In this article, we consider a model that describes the dynamics of Cohen-Grossberg neural networks with unbounded distributed delays, whose state variable are governed by stochastic non-linear integro-differential equations. Without assuming the smoothness, monotonicity and boundedness of the activation functions, by constructing suitable Lyapunov functional, employing the semi-martingale conv...

The stochastic reaction diffusion systems may suffer sudden shocks‎, ‎in order to explain this phenomena‎, ‎we use Markovian jumps to model stochastic reaction diffusion systems‎. ‎In this paper‎, ‎we are interested in almost sure exponential stability of stochastic reaction diffusion systems with Markovian jumps‎. ‎Under some reasonable conditions‎, ‎we show that the trivial solution of stocha...

Freedman’s inequality is a martingale counterpart to Bernstein’s inequality. This result shows that the large-deviation behavior of a martingale is controlled by the predictable quadratic variation and a uniform upper bound for the martingale difference sequence. Oliveira has recently established a natural extension of Freedman’s inequality that provides tail bounds for the maximum singular val...

In this paper we generalize recent comparison results of El Karoui, Jeanblanc-Picqué, and Shreve (1998), Bellamy and Jeanblanc (2000) and Gushchin and Mordecki (2002) to d-dimensional exponential semimartingales S, S∗. Our main result gives sufficient conditions for the comparison of European options w.r.t. martingale pricing measures. The comparison is with respect to convex and also with resp...

SUMMARY A functional central limit theorem for a local square integrable martingale with persistent disconti-nuities is given. By persistent discontinuities, it is meant that the martingale has jumps which do not vanish asymptotically. This central limit theorem is motivated by problems in the analysis of longitudinal and life history data.

A strict local martingale is a local martingale which is not a martingale. There are few explicit examples of “naturally occurring” strict local martingales with jumps available in the literature. The purpose of this paper is to provide such examples, and to illustrate how they might arise via filtration shrinkage, a phenomenon we would contend is common in applications such as filtering, contr...

General martingale theory shows that every martingale can be decomposed into continuous and purely discontinuous parts. In this paper specify a filtration for which the continuous part of the decomposition is 0 a.s. for any Ft martingale. It is a well-known fact that every martingale can be decomposed into continuous and purely discontinuous parts. It is of interest to study the filtrations tha...

The theory of approximate martingale estimating functions for continuous diffusions is well developed and encompasses many estimators proposed in the literature. This paper extends the asymptotic theory for approximate martingale estimating functions to diffusions with finite-activity jumps. The primary aim is to shed light on the question of rate optimality and efficiency of estimators when ob...

We prove an existence of a unique solution of an exponential martingale equation in the class of BMO martingales. The solution is used to characterize optimal martingale measures.

Abstract: We prove a martingale triangular array generalization of the Chow-BirnbaumMarshall’s inequality. The result is used to derive a strong law of large numbers for martingale triangular arrays whose rows are asymptotically stable in a certain sense. To illustrate, we derive a simple proof, based on martingale arguments, of the consistency of kernel regression with dependent data. Another ...