Maximum likelihood estimation of the parameters of a system of stochastic differential equations that models the returns of the index of some classes of hedge funds

In this paper we study an inverse problem for a parabolic partial differential equation. The parabolic partial differential equation considered is the Fokker Planck equation associated to a system of stochastic differential equations and the inverse problem studied consists in finding from suitable data the values of the parameters that appear in the coefficients of this Fokker Planck equation. The data used in the reconstruction of the parameters are observations made at discrete times of the stochastic process solution of the system of stochastic differential equations. That is, the data of the inverse problem are a sample taken at discrete times of some of the components of the random vector solution of the stochastic differential equations and not, as usual, observations made on the solution of the parabolic equation. The choice of the system of stochastic differential equations and of the data used in the inverse problem are motivated by applications in mathematical finance. The stochastic differential equations presented can be used to model the dynamics of the log-returns of the index of some classes of hedge funds, such as, for example, the so called long short equity hedge funds and of some auxiliary variables. The solution of the inverse problem proposed is obtained through the solution of a filtering and of an estimation problem. The solution of these last two problems is based on the knowledge of the joint probability density function of the state variables of the model conditioned to the observations made and to the initial condition. This joint probability density function is solution of an initial value problem for the Kushner equation that in the circumstances considered here can be written as a sequence of initial value problems for the Fokker Planck equation associated to the system of stochastic differential equations with appropriate initial conditions. An integral representation formula for this probability density function is derived and used to develop a numerical procedure to solve the estimation problem using the maximum likelihood method. The Kushner equation provides the relation between the data and the Fokker Planck equation used to solve the inverse problem considered. The computational method proposed has been tested on synthetic data and the results obtained are presented. Some auxiliary material useful to understand this paper including some animations and some numerical experiments can be found in the website http://www.econ.univpm.it/recchioni/finance/w5. A more general reference to the work in mathematical finance of the authors and of their coauthors is the website http://www.econ.univpm.it/recchioni/finance. © de Gruyter 2007