On the method of non-parametric evaluation in statistics of random processes on the basis of ill-posed problem approach

S. A. Vavilov ; K. Yu. Ermolenko

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 12, Tome 351 (2007), pp. 117-128

Résumé

The problem to evaluate integrated volatility on the basis of the observable realization of the stochastic process corresponding to geometrical Brownian motion is considered. In one line with theoretical interest the urgency of the problem inquest is stipulated also by the fact that calculation of integrated volatility for different financial assets is the inseparable part of financial engineering topics. In the present paper the new approach to tackle the problem of integrated volatility evaluation is proposed. The integral equation to provide the calculation of integrated volatility is derived. The solving of this integral equation turns out to be a standard ill-posed problem of mathematical physics. The main idea of the original problem reduction to the ill- posed problem is to make its solution robust towards the presence of anomalous statistical data, for instance, generated by the market microstructure effect such as the bid-ask spread existence.