Nicole Bäuerle

Modeling and Comparing Dependencies in Multivariate Risk Portfolios

In this paper we investigate multivariate risk portfolios, where the risks are dependent. By providing some natural models for risk portfolios with the same marginal distributions we are able to compare two portfolios with different dependence structure with respect to their stop-loss premiums. In particular, some comparison results for portfolios with two-point distributions are obtained. The analysis is based on the concept of the so-called supermodular ordering. We also give some numerical results which indicate that dependencies in risk portfolios can have a severe impact on the stop-loss premium. In fact, we show that the effect of dependencies can grow beyond any given bound.

Portfolio optimization with Markov-modulated stock prices and interest rates

A financial market with one bond and one stock is considered where the risk free interest rate, the appreciation rate of the stock and the volatility of the stock depend on an external finite state Markov chain. We investigate the problem of maximizing the expected utility from terminal wealth and solve it by stochastic control methods for different utility functions. Due to explicit solutions it is possible to compare the value function of the problem to one where we have constant (average) market data. The case of benchmark optimization is also considered.

Benchmark and mean-variance problems for insurers

We consider the classical Cramér-Lundberg model with dynamic proportional reinsurance and solve the problem of finding the optimal reinsurance strategy which minimizes the expected quadratic distance of the risk reserve to a given benchmark. This result is extended to a mean-variance problem.

Inequalities for stochastic models via supermodular orderings

The aim of this paper is to derive inequalities for random vectors by using the supermodular ordering. The properties of this ordering suggest to use it as a comparison for the “ strength of dependence” in random vectors. In contrast to already established orderings of this type, the supermodular ordering has the advantage that it is not necessary to assume a common marginal distribution for the random vectors under comparison. As a consequence we obtain new inequalities by applying it to multivariate normal distributions, Markov chains and some stochastic models

On the waiting time of arriving aircrafts and the capacity of airports with one or two runways

Abstract In this paper we examine a model for the landing procedure of aircrafts at an airport. The characteristic feature here is that due to air turbulence the safety distance between two landing aircrafts depends on the types of these two machines. Hence, an efficient routing of the aircraft to two runways may reduce their waiting time. First, we use M/SM/1 queues (with dependent service times) to model a single runway. We give the stability condition and a formula for the average waiting time of the aircrafts. Moreover, we derive easy to compute bounds on the waiting times by comparison to simpler queuing systems. In particular we study the effect of neglecting the dependency of the service times when using M/G/1-models. We then consider the case of two runways with a number of heuristic routing strategies such as coin flipping, type splitting, Round Robin and variants of the join-the-least-load rule. These strategies are analyzed and compared numerically with respect to the average delay they cause. It turns out that a certain modification of join-the-least-load gives the best results.

More Risk-Sensitive Markov Decision Processes

We investigate the problem of minimizing a certainty equivalent of the total or discounted cost over a finite and an infinite horizon that is generated by a Markov decision process MDP. In contrast to a risk-neutral decision maker this optimization criterion takes the variability of the cost into account. It contains as a special case the classical risk-sensitive optimization criterion with an exponential utility. We show that this optimization problem can be solved by an ordinary MDP with extended state space and give conditions under which an optimal policy exists. In the case of an infinite time horizon we show that the minimal discounted cost can be obtained by value iteration and can be characterized as the unique solution of a fixed-point equation using a “sandwich” argument. Interestingly, it turns out that in the case of a power utility, the problem simplifies and is of similar complexity than the exponential utility case, however has not been treated in the literature so far. We also establish the validity and convergence of the policy improvement method. A simple numerical example, namely, the classical repeated casino game, is considered to illustrate the influence of the certainty equivalent and its parameters. Finally, the average cost problem is also investigated. Surprisingly, it turns out that under suitable recurrence conditions on the MDP for convex power utility, the minimal average cost does not depend on the parameter of the utility function and is equal to the risk-neutral average cost. This is in contrast to the classical risk-sensitive criterion with exponential utility.

MULTIVARIATE COUNTING PROCESSES: COPULAS AND BEYOND

Multivariate stochastic processes with Poisson marginals are of interest in insurance and finance; they can be used to model the joint behaviour of several claim arrival processes, for example. We discuss various methods for the construction of such models, with particular emphasis on the use of copulas. An important class of multivariate counting processes with Poisson marginals arises if the events of a background Poisson process with constant intensity are moved forward in time by a random amount and possibly deleted; here we think of the events of the background process as triggering later claims in different categories. We discuss structural aspects of these models, their dependence properties together with stochastic order aspects, and also some related computational issues. Various actuarial applications are indicated.

Markov Decision Processes with Average-Value-at-Risk criteria

We investigate the problem of minimizing the Average-Value-at-Risk (AVaRτ) of the discounted cost over a finite and an infinite horizon which is generated by a Markov Decision Process (MDP). We show that this problem can be reduced to an ordinary MDP with extended state space and give conditions under which an optimal policy exists. We also give a time-consistent interpretation of the AVaRτ. At the end we consider a numerical example which is a simple repeated casino game. It is used to discuss the influence of the risk aversion parameter τ of the AVaRτ-criterion.

MONOTONICITY RESULTS FOR MR/GI/1 QUEUES

This paper considers queues with a Markov renewal arrival process and a particular transition matrix for the underlying Markov chain. We study the effect that the transition matrix has on the waiting time of the n th customer as well as on the stationary waiting time. The main theorem generalizes results of Szekli et al. (1994a) and partly confirms their conjecture. In this context we show the importance of a new stochastic ordering concept.

Some results about the expected ruin time in Markov-modulated risk models

Abstract In this paper we investigate the expected ruin time of Markov-modulated risk models. It turns out that the expected ruin time ξ(u), depending on the initial risk reserve u ∈ R +, is asymptotically linear. In the two-state model we are able to derive exact formulas. A very interesting result is the monotonicity property of ξ(u). We show that the more slowly the environment changes, the greater is the expected ruin time.