Kacha Dzhaparidze

On Bernstein-type inequalities for martingales

Bernstein-type inequalities for local martingales are derived. The results extend a number of well-known exponential inequalities and yield an asymptotic inequality for a sequence of asymptotically continuous martingales.

Spectral characterization of the optional quadratic variation process

In this paper we show how the periodogram of a semimartingale can be used to characterize the optional quadratic variation process.

Representations of isotropic Gaussian random fields with homogeneous increments

We present series expansions and moving average representations of isotropic Gaussian random fields with homogeneous increments, making use of concepts of the theory of vibrating strings. We illustrate our results using the example of Levys fractional Brownian motion on ℝ N .

On correlation calculus for multivariate martingales

In this paper the correlation between two multivariate martingales is studied. This correlation can be expressed in a nondecreasing process, that remains zero in the case of linear dependence. A key result is an integral representation for this process.

The strong law of large numbers for martingales with deterministic quadratic variation

The strong law of large numbers is proved for multivariate martingales with deterministic quadratic variation, along the same lines as in Lai, Wei and Robbins (1979), though the setting here is more general

On optimality of regular projective estimators in semimartingale models

In the partially specified statistical models the class of regular estimators having a martingale representation is defined and the best among them is sought in the sense of asymptotic second order characteristics

Some aspects of modeling and statistical inference for financial models

Modeling the stock price development as a geometric Brownian motion or, more generally, as a stochastic exponential of a diffusion, requires the use of specific statistical methods. For instance, the observations seldom reach us in the form of a continuous record and we are led to infer about diffusion coefficients from discrete time data. Next, often the classical assumption that the volatility is constant has to be dropped. Instead, a range of various stochastic volatility models is formed by the limiting transition from known volatility models in discrete time towards their continuous time counterparts. These are the main topics of the present survey. It is closed by a quick look beyond the usual Gaussian world in continuous time modeling by allowing a Levy process to be the driving process.

On optimality of regular projective estimators for semimartingale models III. One step improvements

In this paper we consider one step improvement techniques that yield optimal regular projective estimators

On optimality of regular projective estimators for semimartingale models II. Asymptotically linear estimators.

In this paper we consider estimators that (asymptotically) admit a so called linear representation. Using a parametrization of the model, that has been defined in a previous paper [1], and a certain notion of smoothness of the parametrization, it is possible to define a concept of optimality for these estimators and to characterize the optimal estimators. In contrast with the situation in [1], only the compensator is fully parametrized by the parameter we want to estimate. Embedding the problem under consideration in the previously developed framework then requires the introduction of several nuisance parameters, that are needed to describe certain stochastic integrals with respect to the compensator of the jump measure