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Approximative solutions of optimal stopping and selection problems

100%
Rüschendorf L.
Mathematica Applicanda
|
2015
|
tom 44
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nr 1
PL
W artykule opracowano przeglad różnych podejść z ostatnich 15 lat do przybliżonego rozwiazywania zadan optymalnego zatrzymania procesów z czasem dyskretnym, w tym takze zadan wyboru najlepszeg obiektu. Metody te pozwalaja takze na rozwiazywanie niektórych problemów wielokrotnego zatrzymania, a takze radza sobie z rozwiazaniem zadan dla ciagów zaleznych.Podstawa tych analiz jest obserwacja, iz ciag unormowanych obserwacji odwzorowanych na płaszczyzne jest zbiezny według rozkładu do pewnego procesu punktowego. Dla róznych klas zadan metoda prowadzi do uzyskania zamknietych analitycznych formuł lub pozwala na uzyskanie rozwiazan numerycznych.
EN
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 to a Poisson or to a cluster process. For various classes of examples the method leads to explicit or numerically accessible solutions.
2
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Optimal solutions of multivariate coupling problems

100%
Rüschendorf L.
Applicationes Mathematicae
|
1995-1996
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tom 23
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nr 3
325-338
EN
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 $l_p$-type metrics, where fairly explicit and complete characterizations of optimal transportation plans (couplings) are obtained. The results are of interest even in the one-dimensional case. For the first time an explicit criterion is given for the construction of optimal multivariate couplings for the Kantorovich metric $l_1$.
3
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On the optimal reinsurance problem

64%
Kiesel S., Rüschendorf L.
Applicationes Mathematicae
|
2013
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tom 40
|
nr 3
259-280
EN
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 to insurance takers acting individually. Our results extend previously known results from the literature.
4
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Markov morphisms: a combined copula and mass transportation approach to multivariate quantiles

64%
Faugeras O. P., Rüschendorf L.
Mathematica Applicanda
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2017
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tom 45
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nr 1
PL
W artykule zaproponowano pewien sposobu wprowadzania kwantyli wielowymiarowych, jak i metody ich wyznaczania. Z jednej strony podstawą rozważań są podstawowe własności uogólnionego pojęcia wielowymiarowego kwantyla, który jest morfizmem markowskiem, zachowującym podobne własności algebraiczne, topologiczne oraz porządku, jakie znamy dla linii kwantylowych na prostej rzeczywistej. Z drugiej zaś strony, zaproponowano morfiz markowski, który łączy standaryzowaną kopułę (funkcję łącznikową) z zastosowaniem  zagadnienia transportowego (v. Chernozhukov et al.(2017)). Proponowane podejście daje ogólne i jednolite podejście do definicji kwantyli i ich estymacji, zarówno dla ciągłych, jak i dyskretnych rozkładów wielowymiarowych.
EN
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 and the recent optimal mass transportation method of Chernozhukov et al. (2015). Its empirical counterpart has the advantages of being a bandwidth-free, monotone invariant, a.s. consistent transformation. The proposed the approach gives a general and unified framework to quantiles and their corresponding depth areas, for both a continuous or a discrete multivariate distribution.
5
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VaR bounds in models with partial dependence information on subgroups

64%
Rüschendorf L., Witting J.
Dependence Modeling
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2017
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tom 5
|
nr 1
59-74
EN
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 and usable risk bounds resp. estimates of the model risk by including partial dependence information additional to the information on the marginals. In particular we extend an approach suggested in Bignozzi, Puccetti and Rüschendorf (2015) and in Puccetti, Rüschendorf, Small and Vanduffel (2017), which is based on positive dependence resp. on independence information available for some subgroups.
6
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Analysis of Markov chain algorithms on spanning trees, rooted forests, and connected subgraphs

64%
Fehrenbach J., Rüschendorf L.
Applicationes Mathematicae
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2005
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tom 32
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nr 3
341-365
EN
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 time mixing rates for some related Markov chains on the set of forests with roots and on the set of connected spanning subgraphs. We obtain a parametric class of stationary distributions from which we can efficiently sample. For rooted forests this includes the uniform distribution. For connected spanning subgraphs the uniform distribution is not covered.
7
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On the weighted Euclidean matching problem in $ℝ^d$

64%
Anthes B., Rüschendorf L.
Applicationes Mathematicae
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2001
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tom 28
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nr 2
181-190
EN
A partitioning algorithm for the Euclidean matching problem in $ℝ^d$ 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 $n(logn)^{p-1}$ and approximates the optimal matching in the probabilistic sense.
8
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Cost-efficiency in multivariate Lévy models

64%
Rüschendorf L., Wolf V.
Dependence Modeling
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2015
|
tom 3
|
nr 1
EN
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 and uniqueness results for the Esscher parameter and determine cost efficient strategies in particular in the case of price processes driven by multivariate NIG- and VG-processes. From a monotonicity characterization of efficient payoffs we obtain that basket options are generally inefficient in Lévy markets when pricing is based on the Esscher measure.We determine efficient versions of the basket options in real market data and show that the proposed cost efficient strategies are also feasible from a numerical viewpoint. As a result we find that a considerable efficiency loss may arise when using the inefficient payoffs.
9
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VaR bounds for joint portfolios with dependence constraints

51%
Puccetti G., Rüschendorf L., Manko D.
Dependence Modeling
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2016
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tom 4
|
nr 1
EN
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 is illustrated in a series of examples of practical interest.
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