On Quasi-Likelihood Analyses for Stochastic Differential Equations with Jumps
Hiroki Masuda
Graduate School of Mathematics, Kyushu University, Fukuoka, Fukuoka, Japan

We consider parametric estimation of stochastic differential equations with jumps and with or without continuous local-martingale component, when the process is discretely observed at high frequency. The talk begins with some background materials.

Since a closed form transition probability is generally not available, the maximum likelihood estimation cannot be of practical use, so that we have to adopt some other feasible estimation procedure. On the one hand, the Gaussian quasi-likelihood estimation (GQLE), which is based on fitting local mean and local variance, is one of natural candidates as in the case of diffusions. As a matter of fact, the GQLE leads to an asymptotically normally distributed estimator under standard regularity conditions even in the presence of jumps, which may or may not be finite-activity. Nevertheless, we can see that a direct use of the GQLE (without using a jump-detection filter) may lose much asymptotic efficiency, although the structure of the jump component can be of wide variety; in this case, we may regard the driving jump-intensity measure as a nuisance parameter up to a multiplicative constant. On the other hand, especially when the driving noise process is of pure-jump type with positive activity index, we may adopt yet another quasi-likelihood estimation procedure based on non-Gaussian type contrast functions. We show that the resulting estimator may exhibit much better theoretical performances than the one based on the GQLE procedure. Some comparisons will be made in order to clarify the essential differences between the Gaussian and the non-Gaussian quasi-likelihood estimation procedures.

Keywords: Stochastic differential equations with jumps; High-frequency sampling; Gaussian quasi-likelihood analysis; Non-Gaussian quasi-likelihood analysis

Biography: Hiroki Masuda received the B.S. degree in science and engineering from Waseda University, Japan, in 1999, and then M.S. and Ph.D. degrees in mathematical science from University of Tokyo, Japan, in 2001 and 2004, respectively.

From April 2004 to the present, he has been working at Graduate School of Mathematics, Kyushu University, Japan, as a Research Assistant (from April 2004 to March 2007), as an Assistant Professor (from April 2007 to November 2010), and as an Associate Professor (from December 2010 to the present).

His current research interests are in the area of: statistics for stochastic processes, especially for processes with jumps; stability of stochastic differential equations; and practical recipes for random number generation for specific infinitely divisible distributions.