Parameter estimation for differential equations using fractal-based methods and applications to economics

Many problems from the area of economics and finance can be described using dynamical models. For them, in which time is the only independent variable and for which we work in a continuous framework, these models take the form of deterministic differential equations (DEs). We may study these models in two fundamental ways: the direct problem and the inverse problem. The direct problem is stated as follows: given all of the parameters in a system of DEs, find a solution or determine its properties either analytically or numerically. The inverse problem reads: given a system of DEs with unknown parameters and some observational data, determine the values of the parameters such that the system admits the data as an approximate solution. The inverse problem is crucial for the calibration of the model; starting from a series of data we wish to describe them using deterministic differential equations in which the parameters have to be estimated from data samples. The solutions of the inverse problems are the estimations of the unknown parameters and we use fractal-based methods to get them. We then show some applications to technological change and competition models.