We apply the Malliavin calculus to study several non-degeneracy conditions on the coefficients of a stochastic differential equation on the plane, in order to deduce the existence and smoothness of density for the law of the solution.

Let B be a 2-parameter Brownian motion on R+2. Consider the non-Markovian stochastic differential equation in the plane dX(z) = [alpha](z, X) dB(z) + [beta](z, X)dz for z [set membership, variant] R+2, i.e., Xs, t - X0, t - Xs, 0 + X0, 0 = [integral operator]Rz [alpha]([zeta], X) dB[zeta] +...

Asymptotic properties of singularly perturbed Markov chains having measurable and/or continuous generators are developed in this work. The Markov chain under consideration has a finite-state space and is allowed to be nonstationary. Its generator consists of a rapidly varying part and a slowly...

In this paper, we consider a class of nonlinear autoregressive (AR) processes with state-dependent switching, which are two-component Markov processes. The state-dependent switching model is a nontrivial generalization of Markovian switching formulation and it includes the Markovian switching as...

The Robbins-Monto processes in a Hilbert space are considered in this work. A nonlinear mapping f is treated instead of the usual linearity assumption. A modified process with varying truncations is analyzed, and asymptotic properties are investigated. Convergence as well as necessary and...