We show that iterated stochastic integrals can be described equivalently either by the conventional forward adapted, or by backward adapted quantum stochastic calculus. By using this equivalence we establish two properties of triangular (causal) and rectangular double quantum stochastic product integrals, namely a necessary and su¢ cient condition for their unitarity, and the coboundary relatio...

We construct and analyze Gauss-type quadrature rules with complex-valued nodes and weights to approximate oscillatory integrals with stationary points of high order. The method is based on substituting the original interval of integration by a set of contours in the complex plane, corresponding to the paths of steepest descent. Each of these line integrals shows an exponentially decaying behavi...

A definition of concave integral is given for real-valued maps and with respect to Dedekind complete Riesz space-valued “capacities”. Some comparison results with other integrals are given and some convergence theorems are proved.

We present here a method for the study of stochastic neurodynamics in the framework of the "Neural Network Master Equation" proposed by Cowan. We consider a model neural network composed of two{state neurons subject to simple stochastic kinetics. We introduce a method based on a spin choerent state path integral to compute the moment generating function of such a network. A formal construction ...

The study of stochastic approximation algorithms (SAAs) with setvalued mean-fields has been popular during recent times due to their applications to many large scale model-free problems arising in stochastic optimization and control. The analysis of such algorithms requires the almost sure boundedness of the iterates, which can be hard to verify. In this paper we extend the ideas of Abounadi, B...

Many microeconomic and engineering problems can be formulated as stochastic optimization problems that are modelled by Itô evolution systems and by cost functionals expressed as stochastic integrals. Our paper studies some optimization problems constrained by stochastic evolution systems, giving original results on stochastic first integrals, adjoint stochastic processes and a version of simpli...

The introduction of the anticipating (or Skorohod) integral in [8] and of the anticipating stochastic calculus in [7] has opened the question of solving anticipating stochastic differential equations. In general, the existence and uniqueness of the solution for these equations is not known. The difficulty of solving such equations is due to the fact that the classical method of Picard’s iterati...

The paper considers a particular family of set–valued stochastic processes modeling birth–and–growth processes. The proposed setting allows us to investigate the nucleation and the growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set–valued estimators are studied for growth process. Moreover, the nucle...

The paper considers a particular family of fuzzy monotone set–valued stochastic processes. The proposed setting allows us to investigate suitable α-level sets of such processes, modeling birth–and–growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set–valued estimators are studied for growth process. Mor...