Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite-dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.     

Exploding Markov operators

Positivity - A special class of doubly stochastic (Markov) operators is constructed. In a sense these operators come from measure preserving transformations and inherit some of their properties,...     

Ergodicity for a weakly damped stochastic non-linear Schrödinger equation

We study a damped stochastic non-linear Schr?dinger (NLS) equation driven by an additive noise. It is white in time and smooth in space. Using a coupling method, we establish convergence of the Markov transition semi-group toward a unique invariant probability measure. This kind of method was originally developed to prove exponential mixing for strongly dissipative equations such as the Navier-Stokes equations. We consider here a weakly dissipative equation, the damped nonlinear Schr?dinger equation in the one-dimensional cubic case. We prove that the mixing property holds and that the rate of convergence to equilibrium is at least polynomial of any power.     

Discrete chaotic calculus and covariance identities

We show that for the binomial process (or Bernoulli random walk) the orthogonal functionals constructed in Kroeker, J.P. (1980) "Wiener analysis of functionals of a Markov chain: application to neural transformations of random signals", Biol. Cybernetics 36 , 243-248, [14] for Markov chains can be expressed using the Krawtchouk polynomials, and by iterated stochastic integrals. This allows to construct a chaotic calculus based on gradient and divergence operators and structure equations, and to establish a Clark representation formula. As an application we obtain simple infinite dimensional proofs of covariance identities on the discrete cube.     

Stochastic processes with orthogonal polynomial eigenfunctions

Markov processes which are reversible with either Gamma, Normal, Poisson or Negative Binomial stationary distributions in the Meixner class and have orthogonal polynomial eigenfunctions are characterized as being processes subordinated to well-known diffusion processes for the Gamma and Normal, and birth and death processes for the Poisson and Negative Binomial. A characterization of Markov processes with Beta stationary distributions and Jacobi polynomial eigenvalues is also discussed.     

The Genuine Bernstein-Durrmeyer Operator on a Simplex

In 1967 Durrmeyer introduced a modification of the Bernstein polynomials as a selfadjoint polynomial operator on L²[0,1] which proved to be an interesting and rich object of investigation. Incorporating Jacobi weights Berens and Xu obtained a more general class of operators, sharing all the advantages of Durrmeyer’s modification, and identified these operators as de la Vallée-Poussin means with respect to the associated Jacobi polynomial expansion. Nevertheless, all these modifications lack one important property of the Bernstein polynomials, namely the preservation of linear functions. To overcome this drawback a Bernstein-Durrmeyer operator with respect to a singular Jacobi weight will be introduced and investigated. For this purpose an orthogonal series expansion in terms generalized Jacobi polynomials and its de la Vallée-Poussin means will be considered. These Bernstein-Durrmeyer polynomials with respect to the singular weight combine all the nice properties of Bernstein-Durrmeyer polynomials with the preservation of linear functions, and are closely tied to classical Bernstein polynomials. Focusing not on the approximation behavior of the operators but on shape preserving properties, these operators we will prove them to converge monotonically decreasing, if and only if the underlying function is subharmonic with respect to the elliptic differential operator associated to the Bernstein as well as to these Bernstein-Durrmeyer polynomials. In addition to various generalizations of convexity, subharmonicity is one further shape property being preserved by these Bernstein-Durrmeyer polynomials. Finally, pointwise and global saturation results will be derived in a very elementary way.     

Optimal non-linear filtering of a class of partially observable Markov processes

For a class of partially observed Markov processes a representation for the optimal non-linear filter is obtained in which stochastic integrals are absent.     

On surjective second order non-linear Markov operators and associated nonlinear integral equations

It was known that orthogonality preserving property and surjectivity of nonlinear Markov operators, acting on finite dimensional simpleces, are equivalent. It turns out that these notions are no longer equivalent when such kind of operators are considered over on infinite dimensional spaces. In the present paper, we find necessary and sufficient condition to be equivalent of these notions, for the second order nonlinear Markov operators. To do this, we fully describe all surjective second order nonlinear Markov operators acting on infinite dimensional simplex. As an application of this result, we provided some sufficient conditions for the existence of positive solutions of nonlinear integral equations whose domain are not compact.     

Markov dilation of Diffusion Type Processes and Its Application to the Financial Mathematics

The Markov dilation of diffusion type processes is defined. Infinitesimal operators and stochastic differential equations for the obtained Markov processes are described. Some applications to the integral representation for functionals of diffusion type processes and to the construction of a replicating portfolio for a non-terminal contingent claim are considered.     

Local convergence of the (exact and inexact) iterative aggregation method for linear systems and Markov operators

Summary. The iterative aggregation method for the solution of linear systems is extended in several directions: to operators on Banach spaces; to the method with inexact correction, i.e., to methods where the (inner) linear system is in turn solved iteratively; and to the problem of finding stationary distributions of Markov operators. Local convergence is shown in all cases. Convergence results apply to the particular case of stochastic matrices. Moreover, an argument is given which suggests why the iterative aggregation method works so well for nearly uncoupled Markov chains, as well as for Markov chains with other zero-nonzero structures. Received May 25, 1991/Revised version received February 23, 1994     

Potentials Kernels Associated with a Filtration and Forward- Backward SDEs

In this paper, we study some operators associated with a filtration. We prove that these operators are Markov potential kernels closing a sub-Markov resolvent, using the method of continuation inspired by the study of forward-backward stochastic differential equations.     

A family of univariate rational Bernstein operators

We define and study a new family of univariate rational Bernstein operators. They are positive operators exact on linear polynomials. Moreover, like classical polynomial Bernstein operators, they enjoy the traditional shape preserving properties and they are total variation diminishing. Finally, for a specific class of denominators, some convergence results are proved, in particular a Voronovskaja theorem, and some error bounds are given.     

Stability of abstract Markov processes

A necessary and sufficient criterion weaker than the Markoff's hypothesis for asymptotical stability of stochastic semigroups of Markov operators on (L)-spaces is proven.     

Exceptional Laurent biorthogonal polynomials through spectral transformations of generalized eigenvalue problems

A formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second-order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional-type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen–van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar to the cases of exceptional extensions of classical orthogonal polynomials, both state-deletion and state-addition occur.     

Pseudo-differential operators and Markov semigroups on compact Lie groups

We extend the Ruzhansky-Turunen theory of pseudo-differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N. Jacob and collaborators. Feller semigroups, their generators and resolvents are exhibited as pseudo-differential operators and the symbols of the operators forming the semigroup are expressed in terms of the Fourier transform of the transition kernel. The symbols are explicitly computed for some examples including the Feller processes associated to stochastic flows arising from solutions of stochastic differential equations on the group driven by Lévy processes. We study a family of Lévy-type linear operators on general Lie groups that are pseudo-differential operators when the group is compact and find conditions for them to give rise to symmetric Dirichlet forms.     

Successful couplings for a class of stochastic differential equations driven by Lvy processes

By constructing proper coupling operators for the integro-differential type Markov generator,we establish the existence of a successful coupling for a class of stochastic differential equations driven by L’evy processes.Our result implies a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying Markov semigroups,and it is sharp for Ornstein-Uhlenbeck processes driven by α-stable L’evy processes.     

Polynomial operators and local smoothness classes on the unit interval

We obtain a characterization of local Besov spaces of functions on [-1,1] in terms of algebraic polynomial operators. These operators are constructed using the coefficients in the orthogonal polynomial expansions of the functions involved. The example of Jacobi polynomials is studied in further detail. A by-product of our proofs is an apparently simple proof of the fact that the Cesàro means of a sufficiently high integer order of the Jacobi expansion of a continuous function are uniformly bounded.     

Stochastic Systems with Averaging in the Scheme of Diffusion Approximation

We propose a system approach to the asymptotic analysis of stochastic systems in the scheme of series with averaging and diffusion approximation. Stochastic systems are defined by Markov processes with locally independent increments in a Euclidean space with random switchings that are described by jump Markov and semi-Markov processes. We use the asymptotic analysis of Markov and semi-Markov random evolutions. We construct the diffusion approximation using the asymptotic decomposition of generating operators and solutions of problems of singular perturbation for reducibly inverse operators. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 9, pp. 1235–1252, September, 2005.     

Matrix operator equations

This article is devoted to the theory and applications of matrix operator equations in normed spaces. We describe in detail the general properties of matrix operators and their representing matrices. As the indexing set we take an arbitrary countable set. This is related to stochastic applications in which it is difficult to find an enumeration that fits well with the content. Sample applications are the Markov matrices and the operators they define. Comparisons with operators of this type arise in some stochastic problems of integral geometry and tomography.     

Discriminants and Functional Equations for Polynomials Orthogonal on the Unit Circle

We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and q-difference equations for these polynomials. A general functional equation is found which allows one to relate the zeros of the orthogonal polynomials to the stationary values of an explicit quasi-energy and implies recurrences on the orthogonal polynomial coefficients. We also evaluate the discriminants and quantized discriminants of polynomials orthogonal on the unit circle.