Law of Large Numbers for Random LU-Fuzzy Numbers: Some Results in the Context of Simulation of Financial Quantities

Financial problems can be analyzed by several ways. In the standard case, we assume that a given financial quantity in question is a stochastic (or random) variable, ie. it follows some probability distribution and its possible states can get prescribed particular probabilities. However, in financial modeling it can appear that the estimation of parameters of such models (e.g. volatility) does not lead to reliable results, so that it can be fruitful to incorporate some kind of impreciseness. One can recognize an unnatural simplification of parameters that can lead to a loss of important information hidden in data. If one wants to apply Monte Carlo simulation to analyze a financial problem with values expressed by imprecisely defined numbers, it is important to show that random variables with imprecisely defined numbers satisfy the (strong) law of large numbers, as well. Otherwise such approach would have no sense. The aim of the paper is to provide a justification to this novel approach. Besides theoretical proves we show also via empirical example that the law holds for a given type of imprecise number.