study certain types of random selection within a uniform model. Such samples can occur when we select a representative from an audience, when we sample for errors, play lotteries, or generally when we choose between alternatives which are equally probable. The main problem is then to figure out how many combinations there ar...

When we make observations, we often encounter cases where the size of observations appears to change in some systematic way. One possible cause for this is that we have done the observations over a period of time, and as time passes the distribution of values may change. To determine if such changes are random or systematic, we can use linear regre...

In this chapter we will look at some special hypothesis tests. These tests have found widespread applications and are used by almost everyone that need to process statistical data. The tests we consider are: A test for binomial variables, the t-test for expected value, the t-test for comparison of two populations, Wilcoxons’ distribution free tests...

In most cases where we study probabilities, we have some additional information about what has happened. Information of this sort will often pose a restriction on the sample space. In this chapter we will show how to compute probabilities under such restrictions.

We consider a simple stochastic model for fluid flow in a porous medium. The permeability is modelled through a lognormal distribution with a certain correlation structure. Our equations can be interpreted in two ways. We first use ordinary products and then Wick products. The latter can be looked upon as a kind of renormalization procedure. We com...

In this paper we define bounded rationality in terms of probabilistic cost efficiency. This establishes a framework where bounded rationality can be examined rigorously by statistical methods. We apply this theory to data from laboratory experiments on the newsvendor model. We enter into a theoretical discussion of the classical pull-to-center effe...

A process of rural depopulation may be initiated by relocations or closures of basic sector
�rms. Reduced basic sector employment can cause out-migration from a region, leading
to an economic base multiplier process resulting in a new equilibrium with a lower level
of local employment and population. In addition, the out-migration can be enforced b...

We consider explicit formulae for equilibrium prices in a continuous-time vertical contracting model. A manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits. Demand is an Itô-Lévy process, and to increase realism, information is delayed. We provide complete existence and uniqueness proofs for a s...

In this paper we propose a new spatial equilibrium model, and use it to discuss issues related to rural depopulation. The discussion focuses on how investments in transport infrastructure and the spatial distribution of basic-sector jobs can promote a relatively balanced growth of peripheral and central areas of a region. Through interdependencies...

A worker might respond to an unfortunate local labour market situation by commuting or migrating to a zone with better prospects. This decision depends on the labour market accessibility of the worker’s current residential location. A spatial equilibrium model is applied to analyse the interactions between commuting, migration, firm relocation and...

In this paper we establish a link between probabilistic cost efficiency and bounded rationality in the newsvendor model. This establishes a framework where bounded rationality can be examined rigorously by statistical methods. The paper offers a relatively deep theoretical analysis of underorders/overorders in the newsvendor model. The theory is su...

Road pricing is a popular congestion reduction strategy. However, there may be wider impacts associated with a road toll. We consider a factor which is sometimes overlooked, namely that workers and firms may choose to change location in response to changes in the travel costs. A spatial equilibrium model is used to analyse suboptimalities in road p...

In this paper we consider the newsvendor model with real options under discrete demand. We consider a mixed contract where the retailer can order a combination of q units subject to the conditions in a classical newsvendor contract and Q real options on the same items. We provide a closed form solution to this mixed contract when the demand is disc...

Road tolls are a well established way of dealing with problems of congestion. Over recent years, the literature has expanded to take account of how congestion charges might interact with imperfections in other markets. In this paper, we consider the case where congestion occurs within a complex road network, with congestion on multiple links. To de...

In this paper we consider the transfer of risk in a newsvendor model with discrete demand. We view the newsvendor model as a leader/follower problem where the manufacturer (leader) decides the wholesale price and the retailer (follower) decides the quantity ordered. Taking a Pareto-optimal contract as a starting point, the manufacturer wishes to de...

In this paper, we consider Stackelberg games in a multiperiod vertical contracting model with uncertain demand. Demand has a distribution with a mean and variance that depend on the current retail price, and this dependence may vary from period to period. We focus on a class of problems in which the market has a memory-based scaling of demand, and...

Traffic congestion and the policies used to combat it have been studied extensively. One area which has received less attention is the secondary impacts of such policies. This paper uses a micro-simulation framework to study the effect on labour markets of road pricing. The key benefit of our chosen methodology is that it allows a simultaneous cons...

In Western Norway, fjords cause disconnections in the road network, necessitating the use of ferries. In several cases, ferries have been replaced by roads, often part-financed by tolls. We use data on commuting from a region with a high number of ferries, tunnels and bridges. Using a doubly-constrained gravity-based model specification, we focus o...

In this paper, we consider the newsvendor model under partial information, i.e., where the demand distribution D is partly unknown. We focus on the classical case where the retailer only knows the expectation and variance of D. The standard approach is then to determine the order quantity using conservative rules such as minimax regret or Scarf's r...

In this paper we prove a sufficient maximum principle for general stochastic differential Stackelberg games, and apply the theory to continuous time newsvendor problems. In the newsvendor problem a manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits under a random demand rate. Our demand rate is...

In some countries every citizen has the right to obtain a designated general practitioner. However, each individual may have preferences that cannot be fulfilled due to shortages of some kind. The questions raised in this paper are: To what extent can we expect that preferences are fulfilled when the patients "compete" for entry on the lists of pra...

In this paper we consider the newsvendor model with real options. We consider a mixed contract where the retailer can order a combination of q units subject to the conditions in a classical newsvendor contract and Q real options on the same items. We provide a closed form solution to this mixed contract when the demand is discrete and study some of...

As mentioned in the introduction, the framework that we developed in Chapter 2 for the main purpose of solving stochastic partial differential equations, can also be used to obtain new results – as well as new proofs of old results – for stochastic ordinary differential equations. In this chapter we will illustrate this by discussing some important...

This paper aims to construct a comprehensive model capable of simulating spatial unemployment disparities. The key feature of the model is that it simultaneously deals with commuting and migration. Much of the existing literature simply models one adjustment mechanism at a time. This paper adopts a micro-simulation approach to build a model which c...

In this paper we quantify agent preferences in a market. In our framework every agent has a utility level associated with each transaction, and we assume that the probability of a feasible market transaction increases with an increase in total utility. It is surprising to observe that this simple behavioral principle induces a usually unique probab...

This article provides an empirical evaluation of a hierarchical approach to modeling commuting flows. As the gravity family of spatial interaction models represents a benchmark for empirical evaluation, we begin by reviewing basic aspects of these models. The hierarchical modeling framework is the same that Thorsen, Ubøe, and Nævdal (1999) used. Ho...

In this paper we give a survey on some basic ideas related to random utility, extreme value theory and multinomial logit models. These ideas are well known within the field of spatial economics, but do not appear to be common knowledge to researchers in probability theory. The purpose of the paper is to try to bridge this gap.

In this paper it is shown how the benefit efficient patient list model of Ubøe and Lillestøl [1] can be used to infer strength of preferences from patient list data. It is proved that the model allows the cons-truction of unique sets of preferences replicating the observed allocations. To illustrate how this theory can be applied in practice, prefe...

In this chapter we will apply the general theory developed in Chapter 2 to solve various stochastic partial differential equations
(SPDEs) driven by Brownian white noise. In fact, as pointed out in Chapter 1, our main motivation for setting up this machinery
was to enable us to solve some of the basic SPDEs that appear frequently in applications.

In the last decades there has been an increased interest in stochastic models based on other processes than the Brownian motion
B(t).

In western Norway, fjords cause disconnections in the road network, necessitating the use of ferries. In several cases, ferries have been replaced by roads, often part-financed by tolls. We use data on commuting from a region with a high number of ferries, tunnels and bridges. Using a doubly-constrained gravity-based model specification, we focus o...

This paper studies whether gravity model parameters estimated in one geographic area can give reasonable predictions of commuting flows in another. To do this, three sets of parameters are estimated for geographically proximate yet separate regions in south-west Norway. All possible combinations of data and parameters are considered, giving a total...

We consider a setting where a large number of agents are trading commodity bundles. Assuming that agents of the same type have a certain utility attached to each transaction, we construct a statistical equilibrium which in turn implies prices on the different commodities. Our basic question is then the following. Assuming that some commodities come...

Traffic congestion and the policies to combat it have been studied extensively. However, most studies neglect the labour market impacts of congestion. Many also fail to account for the simultaneity between commuting and migration. This paper models impacts such as unemployment disparities, changes in commuting flows and changes in the flow of migra...

In this paper we study bulk shipping of coal between the central regions in the world. We compare the performance of cost-minimizing models with a gravity model approach. The main finding in the paper is that cost minimizing models provide relative poor fits to data. A simple one parameter gravity model, however, provides very satisfactory fits to...

In this chapter we develop the general framework to be used in this book. The starting point for the discussion will be the
standard white noise structures and how constructions of this kind can be given a rigorous treatment. White noise analysis
can be addressed in several different ways. The presentation here is to a large extent influenced by id...

Abstract In this paper we examine the risk of reporting spurious relationships in trip distribution models. We show how to make synthetic data sets that (by construction) are neutral with respect to clustering effects. We study a particular case with two non-interacting groups of jobs/workers. A competing destinations model is applied to 100 random...

This paper explores possible reasons for persistent spatial unemployment disparities using agent-based computational methods. The method relies on observing the actions of thousands of individuals within an artificial society. The paper models the effect of unemployment insurance, wage disparities, region specific amenities and innate residential p...

In this paper we propose a modified gravity model that takes into account that a population generally consists of heterogeneous groups, and we suggest a new statistical test for heterogeneity. We apply our new model to two real world data sets, and it turns out that this new model fits the data surpricingly well. Not only is the effect of heterogen...

In this paper we will show how the patient list model in Ub�e & Lillest�l (2007) can be used to infer strength of preferences from patient list data. We prove that we can construct unique sets of preferences that replicates patient list data, and we also show how to approach cases where we only have partial information of the system. As an illustra...

In this paper, we consider statistical distributions of different types of patients on the patient lists of doctors. In our framework different types of patients have different preferences regarding their preferred choice of doctor. Assuming that the system is benefit efficient in the sense that distributions with larger total utility have higher p...

In this paper we will see how commuter preferences can be revealed from observations of trip distributions. We will explain how to find unique representations of preference structures leading to an observed trip distribution, and will also present a numerical method that can be used to infer preferences from systems of considerable size. The theory...

We consider the multidimensional Burgers equation with a viscosity term and a random force modelled by a functional of time-space white noise, Wick products. Then we show that the nonlinear equation (B) can be transformed into a linear, stochastic heat equation with a noisy potential. This heat equation is solved explicitly in the following two cas...

We study how the value function (minimal cost function) Vc of certain impulse control problems depends on the intervention cost c. We consider the case when the cost of interfering with an impulse control of size ζ ∈ R is given by c + λ|ζ| with c ≤ λt > 0 constants, and we show (under some assumptions) that Vc is very sensitive (non-robust) to an i...

In some countries every citizen has the right to obtain a designated general practitioner. However, each individual may have preferences that cannot be fulfilled due to shortages of some kind. The questions raised in this paper are: To what extent can we expect that preferences are fulfilled when the patients “compete” for entry on the lists of pra...

In this paper we present a model for spatial interaction within a network of towns. This interaction is modeled through equilibrium states for certain Markov chains where, in particular, explicit formulas for these states are given. Our model exploits and intertwines ideas from gravity models, the competing destinations model and the intervening op...

In this paper we present empirical results based on a network model for commuting flows. The model is a modified version of a construction introduced in Thorsen et al. (1999). Journeys-to-work are determined by distance deterrence effects, the effects of intervening opportunities, and the location of potential destinations relative to alternatives...

In this paper we will study statistical equilibria in commodity markets where agents have a specified utility attached to every transaction in their offer sets. A probability measure on the product of all offer sets is called benefit efficient if market transactions with higher total benefit are more probable. We will characterize all such probabil...

In this paper we develop a new simulation procedure that can be used to examine validity of model extensions. Our testing regime is carried out on a number of different trip distribution models. We test the models on synthetic populations contructed from an aggregated set of worker categories, reflecting for instance different qualifications. The a...

In this paper we suggest a microeconomic model for how commuting flows relate to traveling distance in a two-region system. Commuting is the preferred choice of a worker whenever he can obtain an increase in wages greater than the cost of commuting. Our framework is based on an approach where workers apply for jobs according to a strategy that maxi...

In this paper we perform a simulation procedure of testing models for journeys to work. The testing regime is carried out on a number of such models, mainly within the class of gravity models. We test the models on synthetic populations constructed from an aggregated set of a large number of worker subcategories, reflecting for instance different q...

This paper deals with the use of gravity models to examine journeys to work. The purpose of the paper is to study very simple examples demonstrating that gravity models may be subject to serious misspecification in aggregate systems. As an outcome of the theoretical analysis, the paper has several implications to empirical work. It suggests a varie...

In this paper we will consider gravity models for journeys to work. In applications of the theory it is sometimes assumed that the parameters in such models are fixed. We will provide examples to show that this is not always a reasonable assumption, for instance, when the model is applied to predict how changes in the road transportation network in...

In this paper we present a model for commuting in a network of towns. A basic assumption is that all individuals have a given residential location and that every node in the network has a fixed number of jobs. We then propose a general model for the commuting of labor between the nodes in the network.

A model is presented for residential location choice in rural areas with spatial barriers. We address the problem through comparative static analysis focusing on how residential location choices are affected by a new road link across the spatial barrier. We proceed through a probability theoretical approach: choose a family of utility functions rep...

We use white noise calculus and the Donsker Delta Function to find explicit formulas for the replicating portfolios in a Black–Scholes market for a class of contingent T-claims.

In this paper we suggest a microeconomic model for how commuting ∞ows relate to traveling distance. Commuting is the preferred choice of a worker whenever he can obtain an increase in wages greater than the cost of commuting. Our framework is based on an approach where workers apply for jobs according to a strategy that maximizes their expected pay...

Solutions to stochastic differential equations depends on the method of approximation. In this paper we give a very simple demonstration that ordinary differential equations, too, exhibit this kind of behavior when the coefficients are measure-valued distributions. We then proceed to show that the Itô and the Stratonovich solutions can be viewed as...

We use a white noise approach to Malliavin calculus to prove the following white noise generalization of the Clark-Haussmann-Ocone formula
\[F(\omega)=E[F]+\int_0^TE[D_tF|\F_t]\diamond W(t)dt\]
Here E[F] denotes the generalized expectation, $D_tF(\omega)={{dF}\over{d\omega}}$ is the (generalized) Malliavin derivative, $\diamond$ is the Wick produ...

[eng] Transportation costs and monopoly location in presence of regional disparities. . This article aims at analysing the impact of the level of transportation costs on the location choice of a monopolist. We consider two asymmetric regions. The heterogeneity of space lies in both regional incomes and population sizes: the first region is endowed...

In this paper we consider Wick products of complex valued random variables. We prove that Wick products of such variables coincide with the ordinary product in a variety of cases. Ordinary SDEs are considered in relation to their Wick versions. We present examples where these notions are equivalent in the complex case.

As mentioned in the introduction, the framework that we developed in Chapter 2 for the main purpose of solving stochastic partial differential equations, can also be used to obtain new results — as well as new proofs of old results — for stochastic ordinary differential equations. In this chapter we will illustrate this by discussing some important...

In this chapter we will apply the general theory developed in Chapter 2 to solve various stochastic partial differential equations (SPDEs). In fact, as pointed out in Chapter 1, our main motivation for setting up this machinery was to enable us to solve some of the basic SPDEs that appear frequently in applications.

An equation modelling the pressurep(x) =p(x, w) atx ∈D ⊂R
d
of an incompressible fluid in a heterogeneous, isotropic medium with a stochastic permeabilityk(x, w) ≥ 0 is the stochastic partial differential equation
$\left\{ {{*{20}c} {div(k(x,{\mathbf{ }}\omega )\diamondsuit \nabla p(x,\omega )){\mathbf{ }} = {\mathbf{ }}--f(x);{\mathbf{ }}x \in D...

In this paper we consider continuity properties of a stochastic heat equation of the form [not partial differential]u(t,x)/[not partial differential]t = [not partial differential]2u(t,x)/[not partial differential]x2 + f(u(t,x))Wx,t. We prove that the solutions of this equation depend continuously on the function f and give some new estimates for th...

Stochastic partial differential equations can be used in many areas of science to model complex systems that evolve over time. Their analysis is currently an area of much research interest. This book consists of papers given at the ICMS Edinburgh meeting held in 1994 on this topic, and it brings together some of the world's best known authorities o...

In this paper we will consider the properties of various stochastic integrals over a complex valued, two parameter Wiener process. Integrals of this type exhibit pleasant features which do not appear within the real framework. They are stable under approximation and viewed in relation with analytic functions, they typically satisfy an ordinary chai...

In this paper we prove a tail estimate on the form: Xt is supposed to be a general Dirichlet process taking values on some Banach space where, in particular, we do not assume any smoothness conditions on the operator function associated with the process

Stochastic partial differential equations (SPDEs) often have solutions that are known to be pure Schwartz distributions i.e. not functions. To make sense of such equations one needs to introduce some kind of smoothing parameters. This paper is concerned with stability properties of the solutions as one lets the smoothing parameters approach some ki...

We find an explicit expression for the (unique) solution u = u(t,x, w) of the stochastic partial differential equation $$\begin{array}{*{20}{c}} {\frac{{\partial u}}{{\partial t}} = \frac{1}{2}{{v}^{2}}\Delta u + {{{\vec{W}}}_{x}}\diamondsuit \nabla u;\;\left( {t,x} \right) \in {{\mathbb{R}}^{ + }} \times {{\mathbb{R}}^{d}},} \\ {u\left( {0,\cdot }...

We give a program for solving stochastic boundary value problems involving functionals of (multiparameter) white noise. As an example we solve the stochastic Schrdinger equation {ie391-1} whereV is a positive, noisy potential. We represent the potentialV by a white noise functional and interpret the product of the two distribution valued processesV...

## Network

Cited