Volume 15, Issue 3

Approximation of one-dimensional stochastic differential equations and their additive functionals by dynamical systems with piecewise-constant random coefficients is obtained. We calculate asymptotic expansion of solution in terms of the step of discretization Δ.

In this article we determine the joint distribution of the first exit time from an interval and the value of the overshoot and the joint distribution of the supremum, infimum and the value of the semi-Markov walk with a linear drift. The results obtained are applied for the case when the jumps of the process are exponential. The corresponding distributions is found in terms of the resolvent of the process. We established the weak convergence of the distributions of two-boundary functionals of the semi-Markov walk with a linear drift to the corresponding functionals of the Wiener process.

The problem of the spectral analysis of random matrizant (the product of random matrices), which is the solution of a recurrent system of equations with random coefficients, or the system of stochastic linear differential equations of growing dimension is considered. The growing dimension means that the dimension of matrices and the number of matrices have the same order and both (dimension and number of matrices) tend to infinity. In this paper we give new method of deriving self averaging property for the V.I.C.T.O.R.I.A. -transform of normalized spectral functions (n.s.f.) of random matrizant or the product of independent random matrices. We apply the REFORM method for normalized spectral functions of this matrizant, where random matrices belong to the domain of attraction of the Strong Circular Law.

We prove a Yamada–Watanabe theorem for forward-backward stochastic differential equations (FBSDEs). As a consequence, we show that weak existence and pathwise uniqueness of the solution imply the existence of a strong solution. We use this result to establish existence and uniqueness of weak solutions for a large class of FBSDEs. We also give a probabilistic interpretation of the solutions of certain nonlinear partial differential equations (PDEs).

Suppose we have observations Y t = m t (θ) + e t in ℝ for t = 1, 2, …, n where each m t = m t (θ) is a smooth function of an unknown vector θ , and the noise { e t } is stationary with unknown marginals. We obtain asymptotic normality of the M-estimate θ with respect to any suitable smooth function ρ ( e ). Hence we obtain confidence regions for any smooth vector function t ( θ ) with ∂t(θ)/∂θ' of full rank. Extensions are given to the model Y t = m t (θ) + σ t (θ)e t in ℝ. Heuristic proofs are given.

A simple (hitherto unknown) representation is derived for parabolic cylinder functions.

We use a class of generalized backward stochastic differential equations to prove an existence result for a viscosity sub and super solutions to a class of PDEs with nonlinear Neumann boundary conditions.