Weak convergence of partial sums and multilinear forms in independent random variables and linear processes to stochastic integrals now plays a major role in nonstationary time series and has been central to the development of unit root econometrics. The present paper develops a new and...

Consider a d-dimensional Brownian motion X (Xl, ... ,Xd ) and a function F which belongs locally to the Sobolev space W 1,2. We prove an extension of Ito's formula where the usual second order terms are replaced by the quadratic covariations [fk(X), Xkj involving the weak first partial...

Consider a d-dimensional Brownian motion X (Xl, ... ,Xd ) and a function F which belongs locally to the Sobolev space W 1,2. We prove an extension of Ito's formula where the usual second order terms are replaced by the quadratic covariations [fk(X), Xkj involving the weak first partial...

Abstract We consider a sequential testing problem of three hypotheses that the unknown drift of a Brownian motion takes one of three values. We show that this problem can be solved by a reduction to an optimal stopping problem for local times of the observable process. For the case of...

We provide a point estimate for integrals on R, based on the standard Brownian motion. We prove the consistency of the estimator and limit theorems for the fluctuations. The proof relies on computing the distribution of the local time of a Brownian motion at a specific stopping time.

A local limit theorem is proved for sample covariances of nonstationary time series and integrable functions of such time series that involve a bandwidth sequence. The resulting theory enables an asymptotic development of nonparametric regression with integrated or fractionally integrated...

An asymptotic theory for stochastic processes generated from nonlinear transformations of nonstationary integrated time series is developed. Various nonlinear functions of integrated series such as ARIMA time series are studied, and the asymptotic distributions of sample moments of such...

An asymptotic theory is developed for nonlinear regression with integrated processes. The models allow for nonlinear effects from unit root time series and therefore deal with the case of parametric nonlinear cointegration. The theory covers integrable, asymptotically homogeneous and explosive...

We compute a closed-form expression for the moment generating function fˆ(x;λ,α)=1λEx(eαLτ), where Lt is the local time at zero for standard Brownian motion with reflecting barriers at 0 and b, and τ∼Exp(λ) is independent of W. By analyzing how and where fˆ(x;⋅,α) blows up in λ, a...