MSI Weekly Bulletin - Week starting Monday 12 June, 2006

Approximations to the modified signed likelihood ratio statistic are asymptotically standard normal with error of order n?1 where n is sample size. Proofs of this fact generally require that the sufficient statistic of the model be written as ($\hat \theta$, $\alpha$) where $\hat \theta$ is the maximum likelihood estimator of parameter $\theta$ of the model and $\alpha$ is an ancillary statistic. This condition is very difficult or impossible to verify for many models. However, calculation of the statistics themselves do not require this assumption. This paper is devoted to exploring higher-order asymptotic normality of these statistics under general conditions. It focuses on the case that $\theta$ may be parameterized as $\theta$ = ( $\psi$ , $\lambda$), where $\psi$ is the scalar parameter of interest and $\lambda$ is a nuisance parameter vector. Under general assumptions, the asymptotic properties of the statistics are proved. These proofs do not put any requirements of the sufficient statistic, they just assume general conditions which are easy to verify for and satisfied by commonly used models. Therefore this research removes the theoretical obstacle for applying these statistics to commonly used models.

We consider a financial market and we denote by the wealth of an investor, investing in a self-financing way. We consider the problem of maximization of expected utility of wealth at a random terminal horizon witha random time anda utility function. We emphasize that hereis not a stopping time. We solve the problem explicitely, using backward stochastic differential equation, in the case of CRRA utility functions