G. Ch. Pflug

Scenario tree generation for multiperiod financial optimization by optimal discretization

Abstract.Multiperiod financial optimization is usually based on a stochastic model for the possible market situations. There is a rich literature about modeling and estimation of continuous-state financial processes, but little attention has been paid how to approximate such a process by a discrete-state scenario process and how to measure the pertaining approximation error.¶In this paper we show how a scenario tree may be constructed in an optimal manner on the basis of a simulation model of the underlying financial process by using a stochastic approximation technique. Consistency relations for the tree may also be taken into account.

Stochastic optimization and statistical inference

Abstract If the distribution of the random parameters of a stochastic program is unknown, the empirical distribution based on a sample may be used as a proxy. This empirical approximation is related to the “true” stochastic program in the same way as a statistical estimate is related to the true parameter value. Properties of statistical estimators, like consistency, asymptotical distributions and the construction of confidence regions are reviewed in the realm of stochastic optimization. The entropic size of a stochastic program determines the quality of the approximation. In case that random constraints are present, the notion of epiconvergence replaces in a natural way the notion of uniform convergence of functions. The asymptotic structures are described by the asymptotic stochastic program associated to the sequence of empirical programs.

Optimal stochastic single-machine-tardiness scheduling by stochastic branch-and-bound

Abstract A stochastic branch-and-bound technique for the solution of stochastic single-machine-tardiness problems with job weights is presented. The technique relies on partitioning the solution space and estimating lower and upper bounds by sampling. For the lower bound estimation, two different types of sampling (“within” and “without” the minimization) are combined. Convergence to the optimal solution (with probability one) can be demonstrated. The approach is generalizable to other discrete stochastic optimization problems. In computational experiments with the single-machine-tardiness problem, the technique worked well for problem instances with a relatively small number of jobs; due to the enormous complexity of the problem, only approximate solutions can be expected for a larger number of jobs. Furthermore, a general precedence rule for the single-machine scheduling of jobs with uncertain processing times has been derived, essentially saying that “safe” jobs are to be scheduled before “unsafe” jobs.

Gradient estimates for the performance of Markov chains and discrete event processes

In this paper, an algorithm for the estimation of the gradient of the stationary performance of a Markov chain w.r.t. a real parameter is presented. The method works for discrete and continuous state spaces. A comparison with the efficient score method and extensions to semi-Markov processes and discrete event dynamical systems (DEDS) are made.

The AURORA financial management system: Model and parallel implementation design

The AURORA financial management system under development at the University of Vienna is a modular decision support tool for portfolio and asset–liability management. It is based on a multivariate Markovian birth-and-death factor model for the economic environment, a pricing model for the financial instruments and an objective function which is flexible enough to express risk aversion.The core of the system is a large scale linear or convex program, which due to its size and structure is well suited for parallel optimization methods.As the system is still at an early stage of development, the results are preliminary in nature. Only a few types of financial instruments are handled and just two types of objectives are considered. The parallel optimization modules are still in the development phase.

The asymptotic contour process of a binary tree is a Brownian excursion

The contour process of a random binary tree t with n internal nodes is defined as the polygonal function constructed from the heights of the leaves of t (normalized by ). We show that, as n --> [infinity], the limiting contour process is identical in distribution to a Brownian excursion.

Selected parallel optimization methods for financial management under uncertainty

A review of some of the most important existing parallel solution algorithms for stochastic dynamic problems arising in financial planning is the main focus of this work. Optimization remains the most difficult, time and resource consuming part of the process of decision support for financial planning under uncertainty. However, other parts of a specialized decision support system (DSS) are also briefly outlined to provide appropriate background. Finally, financial modeling is but one of the possible application fields of stochastic dynamic optimization. Therefore the same fairly general methods described here are also useful in many other contexts. Authors hope that the overview of this application field may be of interest to readers concerned with development of parallel programming paradigms, methodology and tools. Therefore special care was taken to ensure that the presentation is easily understandable without much previous knowledge of theory and methods of operations research.

Multi-Stage Stochastic Electricity Portfolio Optimization in Liberalized Energy Markets

In this paper we analyze the electricity portfolio problem of a big consumer in a multi-stage stochastic programming framework. Stochasticity enters the model via the uncertain spot price process and is represented by a scenario tree. The decision that has to be taken is how much energy should be bought in advance, and how large the exposition to the uncertain spot market, as well as the relatively expensive production with an own power plant should be. The risk is modeled using an Average Value-at-Risk (AVaR) term in the objective function. The results of the stochastic programming model are compared with classical fix mix strategies, which are outperformed. Furthermore, the influence of risk parameters is shown.

Optimal allocation of simulation experiments in discrete stochastic optimization and approximative algorithms

Approximate solutions for discrete stochastic optimization problems are often obtained via simulation. It is reasonable to complement these solutions by confidence regions for the argmin-set. We address the question how a certain total number of random draws should be distributed among the set of alternatives. Two goals are considered: the minimization of the costs caused by using a statistical estimate of the true argmin, and the minimization of the expected size of the confidence sets. We show that an asymptotically optimal sampling strategy in the case of normal errors can be obtained by solving a convex optimization problem. To reduce the computational effort we propose a regularization that leads to a simple one-step allocation rule.

Asset-liability Optimization for Pension Fund Management

We describe the problem of optimal asset allocation for a pension fund as a stochastic dynamic programming problem. We discuss the model generation, the formulation of the objective and solution methods. The original problem is nonconvex, but convex approximations can be found.