One approach to the problem of nonparametric estimation in statistics of random processes based on the method of ill-posed problem

We consider the problem of estimation of integrated volatility, i.e., of the integral of the diffusion coefficient squared, in a stochastic differential equation for a random process that corresponds to geometric Brownian motion. In additon to purely theoretical interest, this problem is of interest for applications since the problem of evaluation of integrated volatility for financial assets is an important part of financial engineering topics. In the present paper, we suggest a new approach to the above-mentioned problem. We derive an integral equation whose solution determines the value of integrated volatility. This integral equation is a typical ill-posed problem of mathematical physics. The main idea of the proposed reduction of the original problem to an ill-posed problem consists of making its solution robust with respect to anomalous values of statistical data which are generated, for example, by market microstructure effects, such as the bid-ask spread. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 117–128.