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  • 1
    Online Resource
    Online Resource
    Intertemporal Surplus Management (2000)
    Berlin : Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
    Details
    UID:
    edochu_18452_8875
    Format: 1 Online-Ressource (19 Seiten)
    Series Statement: Stochastic Programming E-Print Series 1999,2000,7
    Content: This paper presents an intertemporal portfolio selection model for pension funds that maximize the intertemporal expected utility of the surplus of assets net of liabilities. Following Merton (1973) it is assumed that both the asset and the liability return follow Ito processes as functions of a state variable. The optimum occurs for investors holding four funds: the market portfolio, the hedge portfolio for the state variable, the hedge portfolio for the liabilities, and the riskless asset. It is shown that pension funds should purchase hedging for liabilities. In contrast to Merton's result in the assets only case, this hedge depends exclusively on the funding ratio of a specific pension fund and not on preferences. With HARA utility the investments in the state variable hedge portfolios are also preference independent. With log utility the market portfolio investment depends only on the current funding ratio.
    Language: English
    DOI: 10.18452/8223
    URN: urn:nbn:de:kobv:11-10057396
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  • 2
    Book
    Book
    On Rate of Convergence of Optimal Solutions of Monte Carlo Approximations of Stochastic Programs (1999)
    Berlin : Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
    Details
    UID:
    edochu_18452_8870
    Series Statement: Stochastic Programming E-Print Series 1999,1999,2
    Content: In this paper we discuss Monte Carlo simulation based approximations of a stochastic programming problem. We show that if the corresponding random functions are convex piecewise smooth and the distribution is discrete, then (under mild additional assumptions) an opitmal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size. Moreover, by using theory of Large Deviations, we show that the probability of such an event approaches one exponentially fast with increase of the sample size. In particular, this happens in the case of two stage stochastic programming with recourse if the corresponding distributions are discrete. The obtained results suggest that, in such cases, Monte Carlo simulation based methods could be very efficient. We present some numerical examples to illustrate the involved ideas.
    Language: English
    DOI: 10.18452/8218
    URN: urn:nbn:de:kobv:11-110-18452/8870-9
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  • 3
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    Online Resource
    On two-stage convex chance constrained problems (2006)
    Berlin : Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
    Details
    UID:
    edochu_18452_9005
    Format: 1 Online-Ressource (26 Seiten)
    Series Statement: Stochastic Programming E-Print Series 2006,2006,4
    Content: In this paper we develop approximation algorithms for two-stage convex chance constrainedproblems. Nemirovski and Shapiro [16] formulated this class of problems and proposed anellipsoid-like iterative algorithm for the special case where the impact function f (x, h) is bi-affine.We show that this algorithm extends to bi-convex f (x, h) in a fairly straightforward fashion.The complexity of the solution algorithm as well as the quality of its output are functions of theradius r of the largest Euclidean ball that can be inscribed in the polytope defined by a randomset of linear inequalities generated by the algorithm [16]. Since the polytope determining ris random, computing r is diffiult. Yet, the solution algorithm requires r as an input. Inthis paper we provide some guidance for selecting r. We show that the largest value of r isdetermined by the degree of robust feasibility of the two-stage chance constrained problem –the more robust the problem, the higher one can set the parameter r. Next, we formulate ambiguous two-stage chance constrained problems. In this formulation,the random variables defining the chance constraint are known to have a fixed distribution;however, the decision maker is only able to estimate this distribution to within some error. Weconstruct an algorithm that solves the ambiguous two-stage chance constrained problem whenthe impact function f (x, h) is bi-affine and the extreme points of a certain “dual” polytope areknown explicitly.
    Language: English
    DOI: 10.18452/8353
    URN: urn:nbn:de:kobv:11-10066216
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  • 4
    Online Resource
    Online Resource
    Random lsc functions : Scalarization (2000)
    Berlin : Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
    Details
    UID:
    edochu_18452_8883
    Format: 1 Online-Ressource (14 Seiten)
    Series Statement: Stochastic Programming E-Print Series 2000,2000,7
    Content: Random lsc (lower semicontinuous) functions can be indentified with a vector-valued random variable by means of an appropriate scalarization. It is shown that stationarity, ergodicity and independence properties are preserved by this scalarization. The scalarization is exploited to obtain an lsc version of the conditional expectation of a random lsc function.
    Language: English
    DOI: 10.18452/8231
    URN: urn:nbn:de:kobv:11-10057569
    URN: urn:nbn:de:kobv:11-10057579
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  • 5
    Book
    Book
    A Dynamic Asset Allocation Model with Downside Risk Control (2000)
    Berlin : Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
    Details
    UID:
    edochu_18452_8876
    Series Statement: Stochastic Programming E-Print Series 1999,2000,8
    Content: This paper presents a new stochastic model for investment. The investor's objective is to maximize the expected growth rate while controlling for downside risk. Assuming lognormally distributed prices, the strategy that determines the o optimal dynamic portfolio weights by changing risk neutral excess rate is determined by a stochastic differential equation. The maximum loss can be limited almost surely. A constrained optimization model is developed given investors' preference on the minimum subsistence reward among all possible scenarios. The relative changes in the expected terminal wealth, minimum subsistence and the risk aversion are studied. Taking VaR as the risk measure, the return/risk tradeoff efficient frontier is constructed. A comparison of the downside risk control model for a typical example to Buy and Hold (BH) and Fixed Mix (FM) strategic asset allocation models shows that the downside risk control model has superior performance in the return/VaR framework.
    Language: English
    DOI: 10.18452/8224
    URN: urn:nbn:de:kobv:11-110-18452/8876-4
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  • 6
    Online Resource
    Online Resource
    Stochastic programming by Monte Carlo simulation methods (2000)
    Berlin : Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
    Details
    UID:
    edochu_18452_8878
    Format: 1 Online-Ressource (30 Seiten)
    Series Statement: Stochastic Programming E-Print Series 2000,2000,3
    Content: We consider in this paper stochastic programming problems which can be formulated as an optimization problem of an expected value function subject to deterministic constraints. We discuss a Monte Carlo simulation approach based on sample average approximations to a numerical solution of such problems. In particular, we give a survey of a statistical inference of the sample average estimators of the optimal value and optimal solutions of the true problem. We also discuss stopping rules and a validation analysis for such sample average approximation optimization procedures and give some illustration examples.
    Language: English
    DOI: 10.18452/8226
    URN: urn:nbn:de:kobv:11-10046191
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  • 7
    Online Resource
    Online Resource
    Variable-sample methods and simulated annealing for discrete stochastic optimization : (revised version) (2000)
    Berlin : Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
    Details
    UID:
    edochu_18452_8879
    Format: 1 Online-Ressource (36 Seiten)
    Series Statement: Stochastic Programming E-Print Series 2000,2000,4
    Content: In this paper we study a modifcation of the well-known simulated annealing method, adapting it to discrete stochastic optimization problems. Our algorithm is based on a variable-sample Monte Carlo technique, in which the objective function is replaced, at each iteration, by a sample average approximation. The idea is to obtain independent estimates of the objective function, to avoid getting "trapped" in a single sample-path. We first provide general results under which variable-sample methods yield consistent estimators as well as bounds on the estimation error. Then, we concentrate on the simulated annealing algorithm, and derive a proper schedule of sample sizes that guarantees convergence of the overall algorithm w.p.1. Because the convergence analysis is done sample-path wise (by means of the law of the iterated logarithm), we are able to obtain our results in a exible setting, which includes the possibility of using different neighborhood structures and different sampling distributions along the algorithm, without making strong assumptions on the underlying distributions.
    Language: English
    DOI: 10.18452/8227
    URN: urn:nbn:de:kobv:11-10046200
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  • 8
    Online Resource
    Online Resource
    Creating Synthetic Option Strategies for Asset Allocation with Transaction Costs Using Multi-Period Stochastic Programming (1999)
    Berlin : Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
    Details
    UID:
    edochu_18452_8869
    Format: 1 Online-Ressource (22 Seiten)
    Series Statement: Stochastic Programming E-Print Series 1999,1999,1
    Content: We discuss a new approach to asset allocation with transaction costs. A multi-period stochastic linear programming model is developed where the risk is based on the worst case payoff which is endogenously determined by the model. Utilizing portfolio protection and dynamic hedging, an investment strategy similar to that of a "multiple asset option" on the initial investment portfolio is characterized. The relative changes in the expected terminal wealth, planning target and risk aversion are studied theoretically and illustrated by a numerical example. This model dominates a static mean-variance model when the optimal portfolio is measured by the Sharpe ratio.
    Language: English
    DOI: 10.18452/8217
    URN: urn:nbn:de:kobv:11-10057235
    URN: urn:nbn:de:kobv:11-10057244
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  • 9
    Book
    Book
    Power management in a hydro-thermal system under uncertainty by Lagrangian relaxation (2000)
    Berlin : Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
    Details
    UID:
    edochu_18452_8880
    Series Statement: Stochastic Programming E-Print Series 2000,2000,1
    Content: We present a dynamic multistage stochastic programming model for the cost-optimal generation of electric power in a hydro-thermal system under uncertainty in load, inflow to reservoirs and prices for fuel and delivery contracts. The stochastic load process is approximated by a scenario tree obtained by adapting a SARIMA model to historical data, using empirical means and variances of simulated scenarios to construct an initial tree, and reducing it by a scenario deletion procedure based on a suitable probability distance. Our model involves many mixed-integer variables and individual power unit constraints, but relatively few coupling constraints. Hence we employ stochastic Lagrangian relaxation that assigns stochastic multipliers to the coupling constraints. Solving the Lagarangian dual by a proximal bundle method leads to successive decomposition into single thermal and hydro unit subproblems that are solved by dynamic programming and a specialized descent algorithm, respectively. The optimal stochastic multipliers are used in Lagrangian heuristics to construct approximately optimal first stage decisions. Numerical results are presented for realistic data from a German power utility, with a time horizon of one week and scenario numbers ranging from 5 to 100. The corresponding optimization problems have up to 200,000 binary and 350,000 continuous variables, and more than 500,000 constraints.
    Language: English
    DOI: 10.18452/8228
    URN: urn:nbn:de:kobv:11-110-18452/8880-5
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  • 10
    Online Resource
    Online Resource
    Optimization of simultaneous power production and trading by stochastic integer programming (2002)
    Berlin : Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, Institut für Mathematik
    Details
    UID:
    edochu_18452_8925
    Format: 1 Online-Ressource (9 Seiten)
    Series Statement: Stochastic Programming E-Print Series 2002,2002,9
    Content: We develop a two-stage stochastic integer programming model for the simultaneous optimization of power production and day-ahead power trading. The model rests on mixed-integer linear formulations for the unit commitment problem and for the price clearing mechanism at the power exchange. Foreign bids enter as random components into the model. We solve the stochastic integer program by a decomposition method combining Lagrangian relaxation of nonanticipativity with branch-and-bound in the spirit of global optimization. Fianlly, we report some first computational experiences.
    Language: English
    DOI: 10.18452/8273
    URN: urn:nbn:de:kobv:11-10058419
    URN: urn:nbn:de:kobv:11-10058424
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