IFIP/IIASA/GAMM-Workshop on
DYNAMIC STOCHASTIC OPTIMIZATION
March 11-14, 2002International Institute for Applied Systems Analysis - IIASA Laxenburg/Vienna, Austria
A B S T R A C T S
SPONSORED BY
- IFIP - International Federation for Information Processing - IIASA - International Institute for Applied Systems Analysis
- GAMM - International Association for Applied Mathematics and Mechanics - UniBwM - Federal Armed Forces University Munich
SCOPE:
The objective of this Workshop is to bring together scientists from Stochastic Programming and from Application areas, dealing with dynamic (multi-stage) optimization problems un-der stochastic uncertainty. The Workshop will cover all aspects of approximations of (linear and nonlinear) stochastic optimization problems for designing appropriate numerical solu-tion procedures for dynamic (multi-stage) stochastic programs. Furthermore, applicasolu-tions in engineering, economics and environment will be treated.
TOPICS:
• Approximation Methods for Dynamic Stochastic Optimization Problems
• Multi-stage Stochastic Programming
• Numerical Methods: Deterministic and Stochastic Methods
• Applications in Engineering, Economics and Environment
INTERNATIONAL PROGRAM COMMITTEE:
Y. Ermoliev (A), P. Kall (CH), K. Marti (D), G. Pflug (A), S. Sen (USA), M.H. van der Vlerk (NL)
HOMEPAGE OF THE WORKSHOP:
http://www.unibw-muenchen.de/campus/LRT/LRT1/MARTI/WS 2/workshop.html CONFERENCE SECRETARIAT:
Ms. Helene Pankl
IIASA - International Institute for Applied Systems Analysis A-2361 Laxenburg/Vienna, Austria
Tel.: +43 2236 807 456 Fax: +43 2236 807 466 e-mail: [email protected]
Arkin, Vadim; Slastnikov, Alexander CEMI Russian Academy of Sciences
Optimal Stopping Problem and Investment Models
The paper is devoted to the description of a new approach to solving an optimal stopping problem for multidimensional diffusion processes. This approach is based on intimate connection between boundary problem for diffusion processes and Dirichlet problem for PDE of an elliptic type (Feynman-Kac formula). The solution of a Dirichlet problem is considered as a functional of the continuation region. The optimization of this functional on the set of all available continuation regions will be carried out by variational methods. Unlike the heuristic “smooth pasting” method the proposed approach allows to obtain, in principle, to find necessary and sufficient conditions for optimality of stopping time in a given class of continuation regions. The approach which was described, is applied to the solving an optimal stopping problem for a two-dimensional geometric Brownian motion with objective functional, which is an expectation of homogeneous (of any non-negative degree) function of the process at the stopping moment. We intend to discuss an application of this optimal stopping problem to real option theory, optimal timing of investment and models of sequential investments.
Aurnhammer, Andreas
Federal Armed Forces University Munich
Real-time Robust Optimal Trajectory Planning of Industrial Robots
The standard engineering approach for optimal control of industrial or service robots as-sumes that all model parameters are exactly known, but due to stochastic variations of the material, manufacturing errors, modeling errors and stochastic variations of the workspace environment (e.g. stochastic payload) they need to be modeled via stochastic methods instead. This leads to optimal control/variational problems under stochastic disturbances, that can not be solved directly but have to be replaced by suitable substitute problems from Stochastic Optimization that can be solved approximately by means of a reduction to a finite nonlinear program. Due to the short cycle times of robots these nonlinear pro-grams have to be solved in real-time in order to update the optimal controls, whenever new information about the uncertain parameters involved is available. A task impossible because of the complexity of the control problem. Hence, a sensitivity analysis based real-time solution strategy is introduced, that takes advantage of an optimal reference solution calculated before-hand off-line and determines a neighboring optimal solution. In this talk some numerical simulation results based on the industrial robot Manutec r3 are presented.
Berglann, H.; Fl˚am, S. D.
University of Bergen and Norwegian School of Economics and Business Ad-ministration
Optimization under Uncertainty using Momentum
Main objects here are stochastic programs, possibly nonconvex. We develop and illustrate an algorithm that combines gradient projection with the heavy-ball method. What emerges is a constrained, stochastic second-order process. Some friction feeds there into - and stabilizes - a myopic mode of stochastic approximation. Convergence obtains under weak and natural conditions. An important condition is that marginal payoff, accumulated along the trajectory, yields a sum which is bounded above.
De´ak, Istv´an
Technical University of Budapest
Computational Experiences with Two-Stage Stochastic Problems: Correlated Normal Variables
Two-stage models belong to the most popular model-family of stochastic programming. Computational techniques to solve problems with continuously distributed random vari-ables are still somewhat week. The case of normally distributed right hand side vector – with correlated components – is treated here. Successive regression approximations tech-nique is used for computing the optimal solution of the problem. Computational issues of the algorithm are discussed and improvements are proposed. Recent numerical results are presented for random right hand side and random matrix T in the second stage problems, having 10-30 normal variables.
Dempster, M. H. A.; Scott, J. E.
Judge Institute of Management, University of Cambridge
Two nonconvex Stochastic Optimization Problems in Finance
This paper studies two nonconvex variants of dynamic strategic financial planning problems concerned with asset liability management for pension funds. These arose as part of a project in this area sponsored by Pioneer Investments. Defining a benchmark portfolio strategy against which more active strategies may be compared leads to the optimization of the fixed-mix portfolio rebalancing strategy – our first nonconvex optimization problem – in terms of the portfolio weights of the fixed mix portfolio. Such a benchmark problem may be used for either pure asset allocation or for asset liability management. Defined contribution pension plans are simply mutual funds with a guaranteed return to investors. From the funds’ viewpoint the guarantees can be met from the current fund level only with some high probability. Violating such a chance constrait involving the probabilistic achievment of the guarantees constitutes a fund liability. This constraint can be modelled for a finite number of data path scenarios as either a MIP scenario counting problem or directly. Both models lead to versions of our second nonconvex NP hard problem. Current progress in modelling and optimization of these two practical problems is reported and computational results are given.
Dentcheva, Darinka; R¨omisch, Werner
Stevens Institute of Technology Hoboken and Humboldt-Universit¨at Berlin
Dualizations, duality gaps and dual decomposition of nonconvex multistage stochastic programs
We consider nonconvex multistage stochastic programs and discuss dualization approaches leading to scenario, nodal and geographical decomposition schemes for computing lower bounds of optimal values in case of a discrete underlying probability distribution. We compare the size of duality gaps for these decomposition approaches and draw some con-clusions on the design of solution methods for applied stochastic integer programming models.
Dupaˇcov´a, Jitka
Charles University, Prague
Reflections on Discrete Time Dynamic Stochastic Decision Problems
When solving a dynamic decision problem under uncertainty it is essential to choose or to build a suitable model taking into account the nature of the real-life problem, the character and availability of the input data, etc. There exist hints when to use stochastic dynamic programming models or multiperiod and multistage stochastic programs. Still, it is difficult to provide a general recipe. With reference to recent papers [1], [2] we shall characterize the main features and basic requirements of these models and indicate the cases which allow for multimodeling and comparisons or for exploitation of different approaches within one decision problem.
References:
[1] Bertocchi, M., Dupaˇcov´a, J., and Moriggia, V.: “Horizon and stages in applications of stochastic programming in finance,” Preprint 2001.
[2] Dupaˇcov´a, J. and Sladk´y, K.: “Comparison of multistage stochastic programs with recourse and stochastic dynamic programs with discrete time, submitted to ZAMM.
Ermoliev, Yuri M.
IIASA, International Institute for Applied Systems Analysis, Laxenburg, Aus-tria
Optimization of Dependent Risk Processes
This talk provides an overview of approaches for the optimization of catastrophic risk processes being under development at IIASA in collaboration with other organizations. A rather general dynamic stochastic optimization problem is formulated as follows [1]-[13]. There are a number of mutually dependent random processes with stopping times. These processes describe the dynamics of losses and gains in a system affected by catastrophic events that generate shocks in the system and, hence, jumps of random processes.
The timing and sizes of jumps depend on geographically explicit patterns of shocks (say, flood damages), which can be controlled by different decisions (say, a dike). In our stud-ies of insurance programs in Russia, Italy [7], [8] and Hungary [10] the random processes describe the accumulation of risk reserves by insurers, catastrophe funds or pools, the pay-ments of governpay-ments and ”individuals” (cells/locations in risk prone areas), etc. Stopping times are associated with the occurrence of a first catastrophe (first shock) and/or the in-solvency of the insurers or pools. Such decisions as to build up a new dike, increase or reduce coverage/compensation of catastrophic losses, create a pool coupled with various reinsurance arrangements, ex-ante (contingent) or ex post credits significantly modify the structure and the jumps of random processes.
The discontinuity of sample performance functions is a key feature of the optimization problem. Stopping times, in general, cause analytical intractability of these functions and possible discontinuity of their expected values. We analyze new optimality conditions and derive non-parametric estimated of ”gradients” for the expectation functionals. For this we use new approximations via random perturbations of parameters. Rare catastrophic events and the coexistence of ex ante and ex post decisions result in skewed and even multimode distributions of sample performance functions that require the use of appropri-ate risk measures. We show that such natural performance functions as ”overpayments” and ”underpayments” have strong connections with the Insolvency constraints, VaR, and CVaR. We discuss Adaptive Monte Carlo optimization methods aimed at adaptive gener-ation of scenarios with respect to the current approximate solution and the reduction of large deviations [2], [6].
References
- 1. Ermolieva, T., Ermoliev Y., Norkin V. 1997, On the role of advanced modeling in managing catastrophic risks, in Drottz-Sjberg, B.-M. (ed.). Proc. New Risk Frontiers: Conference for the 10th Anniversary of the Society for Risk Analysis -Europe, The Center for Risk Research, Stockholm, pp. 68-74.
- 2. Ermolieva, T., The Design of Optimal Insurance Decisions in the Presence of Catastrophic Risks. IIASA Interim Report IR-97-068, Int. Inst. For Applied Systems Analysis, Laxenburg, Austria, 1997.
- 3. Ermoliva, T., Ermoliev, Y., Norkin, V., Spatial Stochastic Model for Optimization Capacity of Insurance Networks Under Dependent Catastrophic Risks: Numerical Experiments. IIASA Interim Report, IR-97- 028, 1997.
- 4. Digas, B., Ermoliev, Y., Kryazhimskii, A., Guaranteed Optimization in Insurance of Catastrophic Risks, IIASA Interim Report IR-98-082, Int. Inst. For Applied Systems Analysis, Laxenburg, Austria, 1998.
- 5. Ermoliev, Y., Norkin, V., On Nonstationary Law of Large Numbers for Dependent Random Variables and its Application in Stochastic Optimization, Cybernetics and Systems Analysis, 34(4), 1998.
- 6. Ermoliev, Y., Ermolieva, T., MacDonald, G., Norkin, V., Insurability of catas-trophic risks: the stochastic optimization model, Optimization Journal, Volume 47, 2000, pp. 251-265.
- 7. Amendola, A., Ermoliev, Y., Ermolieva, T., Gitis V., Koff, G., Linnerooth- Bayer, J., 2000. A Systems Approach to Modeling Catastrophic Risk and Insurability. Nat-ural Hazards, 21: 2/3, 381-393
- 8. Amendola A., Ermoliev Y., Ermolieva T., Earthquake Risk Management: A Case Study for an Italian Region. In the Proceedings of the ESREL Conference ”Towards a safer world”, Eds. E. Zio, M. Demichela and N. Piccinini, Torino, 2001.
- 9. Ermoliev, Y., Ermolieva, T., MacDonald, G., Norkin, V., Stochastic Optimization of Insurance Portfolios for Managing Exposure to Catastrophic Risks. Annals of Operations Research, 99, 207-225, 2000.
- 10. Ermolieva, T., Ermoliev, Y., Linnerooth-Bayer, J., Galambos, I., The Role of Financial Instruments in Integrated Catastrophic Flood Management. In the Proceedings of the 8-th Annual Conference of the Multinational Financial Society, Garda, Italy, July 2001, (forthcoming in the Multinational Finance Journal).
- 11. Ermoliev, Y., Flam, S., Finding Pareto Optimal Insurance Contracts, IIASA Interim IR-00-033, Int. Inst. For Applied Systems Analysis, Laxenburg, Austria, 2000, (forthcoming in Geneva Papers Journal, Theory and Methods).
- 12. Ermoliev, Y., Flam, S., On Mutual Insuracne, IIASA Interim IR-00-002, Int. Inst. For Applied Systems Analysis, Laxenburg, Austria, 2000.
- 13. Ermoliev, Y., Norkin, V., Stochastic Optimization of Risk Functionals via Para-metric Smoothing, (forthcoming).
Frauendorfer, Karl; Sch¨urle, Michael University of St. Gallen
The Approximation of Multistage Stochastic Programs
We consider approximation techniques within multistage stochastic programming. Exact upper and lower bounds may be derived by exploiting the saddle property of value func-tions. This allows to analyze the discretization error in the nodes of the corresponding scenario trees. Tight bounds can be achieved if the support of the underlying random data is partitioned into subcells. We discuss computational issues and present some new results in the context of financial decision problems under uncertainty.
Henrion, Ren´e
Weierstrass-Institute Berlin, Germany
Polyhedral Chance Constraints: Computational and Stability Aspects
Many problems of engineering can be modeled by means of polyhedral chance constraints of the type
P(A(x)ξ≤b(x))≥p,
whereP is a probability measure,ξ is a random vector,xis a vector of decisions,A(x), b(x) are (nonlinear) matrix or vector functions, respectively, and p∈ (0,1) is some probability measure. We present a computational approach (joint work with T. Szantai (Budapest) and J. Bukszar (Miskolc)) for dealing with such constraints which - at least in moderate dimensions - appears to be quite efficient in comparison with existing alternatives. Fur-thermore, the issue of solution stability w.r.t. perturbations of the probability measure is discussed for problems involving polyhedral chance constraints.
Kallio, Markku
Helsinki School of Economics
Real Option Valuation via Stochastic Optimization
A multi-stage stochastic optimization model is presented to aid valuation of real as well as financial options. An option is specified by a compact set of alternative cash flow streams over the scenario tree of the stochastic program. The allowable choices in the model are specified by the set of option cash flows under consideration as well as by investments in competing assets, such as publicly traded financial instruments. Given preferences of the investor, the model determines simultaneously an optimal strategy for utilizing the option as well as an optimal investment strategy in competing assets. Option value is determined as the maximum price which the investor is willing to pay for the option. Such value is consistent with arbitrage theory of financial derivatives. If the competing assets constitute a complete market within the scenario tree, then the option valuation results from a dynamic programming recursion. For illustration, we present application to the forest industries.
Kaut, Michal
IØT NTNU, Trondheim, Norway
A Heuristic for Generating Scenario Trees for SP
The presented results are a join work with Stein W. Wallace from Molde University College and Kjetil Høyland from Gjensidige NOR Asset Management.
In stochastic programming models we always face the problem of how to represent the uncertainty. When dealing with multidimensional distributions, the problem of generating scenarios can itself be difficult.
We will present a heuristic algorithm for generating scenario trees with specified first four moments and correlations. The algorithm generates a discrete distribution specified by the first four marginal moments and correlations. The scenario tree is constructed by decom-posing the multivariate problem into univariate ones, and using an iterative procedure that combines simulation, Cholesky decomposition and various transformations, to achieve the correct correlations without changing the marginal moments.
Our testing shows that the algorithms is faster than the Høyland and Wallace method. The speed-up increases with the size of the tree and is more than 100 in a case of 20 random variables and 1000 scenarios. This allows us to generate far larger trees and solve thus more realistical problems.
King, Alan; Pennanen, Teemu
Mathematical Sciences Department IBM Thomas J. Watson Research Center and Helsinki School of Economics
Non-parametric Calibration and Pricing of Contingent Claims using Stochastic Programming
Theorems that link the valuation of contingent claims to the existence of a martingale measure on the price process of the underlying assets can be easily developed using an approach based on stochastic programming duality. This approach can be extended to a practical method for the simultaneous market calibration and pricing of a claim, based on the solution of a stochastic program related to the problem of finding a risk-neutral martingale measure that both explains observed market prices and optimizes the risk-neutral expected value of the claim. The only input to the method is the actual market quotes for the bid-ask prices for the options series on the underlyings. Tests of the method using Nokia options data will also be presented.
Korf, Lisa A.
Department of Mathematics, University of Washington
Stochastic Programming, Singular Multipliers, and Applications to Finance This lecture will explore the application and interpretation of stochastic programming duality to the problem of pricing contingent claims. In particular, multipliers in the dual of L∞ have an L1 component and a singular component, each serving as part of a pricing
measure. The singular component, which is known to multiply the ‘induced constraints’, may be interpreted as the extra price associated with exceeding a certain ratio relating the contingent claim to the market. The analysis seems to lead to classes of contingent claims which may be priced via an attainable linear pricing rule in incomplete markets. This is work in progress.
Marti, Kurt
Federal Armed Forces University Munich
Stochastic Linear Programming Methods in Plastic Analysis and Optimal Plas-tic Structural Design Under StochasPlas-tic Uncertainty
Problems from plastic limit or shakedown analysis and optimal plastic design are based on the convex yield criterion and the linear equilibrium equation for the stress state vector. Having to take into account stochastic variations of the model parameters, e.g. yield stresses, external loadings, cost coefficients, etc., the basic stochastic plastic analysis or optimal plastic design problem must be replaced by an appropriate deterministic substitute problem in order to get robust optimal solutions. For this purpose, the existence of a statically admissible (safe) stress state vector is described by means of a scalar function s* being the minimum value function of a convex or linear program based on the conditions of plasticity theory for a safe stress state. Depending on the chosen cost structure, the failure costs c* can be described then by the minimum value of a related convex or linear program. The properties of the functions s* and c* are discussed. The minimization of the expected total costs (initial/primary and failure costs), can be described then in the framework of a Stochastic Optimization Problem with Recourse. Linearizing piecewise the yield condition, a ”Stochastic (Linear) Program (SLP) with Complete Fixed Recourse” is obtained. For discretely distributed random parameters, e.g., after a discretization of a more general distribution of the parameters, and a possible linearization with respect to the vector of design variables, a linear program (LP) with a so-called dual decomposition data structure is obtained. For an LP of this type many efficient special purpose LP-solvers are available, where the (global) equilibrium matrix C may be determined for different types of structures by a special FORTRAN module.
Mayer, J´anos
Institute for Operations Research, University of Z¨urich
Modeling Support for Multistage Recourse Problems
We consider multistage recourse problems (MSR) from the modeling point of view and fo-cus on the additional features which appear when passing from a two stage problem to the multistage case. As a typical modeling situation let us consider a deterministic dynamic LP problem. There are several possibilities for imposing a decision stages structure upon this LP when building stochastic variants of it, e.g. multiperiod two stage problem, MSR problem where the decision stages coincide with the time periods and MSR problems where the decision stages correspond to aggregated time periods. In the discretely distributed case the joint distribution in an MSR problem should be handled as a scenario tree and for continuous distributions discretization procedures (called scenario generation) should be provided. Another distinguishing feature is that multistage problems are much harder from the numerical point of view as their two stage counterparts. Clearly a modeling system for MSR should include facilities for supporting these features.
Recently we have extended the capabilities of our model management system SLP–IOR to include also the MSR case. We endow SLP–IOR with the facilities discussed above in a step–by–step fashion. In our lecture we give an outline of the overall system design for the MSR case and report in detail on the features which have been implemented corresponding to the current stage of development.
Medova, E. A.
Judge Institute of Management, University of Cambridge
Corporate Risk Management: A Stochastic Programming Approach
Corporate risk management serves the unique requirements of a company’s business by looking at risks, both financial and operational, on an integrated basis in relation to the decision-making process. For oil companies it involves the transformation of complete de-pendence on spot oil prices into a variety of exposures to the forward, futures, options and swap markets, i.e. a choice of hedging policy. It also involves the choice of the optimal levels for physical activities such as supply, storage, transportation and refining. The rela-tion between suchasset-lightand asset-heavy activities constitute the integrated corporate strategic plan corresponding to a certain risk management objective. In this presentation we propose an integrated corporate policy (CORPLAN) model for joint optimization of real and financial activities. The volatile trading environment is generated by simulating future and spot prices for both contango and backwardation regimes of the commodity futures markets. Random customer demand is simulated across a number of locations ad-ditional to demands which are fixed by forward supply contracts. The generic stochastic programming software STOCHASTICSTMhas been used to solve a number of instances of the model in a reasonable time. Our model provides robust current decisions in the pres-ence of price and demand uncertainty, and demonstrates the advantages in using derivative contracts for risk management.
Norkin , Vladimir
Glushkov Institute of Cybernetics, Kiev, Ukraine
Constrained recursive kernel density and regression estimation by stochastic gradient method
There is an extensive literature on nonparametric density and regression estimation (Parzen (1962), Nadaraya (1983), Devroy and Gyorfi (1985), Silverman (1986), Katkovnik (1985), Vapnik (1988), H¨ardle (1990), Scott (1992), Wets (1998), Yatchew (1998) and others). Recursive kernel density estimators were considered by Yamato (1971), Davies (1973), De-vroy (1979), Isogai (1982) and recursive kernel regression estimates were studied by Revesz (1976, 1977), Tsybakov (1983). Recursive estimation requires minimum computational efforts and makes minimum demands to computer memory and thus allows to update estimates with each new observation at a dense net of points. Our approach to nonpara-metric estimation is close to recursive kernel estimation but is based on nonstationary (limit) stochastic optimization (Ermoliev and Nurminski (1973), Gaivoronski (1978)) and a stochastic nonmonotonic version of Lyapunov function method (Ermoliev and Norkin (1998)). We formulate nonparametric (kernel) estimation problem through a family of some constrained stochastic optimization problems, and thus take into account a priori knowledge (bounds and other relations) on the regression values to obtain more robust estimates. To solve these optimization problems we apply stochastic quasi-gradient (SQG) method (Ermoliev (1976)) and get SQG-estimates. We prove strong point-wise consistency of the obtained SQG-estimators by means of stochastic Lyapunov function method. To get more precise and stable estimates we apply Ces`aro averaging to SQG-estimates, that is a common technique in stochastic optimization (Nemirovski and Yudin (1983), Mikhalevich, Gupal and Norkin (1987), Uryas’ev (1990)) but has not been applied before in nonpara-metric estimation, and then obtain accuracy and rate of convergence of Ces`aro estimates for any adjustment and smoothing parameters (even not necessarily tending to zero) and any number of observations. On this basis we find optimal values of parameters that give maximal rate convergence for asymptotic and nonasymptotic cases.
The report is partly based on a forthcomming paper by Ermoliev, Keyser and Norkin (2002).
Pennanen, Teemu
Helsinki School of Economics
An Approach to Scenario Generation
The objective of this talk is to discretize a linear time series model in a manner suitable for stochastic programming. Our approach is based on numerical quadratures developed for high-dimensional integration. We obtain a procedure that is just as easy to implement as the traditional Monte Carlo method, but which tends to cover the underlying probability space more uniformly. Preliminary numerical results are presented.
Pflug, Georg
Department of Statistics and Decision Support Systems, University of Vienna, Austria
Risk Measures for Multiperiod Stochastic Incomes
A new type of risk measure is introduced for a sequence of random incomes adapted to some filtration. This measure is formulated as the optimal net present value of a stream of adaptively planned commitments for consumption.
The calculation of the new measure done by solving a stochastic dynamic linear optimiza-tion problem, which, in case of a finite filtraoptimiza-tion, reduces to a simple deterministic linear program.
We show properties of the new measure by exploiting the convexity and duality structure of the stochastic dynamic linear problem. The measure depends on the full distribution of the income process (not only on its marginal distributions) as well as on the filtration, which is interpreted as the information available on the future.
Robinson, Stephen M.
University of Wisconsin-Madison, Madison, WI 53706 USA
Approximation and Optimization for Stochastic Networks
Operations research analysts frequently confront problems in which they must recommend ways to improve a networked, interacting system containing significant uncertainties. One example of this situation is a transportation network in which the time needed to move personnel or cargo through the network is uncertain because of factors such as breakdowns, or limited maintenance or service facilities at intermediate points. Given a limited budget for improving the performance of such a system, how should it be allocated to give the best improvement?
One of the most useful tools in analyzing such systems is stochastic simulation, but if one wants to improve the system rather than just to predict its performance ’as-is’ then repeated simulations are usually necessary. If the system is complex these simulations often require long running times, and therefore such analyses can require very large amounts of time.
We will describe an experimental method using a two-phase approach, with the aim of improving or optimizing the network in much less time. The first phase uses stochastic network approximations in place of repeated simulations to predict good ways to improve the network’s performance, while the second phase uses one simulation run to validate the predicted improvement. The method can stop there, or it can iterate toward complete optimization by repeating these two phases. We will outline the procedure, describe pre-liminary work on analyzing it, and present results of some tests on airlift networks, done in joint work with Julien Granger and Ananth Krishnamurthy. These have shown good improvement capability with a very substantial reduction in running times as compared to standard repeated simulation methods.
Ruszczynski, Andrzej Rutgers University
Stochastic Dominance and Mean–Risk Models
We analyze relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean–risk approaches.
We develop new necessary conditions for stochastic dominance. These conditions compare values of a certain functional, which contains two components: the expected value of a random outcome and a risk term charaterizing the variability of the outcomes. Two groups of conditions will be presented: primal and dual.
In the primal conditions the risk is represented by the central semideviation of the cor-responding degree. If the weight of the semideviation in the composite objective does not exceed the weight of the expected value, the maximization of such a functional yields solutions which are efficient in terms of stochastic dominance.
Next, by exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions. This allows us to show that several models using quantiles and tail characteristics of the distribution as risk measures are in harmony with the stochastic dominance relation, if the weight of the risk term in the composite objective is bounded by one.
We also provide stochastic linear programming formulations of all these models and analyse their properties.
Sladk´y, Karel
Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic
Variance Penalized Stochastic Optimization
Mean variance selection rules were originally proposed for the portfolio selection problem. Following the mean variance selection rule, the investor selects from among a given set of investment alternatives only investments with a higher mean and lower variance than a member of the given set.
In this contribution we investigate how the mean variance selection rule can work in dis-crete dynamic stochastic models. To this end we adapt notions and notations used in Markov decision processes and in contrast to the classical models (where usually only the expected long run discounted reward or mean reward is minimized or maximized) we shall also consider variance of the obtained total reward. Alternative definitions of the reward variance along with their mutual connections will be discussed. Finally, attention will be focused on finding policies minimizing the long run reward variance on condition that the mean reward is not less than a given value.
Steinbach, Marc
Konrad-Zuse-Zentrum f¨ur Informationstechnik Berlin
Solving Stochastic Programs with Automatically Generated Sparse Factorizations
Dynamic stochastic programs exhibit diverse sparsity patterns determined by information structure and dynamics (block level) and by model-specific properties (sub-block level). We distinguish three block-level scenario tree formulations that yield favorable sparsity in the KKT systems of interior methods. To increase performance of the computationally expensive KKT factorization, we propose the automatic generation of source code that implements model-specific data structures and node operations. Experience with a financial engineering problem will be presented.
Sz´antai, Tam´as; Long, Janmin; Pr´ekopa, Andr´as
Budapest University of Technology and Economics and Rutgers University
New Bounds and Approximations for the Probability Distribution of the Length of the Critical Path
We consider a PERT network and assume that the durations of the activities are bounded random variables. The number of all paths between the origin and terminal nodes, that we obtain by enumeration, is usually very large. We present methods that eliminate those which may not become critical under any realization of the random activity lengths. Numerical experimentation shows that in many cases only a few paths remain to compete for being critical. We present lower and upper bounds for the probability distribution function of the critical path length, using some classical and recently developed probability bounds. Assuming that the joint probability distribution of the lengths of these paths is approximately multivariate normal, we also present a method to approximate the values of the same function. The paper concludes with illustrations and reports on computational results.
Vladimirou, Hercules; Topaloglou, Nikolas; Zenios, Stavros A.,
HERMES Center of Excellence on Computational Finance and Economics, University of Cyprus
Stochastic Programming Models for International Portfolio Management We develop stochastic programming models for managing multicurrency investment portfo-lios in the context of scenario analysis. Scenario sets that depict discrete joint distributions for the uncertain asset returns and exchange rates constitute the necessary inputs to the stochastic optimization models. The scenario generation procedure is based on principles of moment matching and is calibrated using historical market data. The scenario-based op-timization models incorporate alternative selective hedging strategies. Thus, the portfolio optimization models determine jointly the portfolio composition and the appropriate level of currency hedging for each market via forward currency exchange contracts. The mod-els apply the conditional value-at-risk (CVaR) metric to control total risk exposure. The performance of alternative models on international portfolios of stock and bond indices is assessed through empirical experiments using real market data. The models are contrasted with respect to the impact of hedging policies on the risk-return profiles of portfolios in static tests, and their ex-post realized in backtests over past time periods. 1
van der Vlerk, Maarten H. University of Groningen
Simplification of Recourse Models by Modification of Distributions
We show that several integer recourse models can be approximated by continuous recourse models. Moreover, we show that multiple simple recourse models can be represented as simple recourse models.
In each case, the simplified model is obtained by applying a suitable transformation to the distribution of the random right-hand side parameters.
Wallace, Stein W.
Molde University College, Servicebox 8, N-6405 Molde
Scenario Generation - What makes a Scenario Tree good?
The purpose of this talk is to discuss our practical experience with scenario tree generation in different applications. On one hand, there is the issue of efficiently creating trees. On the other hand, the issue of whether or not the tree is a good one – in particular what we should mean by a good tree.
Wets, Roger
University of California, Davis
Serious Zero-curves
All valuations (discounted cash flow, instrument pricing, option pricing) and other financial calculations require an estimate of the evolution of the risk-free rates as implied by the term structure. This presumes that one has, if not complete knowledge, at least a very good estimate of the term structure, for the so-called zero-curves (spot and forward rate curves, discount factor curve, etc.). This paper is concerned with the methodology of deriving these zero-curves. It reviews the methodology and the limitations of standard BootStrapping and proposes a quite different approach based on Approximation Theory that has been implemented at EpiSolutions Inc.
Ziemba, William T.
University of British Columbia, Vancouver, B.C., Canada
The Innovest Austrian Pension Fund Financial Planning Model InnoALM This talk describes the financial planning model InnoALM developed by Innovest for Aus-trian pension funds including their own managed for the AusAus-trian employees of the elec-tronics firm Siemens. The model is one tool in the analysis of the growing worldwide problem of ageing and the growing number of pensioners in an environment of increased demand for government services such as national pensions. The model uses a multiperiod stochastic linear programming framework with a flexible number of time periods of varying length. Various forecasting models yield inputs that provide the generation and aggregation of multiperiod discrete probability scenarios for random return and other model param-eters. The correlations across asset classes, of bonds, stocks, cash and other financial instruments, are scenario dependent using multiple covariance matrices that correspond to differing market conditions. This feature allows InnoALM to anticipate and react to severe as well as normal market conditions. Austrian pension law and policy considerations are modeled as constraints in the optimization. The concave risk averse preference function is to maximize the expected present value of terminal wealth at the specified horizon net of expected convex (piecewise linear) penalty costs for wealth and benchmark targets in each decision period. InnoALM has a user interface that allows for visualization of key model outputs, the effect of input changes, growing pension benefits from increased deterministic wealth target violations, stochastic benchmark targets, security reserves, policy changes, etc. The solution process using the IBM OSL stochastic programming code is fast enough to generate virtually online decisions and results and allows for easy interaction of the user with the model to improve pension fund performance.
