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A Note on Stochastic Optimization Problems with Nonlinear Dependence on a Probability Measure
Kaňková, Vlasta
Nonlinear dependence on a probability measure begins to appear (last time) in a stochastic optimization rather often. Namely, the corresponding type of problems corresponds to many situations in applications. The nonlinear dependence can appear as in the objective functions so in a constraints set. We plan to consider the case of static (one-objective) problems in which nonlinear dependence appears in the objective function with a few types of constraints sets. In details we consider constraints sets “deterministic”, depending nonlinearly on the probability measure, constraints set determined by second order stochastic dominance and the sets given by mean-risk problems. The last case means that the constraints set corresponds to solutions those guarantee an acceptable value in both criteria. To introduce corresponding assertions we employ the stability results based on the Wasserstein metric and L1 norm. Moreover, we try to deal also with the case when all results have to be obtained (estimated) on the data base.
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Mean-Risk Optimization Problem via Scalarization, Stochastic Dominance, Empirical Estimates
Kaňková, Vlasta
Many economic and financial situations depend simultaneously on a random element and on a decision parameter. Mostly it is possible to influence the above mentioned situation by an optimization model depending on a probability measure. We focus on a special case of one-stage two objective stochastic “Mean-Risk problem”. Of course to determine optimal solution simultaneously with respect to the both criteria is mostly impossible. Consequently, it is necessary to employ some approaches. A few of them are known (from the literature), however two of them are very important: first of them is based on a scalarizing technique and the second one is based on the stochastic dominance. First approach has been suggested (in special case) by Markowitz, the second approach is based on the second order stochastic dominance. The last approach corresponds (under some assumptions) to partial order in the set of the utility functions.\nThe aim of the contribution is to deal with the both main above mentioned approaches. First, we repeat their properties and further we try to suggest possibility to improve the both values simultaneously with respect to the both criteria. However, we focus mainly on the case when probability characteristics has to be estimated on the data base.
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Multistage Stochastic Programming Problems - Decomposition
Lapšanská, Alica ; Kaňková, Vlasta (vedoucí práce) ; Lachout, Petr (oponent)
V práci je představena úloha vícestupňového stochastického programování a její aplikace na několik praktických problémů. Detailněji je rozebírán případ, kdy má náhodný prvek autoregresní vlastnost a množiny omezení jsou ve tvaru individuálních pravděpodobnostních omezení. Pro tyto úlohy jsou uvedeny podmínky, které musí dobře definovaný problém splňovat. Dále je řešena aproximace úlohy a rychlost její konvergence při empirickém odhadu distribuční funkce. Nakonec je vyřešen problém týkající se investování do finančních instrumentů, který je definován jako úloha dvoustupňového stochastického programování s pravděpodobnostným omezením a náhodným prvkem řídícím se autoregresní posloupností. Powered by TCPDF (www.tcpdf.org)
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Problém prodavače novin
Šedina, Jaroslav ; Dupačová, Jitka (vedoucí práce) ; Lachout, Petr (oponent)
Tato práce se zabývá problémem prodavače novin a jeho různými obměnami. V první kapitole je zaveden aparát, který je třeba k vyšetřování optimálního řešení úlohy. Druhá kapitola obsahuje různé formulace problému prodavače novin a jejich řešení, například metodou SAA. V závěru jsou výsledky aplikovány na výpočet podmíněné míry rizika CVaR a v programu R je uvedena numerická studie, která porovnává parametrický a neparametrický přístup k úloze. Text je průběžně doplňován grafy. Powered by TCPDF (www.tcpdf.org)
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Multiobjective Stochastic Optimization Problems with Probability Constraints
Kaňková, Vlasta
Rather general multiobjective optimization problems depending on a probability measure correspond often to situations in which an economic or financial process is simultaneously influenced by a random factor and a “decision” parameter; moreover simultaneously it is reasonable to evaluate the process by a few objective functions and it seems reasonable to determine the decision with to the mathematical expectation of objectives. A complete knowledge of the probability measure is a necessary assumption to analyze the problem. However, in applications mostly the problem has to be solved on the data base. A relationship between “characteristics” obtained on the base of complete knowledge of the probability measure and them obtained on the above mentioned data base has been already investigated in the case when constraints are not depending on the probability measure. The aim of the talk will be to relax this condition.
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Economic and Financial Problems via Multiobjective Stochastic Optimization
Kaňková, Vlasta
Multiobjective optimization problems depending on a probability measure correspond to many economic and financial activities. Evidently if the probability measure is completely known, then we can try to influence economic process employing methods of multiobjective deterministic optimization theory. Since this assumption is fulfilled very seldom we have mostly to analyze the mathematical model and consequently also economic process on the data base. The aim of the talk will be to investigate a relationship between ``characteristics" obtained on the base of complete knowledge of the probability measure and them obtained on the above mentioned data base. To this end, the results of the deterministic multiobjective optimization theory and the results obtained for stochastic one objective problems will be employed.
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Empirical Estimates in Economic and Financial Problems via Heavy Tails
Kaňková, Vlasta
Optimization problems depending on a probability measure correspond to many economic and financial applications. Complete knowledge of this measure is necessary to solve exactly these problems. Since this condition is fulfilled only seldom, the problem has to be usually solved on the data basis to obtain satistical estimates of an optimal value and optimal solutions. Great effort has been paid to investigate properties of these estimates; first under assumptions of disribution with thin tails and linear dependence on the probability measure. Recently, it has appeared an investigation in the case of nonlinear dependence on the probability measure and heavy tailed distributions with shape parameter greater two. We focus on the case of the stable and Pareto distributions with a shape parameter in the inteval (1, 2).
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Dependent Data in Economic and Financial Problems
Kaňková, Vlasta
Optimization problems depending on a probability measure correspond to many economic and financial applications. The paper deals with the case when an empirical measure substitutes the theoretical one. Especially the paper deals with a convergence rate of the corresponding estimates. ``Classical" results for independent samples are recalled, situations in which the case of dependent sample can be (from the mathematical point of view) reduced to independent case are mentioned. A great attention is paid to weak dependent samples fulfilling the Phi-mixing condition.
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Empirical Estimates in Stochastic Optimization: Special cases
Kaňková, Vlasta
Classical optimization problems depending on a probability measure belong mostly to nonlinear deterministic optimization problems that are relatively complicated. On the other hand, these problems fulfil very often "suitable" mathematical properties guaranteing the stability (w.r.t. probability measure) and, moreover, giving a possibility to replace the "underlying" probability measure by an empirical one to obtain "good" stochastic estimates of the optimal value and the optimal solution. Properties of thess estimates have been investigated mostly for standard types of probability measures with suitable (thin) tails and independent random samples. However distributions with heavy tails correspond to many economic problems and, moreover, many applications do not correspond to the "classical" problems. The aim of the paper is, first, to try to recall stability results including also heavy tails and more general problems.
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Ramsey Stochastic Model via Multistage Stochastic Programming
Kaňková, Vlasta
Ramsey model belongs to ``classical" economic dynamic models. It has been (1928)originally constructed (with a farmer interpretation)in a deterministic setting. Later this model has been generalized to a stochastic version. Time horizont in the original deterministic model as well as in modified stochastic one can be considered finite or infinite. The contribution deals with the stochastic model and finite horizont. However, in spite of the classical approach to analyze it we employ a stochastic programming technique. This approach gives a possibility to employ well known results on stability and empirical estimates also in the case of Ramsey model. However, first, we introduce some confidence intervals. To obtain the new assertions we restrict our consideration mostly to the case when the ``underlying" random element follows autoregressive (or at least Markov) sequence.
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