An approximate martingale estimating function with an eigenfunction is proposed for an estimation problem about an unknown drift parameter for a one-dimensional diffusion process with small perturbed parameter [epsilon] from discrete time observations at n regularly spaced time points k/n,...

We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H1/2. For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved by arbitrary approximation methods that are based on an...

We investigate the high resolution quantization and entropy coding problem for solutions of stochastic differential equations under Lp[0,1]-norm distortion. We find explicit high resolution formulas in terms of the average diffusion coefficient seen by the process. The proof is based on a...

We consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional Itô diffusion. The control effort that can be applied to this system takes the form that is associated with the so-called monotone follower problem of singular stochastic control. The...

We consider the critical Sobolev isotropic Brownian flow in . On the basis of the work of LeJan and Raimond [Y. LeJan, O. Raimond, Integration of Brownian vector fields, Ann. Probab. 30 (2002) 826-873], we prove that the corresponding flow is a flow of homeomorphisms. As an application, we...

Let Xt be any additive process in . There are two lower indices and an upper index [beta]T for T[set membership, variant](0,[infinity]) such that for any Borel set , if , and dimHX(E)=[beta]TdimHE, where X(E)={Xs:s[set membership, variant]E} for and dimH denotes the Hausdorff dimension. When Xt...

The main aim of this paper is to discuss the almost surely asymptotic stability of the neutral stochastic differential delay equations (NSDDEs) with Markovian switching. Linear NSDDEs with Markovian switching and nonlinear examples will be discussed to illustrate the theory.

Let fn(x) be the non-parametric kernel density estimator of a density function f(x) based on a kernel function K. In this paper, we first prove two moderate deviation theorems in for {fn(x),n=1}. Then, as an application of the moderate deviations, we obtain a law of the iterated logarithm for...

We extend the notion of positive continuous additive functionals of multidimensional Brownian motions to generalized Wiener functionals in the setting of Malliavin calculus. We call such a functional a generalized PCAF. The associated Revuz measure and a characteristic of a generalized PCAF are...

In this paper, we study first the problem of nonparametric estimation of the stationary density f of a discrete-time Markov chain (Xi). We consider a collection of projection estimators on finite dimensional linear spaces. We select an estimator among the collection by minimizing a penalized...