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Some necessary and some sufficient conditions are established for the explicit construction and characterization of optimal solutions of multivariate transportation (coupling) problems. The proofs are based on ideas from duality theory and nonconvex optimization theory. Applications are given to multivariate optimal coupling problems w.r.t. minimal -type metrics, where fairly explicit and complete characterizations of optimal transportation plans (couplings) are obtained. The results are of interest...
In this paper we review a series of developments over the last 15 years in which a general method for the approximative solution of finite discrete time optimal stopping and choice problems has been developed. This method also allows to deal with multiple stopping and choice problems and to deal with stopping or choice problems for some classes of dependent sequences.The basic assumption of this approach is that the sequence of normalized observations when embedded in the plane converges in distribution...
A partitioning algorithm for the Euclidean matching problem in is introduced and analyzed in a probabilistic model. The algorithm uses elements from the fixed dissection algorithm of Karp and Steele (1985) and the Zig-Zag algorithm of Halton and Terada (1982) for the traveling salesman problem. The algorithm runs in expected time and approximates the optimal matching in the probabilistic sense.
We analyse a natural edge exchange Markov chain on the set of spanning trees of an undirected graph by the method of multicommodity flows. The analysis is then refined to obtain a canonical path analysis. The construction of the flow and of the canonical paths is based on related path constructions in a paper of Cordovil and Moreira (1993) on block matroids. The estimates of the congestion measure imply a polynomial bound on the mixing time. The canonical paths for spanning trees also yield polynomial...
In this paper we consider the optimal reinsurance problem in endogenous form with respect to general convex risk measures ϱ and pricing rules π. By means of a subdifferential formula for compositions in Banach spaces we first characterize optimal reinsurance contracts in the case of one insurance taker and one insurer. In the second step we generalize the characterization to the case of several insurance takers. As a consequence we obtain a result saying that cooperation brings less risk compared...
In this paper we determine lowest cost strategies for given payoff distributions called cost-efficient strategies in multivariate exponential Lévy models where the pricing is based on the multivariate Esscher martingale measure. This multivariate framework allows to deal with dependent price processes as arising in typical applications. Dependence of the components of the Lévy Process implies an influence even on the pricing of efficient versions of univariate payoffs.We state various relevant existence...
We derive improved estimates for the model risk of risk portfolios when additional to the marginals some partial dependence information is available.We consider models which are split into k subgroups and consider various classes of dependence information either within the subgroups or between the subgroups. As consequence we obtain improved VaR bounds for the joint portfolio compared to the case with only information on the marginals. Our paper adds to various recent approaches to obtain reliable...
Based on a novel extension of classical Hoeffding-Fréchet bounds, we provide an upper VaR bound for joint risk portfolios with fixed marginal distributions and positive dependence information. The positive dependence information can be assumed to hold in the tails, in some central part, or on a general subset of the domain of the distribution function of a risk portfolio. The newly provided VaR bound can be interpreted as a comonotonic VaR computed at a distorted confidence level and its quality...
Our purpose is both conceptual and practical. On the one hand, we discuss the question which properties are basic ingredients of a general conceptual notion of a multivariate quantile. We propose and argue that the object “quantile” should be defined as a Markov morphism which carries over similar algebraic, ordering and topological properties as known for quantile functions on the real line. On the other hand, we also propose a practical quantile Markov morphism which combines a copula standardization...
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