A duopoly preemption game with two alternative stochastic investment choices

This paper studies a duopoly investment model with uncertainty. There are two alternative irreversible investments. The first firm to invest gets a monopoly benefit for a specified period of time. The second firm to invest gets information based on what happens with the first investor, as well as cost reduction benefits. We describe the payoff functions for both the leader and follower firm. Then, we present a stochastic control game where the firms can choose when to invest, and hence influence whether they become the leader or the follower. In order to solve this problem, we combine techniques from optimal stopping and game theory. For a specific choice of parametres, we show that no pure symmetric subgame perfect Nash equilibrium exists. However, an asymmetric equilibrium is characterized. In this equilibrium, two disjoint intervals of market demand level give rise to preemptive investment behavior of the firms, while the firms otherwise are more reluctant to be the first mover

Sequential Bayesian optimal experimental design for structural reliability analysis

Structural reliability analysis is concerned with estimation of the probability of a critical event taking place, described by $P(g(\textbf{X}) \leq 0)$ for some $n$-dimensional random variable $\textbf{X}$ and some real-valued function $g$. In many applications the function $g$ is practically unknown, as function evaluation involves time consuming numerical simulation or some other form of experiment that is expensive to perform. The problem we address in this paper is how to optimally design experiments, in a Bayesian decision theoretic fashion, when the goal is to estimate the probability $P(g(\textbf{X}) \leq 0)$ using a minimal amount of resources. As opposed to existing methods that have been proposed for this purpose, we consider a general structural reliability model given in hierarchical form. We therefore introduce a general formulation of the experimental design problem, where we distinguish between the uncertainty related to the random variable $\textbf{X}$ and any additional epistemic uncertainty that we want to reduce through experimentation. The effectiveness of a design strategy is evaluated through a measure of residual uncertainty, and efficient approximation of this quantity is crucial if we want to apply algorithms that search for an optimal strategy. The method we propose is based on importance sampling combined with the unscented transform for epistemic uncertainty propagation. We implement this for the myopic (one-step look ahead) alternative, and demonstrate the effectiveness through a series of numerical experiments.Comment: 27 pages, 13 figure

Convex duality and mathematical finance

The theme of this thesis is duality methods in mathematical - nance. This is a hot topic in the eld of mathematical nance, and there is currently a lot of research activity regarding this subject. However, since it is a fairly new eld of study, a lot of the material available is technical and di cult to read. This thesis aims to connect the duality methods used in mathematical nance to the general theory of duality methods in optimization and convexity, and hence clarify the subject. This requires the use of stochastic, real and functional analysis, as well as measure and integration theory. The thesis begins with a presentation of convexity and conju- gate duality theory. Then, this theory is applied to convex risk measures. The nancial market is introduced, and various duality methods, including linear programming duality, Lagrange duality and conjugate duality, are applied to solve utility maximization, pricing and arbitrage problems. This leads to both alternative proofs of known results, as well as some (to my knowledge) new results