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22
Stochastic ordering of risks, influence of dependence and a.s. constructions
 In
, 2005
"... In this paper we review and extend some key results on the stochastic ordering of risks and on bounding the influence of stochastic dependence on risk functionals. The first part of the paper is concerned with a.s. constructions of random vectors and with diffusion kernel type comparisons which are ..."
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Cited by 15 (10 self)
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In this paper we review and extend some key results on the stochastic ordering of risks and on bounding the influence of stochastic dependence on risk functionals. The first part of the paper is concerned with a.s. constructions of random vectors and with diffusion kernel type comparisons which are of importance for various comparison results. In the second part we consider generalizations of the classical Fréchetbounds, in particular for the distribution of sums and maxima and for more general monotonic functionals of the risk vector. In the final part we discuss three important orderings of risks which arise from ∆monotone, supermodular, and directionally convex functions. We give some new criteria for these orderings. For the basic results we also take care to give references to “original sources ” of these results. 1
A Computational Study on Bounding the Makespan Distribution in Stochastic Project Networks
 ANNALS OF OPERATIONS RESEARCH
, 1998
"... Given a stochastic project network with independently distributed activity durations, several approaches to bound the distribution function of the project completion time have been proposed. We have implemented the most promising of these algorithms and compare their behavior on a basis of nearly 20 ..."
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Cited by 14 (1 self)
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Given a stochastic project network with independently distributed activity durations, several approaches to bound the distribution function of the project completion time have been proposed. We have implemented the most promising of these algorithms and compare their behavior on a basis of nearly 2000 instances with up to 1200 activities of different testbeds. We propose a suitable numerical representation of the given distributions which is the basis for excellent computational results.
A Simple Geometric Proof That Comonotonic Risks Have the Convexlargest Sum
, 2002
"... In the recent actuarial literature, several proofs have been given for the fact that if a random vector (X 1 ,X 2 ,...,X n ) with given marginals has a comonotonic joint distribution, the sum X 1 +X 2 + X n is the largest possible in convex order. In this note we give a lucid proof of this fac ..."
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Cited by 12 (6 self)
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In the recent actuarial literature, several proofs have been given for the fact that if a random vector (X 1 ,X 2 ,...,X n ) with given marginals has a comonotonic joint distribution, the sum X 1 +X 2 + X n is the largest possible in convex order. In this note we give a lucid proof of this fact, based on a geometric interpretation of the support of the comonotonic distribution.
A heuristic for optimizing stochastic activity networks with applications to statistical digital circuit sizing
 IEEE Transactions on Circuits and SystemsI
, 2004
"... A deterministic activity network (DAN) is a collection of activities, each with some duration, along with a set of precedence constraints, which specify that activities begin only when certain others have finished. One critical performance measure for an activity network is its makespan, which is th ..."
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Cited by 12 (4 self)
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A deterministic activity network (DAN) is a collection of activities, each with some duration, along with a set of precedence constraints, which specify that activities begin only when certain others have finished. One critical performance measure for an activity network is its makespan, which is the minimum time required to complete all activities. In a stochastic activity network (SAN), the durations of the activities and the makespan are random variables. The analysis of SANs is quite involved, but can be carried out numerically by Monte Carlo analysis. This paper concerns the optimization of a SAN, i.e., the choice of some design variables that affect the probability distributions of the activity durations. We concentrate on the problem of minimizing a quantile (e.g., 95%) of the makespan, subject to constraints on the variables. This problem has many applications, ranging from project management to digital integrated circuit (IC) sizing (the latter being our motivation). While there are effective methods for optimizing DANs, the SAN optimization problem is much more difficult; the few existing methods cannot handle largescale problems.
Multivariate comonotonicity
 Journal of Multivariate Analysis
"... In this paper we consider several multivariate extensions of comonotonicity. We show that naive extensions do not enjoy some of the main properties of the univariate concept. In order to have these properties more structure is needed than in the univariate case. ..."
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Cited by 8 (0 self)
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In this paper we consider several multivariate extensions of comonotonicity. We show that naive extensions do not enjoy some of the main properties of the univariate concept. In order to have these properties more structure is needed than in the univariate case.
Comparison of multivariate risks and positive dependence
 J. Appl. Probab
"... In this paper we extend some recent results on the comparison of multivariate risk vectors w.r.t. supermodular and related orderings. We introduce a dependence notion called ‘weakly conditional increasing in sequence order ’ that allows to conclude that ‘more dependent’ vectors in this ordering are ..."
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Cited by 8 (3 self)
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In this paper we extend some recent results on the comparison of multivariate risk vectors w.r.t. supermodular and related orderings. We introduce a dependence notion called ‘weakly conditional increasing in sequence order ’ that allows to conclude that ‘more dependent’ vectors in this ordering are also comparable w.r.t. the supermodular ordering. At the same time this ordering allows to compare two risks w.r.t. the directionally convex order if the marginals increase convexly. We further state comparison criteria w.r.t. the directionally convex order for some classes of risk vectors which are modelled by functional influence factors. Finally we discuss Fréchetbounds w.r.t. ∆monotone functions when multivariate marginals are given. It turns out that comonotone vectors in the case of multivariate marginals no longer yield necessarily the largest risks but even may in some cases be vectors which minimize risk. Keywords: supermodular ordering, Fréchetbounds, positive dependence, risk vectors 1
Static SuperReplicating Strategies for a Class of Exotic Options
, 2008
"... In this paper, we investigate static superreplicating strategies for Europeantype call options written on a weighted sum of asset prices. This class of exotic options includes Asian options and basket options among others. We assume that there exists a market where the plain vanilla options on the ..."
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Cited by 6 (5 self)
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In this paper, we investigate static superreplicating strategies for Europeantype call options written on a weighted sum of asset prices. This class of exotic options includes Asian options and basket options among others. We assume that there exists a market where the plain vanilla options on the different assets are traded and hence their prices can be observed in the market. Both the infinite market case (where prices of the plain vanilla options are available for all strikes) and the finite market case (where only a finite number of plain vanilla option prices are observed) are considered. We prove that the finite market case converges to the infinite market case when the number of observed plain vanilla option prices tends to infinity. We show how to construct a portfolio consisting of the plain vanilla options on the different assets, whose payoff superreplicates the payoff of the exotic option. As a consequence, the price of the superreplicating portfolio is an upper bound for the price of the exotic option. The superhedging strategy is modelfree in the sense that it is expressed in terms of the observed option prices on the individual assets, which can be e.g. dividend paying stocks with no explicit dividend process known. This paper is a generalization of the work of Simon et al. (2000) who considered this problem for Asian options in the infinite market case. Laurence and Wang (2004) and Hobson et al. (2005) considered this problem for basket options, in the infinite as well as in the finite market case. As opposed to Hobson et al. (2005) who use Lagrange optimization techniques, the proofs in this paper are based on the theory of integral stochastic orders and on the theory of comonotonic risks.
Probabilistic combinatorial optimization: Moments, semidefinite programming and asymptotic bounds
 SIAM Journal on Optimization
, 2003
"... Abstract. We address the problem of evaluating the expected optimal objective value of a 01 optimization problem under uncertainty in the objective coefficients. The probabilistic model we consider prescribes limited marginal distribution information for the objective coefficients in the form of mo ..."
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Cited by 6 (2 self)
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Abstract. We address the problem of evaluating the expected optimal objective value of a 01 optimization problem under uncertainty in the objective coefficients. The probabilistic model we consider prescribes limited marginal distribution information for the objective coefficients in the form of moments. We show that for a fairly general class of marginal information, a tight upper (lower) bound on the expected optimal objective value of a 01 maximization (minimization) problem can be computed in polynomial time if the corresponding deterministic problem is solvable in polynomial time. We provide an efficiently solvable semidefinite programming formulation to compute this tight bound. We also analyze the asymptotic behavior of a general class of combinatorial problems that includes the linear assignment, spanning tree, and traveling salesman problems, under knowledge of complete marginal distributions, with and without independence. We calculate the limiting constants exactly.
Tight bounds on Expected Order Statistics
 Probability in the Engineering and Informational Sciences
, 2006
"... In this article, we study the problem of finding tight bounds on the expected value of the kthorder statistic E @Xk:n # under first and second moment information on n realvalued random variables+ Given means E @X i # � µ i and variances ..."
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Cited by 4 (0 self)
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In this article, we study the problem of finding tight bounds on the expected value of the kthorder statistic E @Xk:n # under first and second moment information on n realvalued random variables+ Given means E @X i # � µ i and variances
Persistence in Discrete Optimization under Data Uncertainty
, 2004
"... An important question in discrete optimization under uncertainty is to understand the persistency of a decision variable, i.e., the probability that it is part of an optimal solution. For instance, in project management, when the task activity times are random, the challenge is to determine a set o ..."
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Cited by 2 (2 self)
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An important question in discrete optimization under uncertainty is to understand the persistency of a decision variable, i.e., the probability that it is part of an optimal solution. For instance, in project management, when the task activity times are random, the challenge is to determine a set of critical activities that will potentially lie on the longest path. In the spanning tree and shortest path network problems, when the arc lengths are random, the challenge is to preprocess the network and determine a smaller set of arcs that will most probably be a part of the optimal solution under different realizations of the arc lengths. Building on a characterization of moment cones for single variate problems, and its associated semidefinite constraint representation, we develop a limited marginal moment model to compute the persistency of a decision variable. Under this model, we show that finding the persistency is tractable for zeroone optimization problems with a polynomial sized representation of the convex hull of the feasible region. Through extensive experiments, we show that the persistency computed under the limited marginal moment model is often close to the simulated persistency value under various distributions that satisfy the prescribed marginal moments and are generated independently.