This paper is concerned with the problem of explosive solutions for a class of stochastic differential equations. Our main results are presented as two theorems. Theorem 1 is concerned with the existence of explosive solutions with positive probability under certain sufficient conditions. With...

We compute the second order correction for the cover time of the binary tree of depth n by (continuous-time) random walk, and show that with probability approaching 1 as n increases, τcov=|E|[2log2⋅n−logn/2log2+O((loglogn)8)], thus showing that the second order correction differs from the...

The problem of exit from a domain of attraction of a stable equilibrium point in the presence of small noise is considered for a class of two-dimensional systems. It is shown that for these systems, the exit measure is 'skewed' in the sense that if S denotes the saddle point in the...

Recursive parameter estimation in diffusion processes is considered. First, stability and asymptotic properties of the global, off-line MLE (maximum likelihood estimator) are obtained under explicit conditions. The MLE evolution equation is then derived by employing a generalized Itô...

An extension of the "prior density for path" (Onsager-Machlup functional) is defined and shown to exist for Gaussian fields generated by solutions of elliptic PDEs driven by white noise. This functional is then used to define and solve the MAP estimation of such fields observed via nonlinear...

The problem of deciding whether the mean of an unknown distribution is in a set A or in its complement based on a sequence of independent random variables drawn according to this distribution is considered. We propose an algorithm which leads to an a.s. correct decision for any A in a class of...

Consider a sequence of m deterministic points in ##R##d, and consider the empirical measure of a random sample (without replacements) of size n = n(m). We prove the large deviation principle and compute the resulting rate function for the latter empirical measure under the assumptions that the...

The "prior density for path" (the Onsager-Machlup functional) is defined for solutions of semilinear elliptic type PDEs driven by white noise. The existence of this functional is proved by applying a general theorem of Ramer on the equivalence of measures on Wiener space. As an application, the...

The exact lower tail of Gaussian seminorms are evaluated, using a refinement of the techniques presented in Mayer-Wolf and Zeitouni (1993).