Michael Sørensen (Copenhagen)
Martingale estimating functions for stochastic differential equations with jumps

Methods are presented for estimating parameters in stochastic differential equations driven not only by a Wiener process, but also by another stochastic mechanism that causes the process to make jumps. This other mechanism can be a Lévy process, or more generally, a random measure on a suitable space. Solutions to such stochastic differential equations, called diffusions with jumps, are often used as models for financial time series. When the data are continuous time observations, likelihood inference for diffusions with jumps has long been well understood; see e.g. Sørensen (1991). However, continuous time observations are not available in practice, and for discrete time observations the likelihood function is not explicitly known and usually extremely difficult to calculate numerically. Therefore alternatives like estimating functions are even more useful for jump diffusions than for classical Wiener driven diffusions. We present a highly flexible class of diffusions with jumps for which explicit optimal martingale estimating functions of the type introduced by Kessler and Sørensen (1999) are available. These are based on eigenfunctions of the generator of the diffusion. The class of Pearson diffusions, investigated in Forman and Sørensen (2008), has the property that the generator maps polynomials into polynomials. Therfore it is easy to find polynomial eigenfunctions. Here we generalize these ideas and consider a class of diffusions with jumps for which the generator has the same property using ideas from Zhou (2003). The generator of a diffusion with jumps is considerably more complicated that that for a classical diffusion: It is a differential-integral operator. However, it turns out that a simple condition on the compensator of the jump measure is enough to ensure that explicit optimal martingale estimating functions can be found. We illustrate the general theory by concrete examples. The talk is based on joint work with Mathias Schmidt.

References:
Forman, J. L. and Sørensen, M. (2008). The Pearson diffusions: A class of statistically tractable diffusion processes. Scandinavian Journal of Statistics, 35, 438 - 465.