Some properties of random fields related to stochastic integrals with respect to strong martingales

We investigate the properties of random fields related to stochastic intergals with respect to a two-parameter strong square-integrable martingale in the case where the martingale and the integrands depend on the limits of integration. We prove inequalities for the moments of the uniform norm and the modulus of continuity of trajectories of modifications of such fields. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 80–96.     

The graphs for which all strong orientations are hamiltonian

We show that the only nontrivial strongly orientable graphs for which every strong oreintation is Hamiltonian are complete graphs and cycles.     

Subrings which are closed with respect to taking the inverse

Let S be a subring of the ring R. We investigate the question of whether S∩U(R)=U(S) holds for the units. In many situations our answer is positive. There is a special emphasis on the case when R is a full matrix ring and S is a structural subring of R defined by a reflexive and transitive relation.     

Stochastic integral with respect to set-valued square integrable martingales

In this paper, we shall firstly illustrate why we should consider integral of a stochastic process with respect to a set-valued square integrable martingale. Secondly, we shall prove the representation theorem of set-valued square integrable martingale. Thirdly, we shall give the definition of stochastic integral of a stochastic process with respect to a set-valued square integrable martingale and the representation theorem of this kind of integrals. Finally, we shall prove that the stochastic integral is a set-valued sub-martingale.     

Integral representation with respect to stopped continuous local martingales

We consider a Markovian jump process θ, with finite state space, feeding the parameters of a nonlinear diffusion process X. We observe θ and X in white noise, and—given a function f—we want to construct a finite filter for the f(X _t )-process. An algorithm is investigated which will produce a finite filter if it halts after a finite number of steps, and we give necessary and. sufficient conditions for this to happen.     

Anticipating stratonovich integral with respect to the Azema's martingales

In this paper we establish the conditions on a L ²-process u for the existence of its anticipating Stratonovich integral with respect to a normal martingale belonging to a certain class. This class includes the Azema's martingales and the compensated Poisson processes.     

Generalized functions which are analytic with respect to some of their variables

For a generalized function which is analytic with respect to some of its variables in the sense of Gording and Malgrange we prove a theorem on representation in the form of a power series in these variables which converges in D′ and a corresponding uniqueness theorem. We also obtain necessary and sufficient conditions for a generalized function to be entire with respect to some of its variables.     

Some remarks about polynomials which are orthogonal with respect to an indefinite weight

In this note Laguerre-, Legendre- and Jacobi-type orthogonal polynomials with respect to an indefinite weight function are considered (see [LK]). In [LK] some values of the parameters are excluded, as in these cases the eigenvectors of the associated operators in the corresponding Pontrjagin spaces do not form a complete system. It is shown, that in these exceptional cases algebraically nonsimple eigenvalues appear, and in order to get a complete system, the system of eigenfunctions has to be augmented by those root functions, which are not eigenfunctions.     

Rings over which all modules are semiregular

For a ring A, it is proved that all A-modules are semiregular if and only if A is an Artinian serial ring and J ²(A) = 0. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 185–194, 2007.