Institute of Information Theory and Automation

Stochastic optimization

Field characteristic

Stochastic programming and Decision in Economy

1. The Wasserstein metric (based on L_1 norm) has been employed to obtain upper stability bounds  (considered with respect to  probability measures space) for ``classical" one-stage stochastic programming problems with operator of mathematical expectation in an objective function and deterministic constrained sets. This result has been generalized to some types of problems in which dependence on probability measure is not linear. Furthermore, the both results have been employed to investigate properties of empirical estimates fot  stochastic programming problems. This approach gives possibility to investigate the case of  ``underlying" dstribution with heavy tailes. The heavy tailes case and the case of ``weakly" dependent data have been investigated also by simulation technique. 

 2.   A great attention has been focused on a special case of multistage stochastic programming problem in which an underlying random element   follows (generally) nonlinear autoregressive sequence and the constraints sets are given by systems of individual probability constraints. The above obtained results on stability and empirical estimates have been employed to these multistage stochastic programming problems.

 3. A stochastic programming technique  has been employed: to investigate the stability and empirical   in the case of version of Ramsey dynamic problem with finite time horizon, for  modelling of an ``nonnegative" influence of highway  building to environment and also  for some financial  problems.

 

Other Applications of Stochastic Programming

Apart from economic applications, the theory of stochastic programming is applied to ecology, unemployment and statistics (especially maximum likelihood estimation).

Mean-Variance Optimality in Markov Decision Processes

Mean variance selection rules, originally introduced by Markowitz for the portfolio selection problem, may very well capture risk sensitive behaviour of a decision maker. For this reason mean variance optimality was also studied stochastic dynamic models, in particular for discrete- and continuous time Markov chains and for semi-Markov reward processes. On the base of the results for the growth rate of the variance of total reward for discrete- and continuous-time Markov reward chains, we were able to estimate the growth rate of the variance for the more general semi-Markov reward processes. Additional attention was paid to the mean-variance optimality in Markov decision processes with respect to various mean-variance optimality criteria. In particular, on employing algorithmic procedures and using a computer program created in the SAS software we were able to solve medium size problems (up to 100 states and 100 admissible actions in each state) within 10 minutes on a standard PC computer.
 
 

Growth Rates and  Average Optimality in Risk-Sensitive Markov Decision Chains 

Attention is focused on characterizations of policies maximizing growth rate of expected utility, along with average of the associated certainty equivalent, in the risk-sensitive Markov decision chains with finite state and action spaces, i.e. controlled Markov chains with exponential utility function. In contrast to the existing literature the problem is handled by methods of stochastic dynamic programming on condition that the transition probabilities are replaced by general nonnegative matrices. Using the block-triangular decomposition of a collection of nonnegative matrices we establish necessary and sufficient conditions guaranteeing independence of optimal values on starting state along with partition of the state space into subsets with constant optimal values. Finally, for models with growth rate independent of the starting state we show how the methods work if we minimize growth rate or average of the certainty equivalent.

 

 

Extended Ramsey Growth Model  

The heart of the seminal paper of F. Ramsey on mathematical theory of saving is an economy producing output from labour and capital and the task is to decide how to divide production between consumption and capital accumulation to maximize the global utility of the consumption. Ramsey's original model is purely deterministic considered in continuous-time setting; Ramsey suggested some variational methods for finding an optimal policy how to divide the production between consumption and capital accumulation. Here we formulate the Ramsey model in the discrete-time setting similarly as in the recent literature on economic growth models. In contrast to the standard Ramsey's model we assume that every splitting of production between consumption and capital accumulation is influenced by some random factor; in particular, governed by transition probabilities depending on the current value of the accumulated capital and possibly on some (costly) decisions. Furthermore, we assume that also some additional (expensive) interventions of the decision maker possible. Finding optimal policy of the extended model can then be formulated as finding optimal policy of a highly structured Markov decision process. Unfortunately, usual optimization criteria for Markov decision chains as total discounted or average rewards cannot reflect variability-risk features of the problem. To this end, we focus attention on policies yielding maximal risk-sensitive rewards, i.e., if the stream of undiscounted one-stage rewards is evaluated by an exponential utility function.  

 

People

  • Vlasta Kaňková
    Stochastic programming problems, stochastic decision procedures
  • Karel Sladký
    Economic dynamics, dynamic programming, stochastic systems
  • Martin Šmíd
    Approximation methods, multistage stochastic programming
  • Michal Houda
    Stochastic programming, stability, empirical estimates, approximations
  • Vadym Omelchenko
    Stochastic dynamic programming.

 

Selected publications

Grants and projects

  • Stochastic Decision Approaches in Nonlinear Economic Models
    Vlasta Kaňková, grant No. 402/04/1294 of the Czech Science Foundation.
  • Validation of Economic Decision Models and Results
    Karel Sladký, grant No.402/05/0115 of the Czech Science Foundation, with Charles University of Prague.
  •  Mathematical modeling of the microstructure of the financial markets with the non-synchronous trading
     Martin Šmíd, grant No.402/06/1417 of the Czech Science Foundation
  • Economic Systems under Uncertainty: Optimization and Approximation
    Vlasta Kaňková, grant No. 402/07/1113 of the Czech Science Foundation
  • Economic Decision Models: Dynamics and Risk
    Karel Sladký, grant No.402/08/0107 of the Czech Science Foundation, with Charles University of Prague.
  •  Rational Decision Making at Markets with Asynchronous Trading: Theory and Empirical Evidence
     Martin Šmíd, grant No. P402/10/1610 of the Czech Science Foundation
  • Stochastic Economic Models under Uncertainty: Development over Time and Optimization
    Vlasta Kaňková, grant No. P402/10/0956 of the Czech Science Foundation.

 

 

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Last modification: 31.01.2011
Institute of Information Theory and Automation