Zvonko Kostanjˇ car, Branko Jeren, Jurica Cerovec Particle Filters in Decision Making Problems under Uncertainty UDK IFAC 65.012.123:004 3.0.3 Original scientific paper In problems of decision making under uncertainty, we are often faced with the problem of representing the uncertainties in a form suitable for quantitative models. Huge databases for the financial system now exist that facilitate the analysis of uncertainties representation. In portfolio management, one has to decide how much wealth to put in each asset. In this paper we present a decision making process that incorporates particle filters and a genetic algorithm into a state dependent dynamic portfolio optimization system. We propose particle filters and scenario trees as a means of capturing uncertainty in future asset returns. Genetic algorithm was used as an optimization method in scenario generation, and for determining the asset allocation. The proposed method shows better results in comparison with the standard mean variance strategy according to Sharpe ratio. Key words: Uncertainty representation, Particle filters, Scenario trees ˇ Cestiˇ cni filtri u problemima odluˇ civanja s prisutnom nesigurnoš´ cu. U problemima odluˇ civanja u kojima je prisutna nesigurnost ˇ cesto se susre´ cemo s problemom predstavljanja nesigurnosti u obliku prikladnom za raˇ cunalnu obradu. Analizu reprezentacije nesigurnosti danas nam olakšavaju velike baze podataka o financijskom sustavu. Kod upravljanja portfeljem potrebno je odluˇ citi koliko novca uložiti u pojedine dionice. U ovom radu predstavljen je proces odluˇ civanja temeljen na ˇ cestiˇ cnim filtrima i genetskom algoritmu. Pomo´ cu razvijenog procesa odluˇ civanja izgrađen je sustav za dinamiˇ cko optimiranje portfelja. ˇ Cestiˇ cni filtri i stabla scenarija predloženi su za predstavljanje nesigurnosti u budu´ cim prinosima dionica. Genetski algoritam se koristi kao optimizacijska metoda u generiranju scenarija i za određivanje optimalnog portfelja. Predložena metoda uspoređena je sa standardnom mean-variance strategijom te prema Sharpeovom omjeru daje bolje rezultate. Kljuˇ cne rijeˇ ci: predstavljanje nesigurnosti, ˇ cestiˇ cni filtri, stabla scenarija 1 INTRODUCTION The development and use of dynamic portfolio opti- mization algorithms is extremely important in financial markets. This is the result of a major growth of financial engineering, including the technological advances, global- ization, increased competition, and ability to solve com- plex financial models [1]. The goal of portfolio optimiza- tion is to automatically determine the optimal percentage of the total investment value allocated to each asset in the portfolio [2]. Optimality is expressed in terms of return maximization or risk minimization. The core of a portfolio optimization problem is a good representation of uncer- tainty. Uncertainties should be represented in a form that reflects the reality and complexity of the financial system, but should also be simple enough for algorithmic imple- mentation [3]. Uncertainty can be represented in a number of ways. One approach is to represent uncertainty by multidimen- sional continuous distributions or discrete distributions with large number of outcomes [4]. In both cases, the prob- lem is how to estimate parameters of the distribution [5]. A naive method would consist of the estimation of parame- ters directly from the historical data. However, such an ap- proach fails to take into account the fact that newer data has more influence on the parameters than older data. In line with that, it is important to note that the problem does not lie in modeling of historical data, but in predicting future uncertainty from the above mentioned data. The most pop- ular approach to parameter estimation is that of Bayesian estimators, developed in [6], [7], and described in [8]. The idea of Bayesian inference is to combine prior information with sample returns. Besides parameter estimation, there is also a problem of selecting the right multivariate distri- bution, especially if statistical properties of uncertainty are time variant. A different method for representing uncertainty is sce- nario trees. The goal of scenario trees is to represent the underlying uncertainty with a small set of discrete out- comes [1]. A scenario is a deterministic realization of all uncertain parameters. There are two approaches in gen- ISSN 0005-1144 ATKAFF 50(3–4), 245–251(2009) AUTOMATIKA 50(2009) 3–4, 245–251 245