Agnieszka Zięba
supervisor: Jacek Wesołowski
Infinitesimal generator of the Markov process is one of the tools which are used to describe this process. The general theory and properties of this mathematical object are well-known, because as the name of infinitesimal generator suggests, it determines transition probabilities of the process uniquely.
In my presentation I will be interested in infinitesimal generators of quadratic harnesses - a special family of square-integrable processes with linear conditional expectations and conditional variances being the polynomials of degree two where the conditioning is with respect to the past-future filtration of the process.
This class contains such well-known processes as Wiener and Poisson as well as some stochastic processes important for quantum mechanics. Typically, quadratic harnesses can parametrized by five numerical constants.
In order to find infinitesimal generators of quadratic harnesses, it is sufficient to find a solution of some equation in a special non-commutative algebraic structure. This solution depends significantly on five parameters of quadratic harness and it is closely related to the celebrated Askey-Wilson scheme orthogonal polynomials in the known cases.