bernstein operators for extended chebyshev systems abstract let u n c n a b be an extended chebyshev space of dimension n 1 suppose that f 0 u n is strictly positive and f 1 u n has the property that f 1 f 0 is strictly increasing we search for conditions ensuring the existence of points t 0 t n a b and positive coefficients 0 n such that for all f c a b the operator b n c a b u n defined by b n f k 0 n f t k k p n k satisfies b n f 0 f 0 and b n f 1 f 1 here it is assumed that p n k k 0 n is a bernstein basis defined by the property that each p n k has a zero of order k at a and a zero of order n k at b

variation of constants for hybrid systems of functional differential equations the objective is to derive a variation of constants formula for systems of functional differential equations or delay differential equations coupled with functional equations or difference equations the difficulties arise because of the constraints imposed by the functional equations

forward backward and symmetric stochastic integration we define three types of non causal stochastic integrals forward backward and symmetric our approach consists in approximating the integrator two optics are considered the first one is based on traditional usual stochastic calculus and the second one on wiener distributions

backward doubly stochastic differential equations and systems of quasilinear spdes we introduce a new class of backward stochastic differential equations which allows us to produce a probabilistic representation of certain quasilinear stochastic partial differential equations thus extending the feynman kac formula for linear spde s

the vanishing of l2 harmonic one forms on based path spaces abstract we prove the triviality of the first l 2 cohomology class of based path spaces of riemannian manifolds furnished with brownian motion measure and the consequent vanishing of l 2 harmonic one forms we give explicit formulae for closed and co closed one forms expressed as differentials of functions and co differentials of l 2 two forms respectively these are considered as extended clark ocone formulae a feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one forms to the exterior differentiation operator used to construct the de rham complex and the self adjoint laplacian on l 2 one forms this laplacian is shown to have a spectral gap

novel criteria for exponential stability of nonlinear differential systems with delay general nonlinear differential systems with time varying delay are considered several explicit criteria for global exponential stability are presented two examples are given to illustrate the obtained results

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

stochastic maximum principle for delayed doubly stochastic control systems and their applications in this paper we investigate the optimal control problems for delayed doubly stochastic control systems we first discuss the existence and uniqueness of the delayed doubly stochastic differential

existence of a smooth density for the filter in nonlinear filtering with infinite dimensional noise abstract in this paper we prove by means of malliavin calculus that under a local hormander condition the unnormalized filter associated with a nonlinear filtering problem in which the system process depends on an infinite dimensional noise admits a smooth density with respect to the lebesgue measure

generalized stochastic integrals and the malliavin calculus the paper first reviews the skorohod generalized stochastic integral with respect to the wiener process over some general parameter space t and it s relation to the malliavin calculus as the adjoint of the malliavin derivative some new results are derived and it is shown that every sufficiently smooth process ut t t can be decomposed into the sum of a malliavin derivative of a wiener functional and a process whose generalized integral over t vanishes using the results on the generalized integral the bismut approach to the malliavin calculus is generalized by allowing non adapted variations of the wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions

linear stochastic differential equations and wick products we establish the existence and uniqueness of the solution to a multidimensional linear skorohod stochastic differential equation with deterministic diffusion matrix using the notions of wick product andstransform if the diffusion matrix is constant and has real eigenvalues the solution is a stochastic process with moments of all orders provided that the initial condition is differentiable up to a suitable order the case of a diffusion matrix in the first wiener chaos is discussed in the last section

estimation de varadhan pour des diffusions a deux parametres sous des conditions de hormander generalisees la loi d une diffusion a parametre planaire possede une densite lisse en dehors des axes dans le cas elliptique nous etablissons une estimation de cette densite pur des petites valeurs du parametre celle ci constitue l analogue de l estimation de varadhan objet qui est bien connu dans le cas des diffusions ordinaires

regularity of skorohod integral processes based on integrands in a finite wiener chaos letf be a square integrable kernel on them dimensional unit cube u the skorohod integral process in themth wiener chaos associated with it isoperimetric inequalities for functions on wiener space yield the exponential integrability of the increments ofu to this result we apply the majorizing measure technique to show thatu possesses a continuous version and give an upper bound of its modulus of continuity

exponential stability of coupled linear delay time varying differential difference equations coupled linear delay time varying differential difference equations are considered explicit criteria for exponential stability of such equations are presented the obtained results are used to derive sufficient conditions for exponential stability in linear time varying differential equations of neutral type and in singular linear delay time varying differential equations some illustrative examples are given

zero sum stochastic differential games and backward equations abstract in this paper we are concerned with the applications to stochastic zero sum differential games of the results on backward stochastic differential equations using these techniques the existence of a saddle point in the bounded case is obtained if the isaacs condition holds furthermore this technique is also a very simple approach for the existence of an optimal strategy in the case of controlled diffusions

malliavin calculus for the stochastic 2d navier stokes equation we consider the incompressible two dimensional navier stokes equation with periodic boundary conditions under the effect of an additive white in time stochastic forcing under mild restrictions on the geometry of the scales forced we show that any finite dimensional projection of the solution possesses a smooth strictly positive density with respect to lebesgue measure in particular our conditions are viscosity independent we are mainly interested in forcing that excites a very small number of modes all of the results rely on proving the nondegeneracy of the infinite dimensional malliavin matrix c 2006 wiley periodicals inc

on a class of weak tchebycheff systems in this paper we study the approximation power the existence of a normalized b basis and the structure of a degree raising process for spaces of the form requiring suitable assumptions on the functions u and v the results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of cagd

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

second order taylor expansion for backward doubly stochastic control system we consider the second order taylor expansion for backward doubly stochastic control system the results are obtained under no restriction on the convexity of control domain moreover the control variable is allowed in the drift coefficient and the diffusion coefficient

linear stochastic differential equations with boundary conditions we study linear stochastic differential equations with affine boundary conditions the equation is linear in the sense that both the drift and the diffusion coefficient are affine functions of the solution the solution is not adapted to the driving brownian motion and we use the extended stochastic calculus of nualart and pardoux 16 to analyse them we give analytical necessary and sufficient conditions for existence and uniqueness of a solution we establish sufficient conditions for the existence of probability densities using both the malliavin calculus and the co aera formula and give sufficient conditions that the solution be either a markov process or a markov field

quasi linear pdes and forward backward stochastic differential equations weak solutions abstract in this paper we study the existence uniqueness and the probabilistic representation of the weak solutions of quasi linear parabolic and elliptic partial differential equations pdes in the sobolev space h 1 r d for this we study first the solutions of forward backward stochastic differential equations fbsdes with smooth coefficients regularity of solutions and their connection with classical solutions of quasi linear parabolic pdes then using the approximation procedure we establish their convergence in the sobolev space to the solutions of the fbsdes in the space l 2 r d r d l 2 r d r k l 2 r d r k d this gives a connection with the weak solutions of quasi linear parabolic pdes finally we study the unique weak solutions of quasi linear elliptic pdes using the solutions of the fbsdes on infinite horizon

anticipated backward doubly stochastic differential equations in this paper we deal with a new type of differential equations called anticipated backward doubly stochastic differential equations anticipated bdsdes the coefficients of these bdsdes depend on the future value of the solution y z we obtain the existence and uniqueness theorem and a comparison theorem for the solutions of these equations besides as an application we also establish a duality between the anticipated bdsdes and the delayed doubly stochastic differential equations delayed dsdes

covariance identities and inequalities for functionals on wiener and poisson spaces we present covariance identities and inequalities for functionals of the wiener and the poisson processes using malliavin calculus techniques an expansion with a remainder term is obtained for the covariance of such functionals our results extend known identities and inequalities for functions of multivariate random vectors

stochastic calculus with respect to gaussian processes in this paper we develop a stochastic calculus with respect to a gaussian process of the form b t int t 0 k t s dw s where w is a wiener process and k t s is a square integrable kernel using the techniques of the stochastic calculus of variations we deduce change of variable formulas for the indefinite integrals and we study the approximation by riemann sums the particular case of the fractional brownian motion is discussed

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

shape preserving properties of generalized bernstein operators on extended chebyshev spaces we study the existence and shape preserving properties of a generalized bernstein operator b n fixing a strictly positive function f 0 and a second function f 1 such that f 1 f 0 is strictly increasing within the framework of extended chebyshev spaces u n the first main result gives an inductive criterion for existence suppose there exists a bernstein operator b n c a b u n with strictly increasing nodes fixing f 0 f 1 in u n if u n subset u n 1 and u n 1 has a non negative bernstein basis then there exists a bernstein operator b n 1 c a b u n 1 with strictly increasing nodes fixing f 0 and f 1 in particular if f 0 f 1 f n is a basis of u n such that the linear span of f 0 f k is an extended chebyshev space over a b for each k 0 n then there exists a bernstein operator b n with increasing nodes fixing f 0 and f 1 the second main result says that under the above assumptions the following inequalities hold b n f geq b n 1 f geq f furthermore b nf is f0 f1 convex for all f0 f1 convex functions f in c a b

a change of variables formula for stratonovich integrals and existence of solutions for two point stochastic boundary value problems existence results for a class of two point non adaptive stochastic boundary value problems are derived by combining a new change of variables formula with optimal control techniques the class of equations treated is significantly wider than that for which previously known existence results apply

euler maruyama approximations for sdes with non lipschitz coefficients and applications abstract in this paper euler maruyama approximation for sde with non lipschitz coefficients is proved to converge uniformly to the solution in l p space with respect to the time and starting points as an application we also study the existence of solution and large deviation principle for anticipative sde with random initial condition

the onsager machlup functional for a class of anticipating processes consider the solution xt 0 t 1 of the equation x t x 0 w t mathop smallint limits 0 t b x s d s wherew wt 0 t 1 is ad dimensional wiener process b is a lipschitz function from d into d andx0 is ad dimensional functional of the wiener process

generalized brownian functionals and the solution to a stochastic partial differential equation abstract a class of generalized wiener functionals related to those of hida and watanabe is introduced and the malliavin calculs associated with these functionals is developed these notions are applied to the derivation of a solution to the stochastic partial differential equation y t ly y where l is a second order partial differential operator in the n space variables denotes a white noise in the n 1 space time variables y denotes a skorohod type stochastic integral and is a non random function of the n 1 independent variables

markov field property of stochastic differential equations the purpose of this paper is to prove a characterization of the conditional independence of two independent random variables given a particular functional of them in terms of a factorization property as an application we discuss the markov field property for solutions of stochastic differential equations with a boundary condition involving the values of the process at times t 0 and t 1

on the optimal stability of bases of univariate functions this paper is concerned with bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation the only bases considered are those whose elements have no sign changes among these an optimally stable basis is characterized under the assumption that the set of points where each basis function is nonzero is an interval a uniqueness result and many examples of such optimally stable bases are also provided

a family of bivariate rational bernstein operators we construct a new family of triangular and tensor product bivariate rational bernstein operators the operators reproduce linear polynomials we analyze the convergence of the operators rational bernstein operators are widely used in approximation theory and geometric modeling but in general they do not reproduce linear polynomials based on the work of p pi ul and p sablonniere we construct a new family of triangular and tensor product bivariate rational bernstein operators which are positive and reproduce the linear polynomials the main result is a proof of convergence of the bivariate rational bernstein operators defined on the square or triangle

stochastic equivariant cohomologies and cyclic cohomology we give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the chen souriau sense and to establish a link with cyclic cohomology with the second one we can establish a stochastic fixed point theorem

stochastic invariant imbedding application to stochastic differential equations with boundary conditions we study stochastic differential equations of the type dx t f t x t dt sum limits k 1 d sigma k t x t bigcirc dw t k x in mathbb r d t in 0 t 0

a two sided stochastic integral and its calculus let x be a forward diffusion and y a backward diffusion both defined on 0 1 x t and y t being respectively adapted to the past of a wiener process w and to its future increments we construct a two sided stochastic integral of the form mathop smallint limits 0 t phi u x u y u dw u which generalizes the backward and forward ito integrals simultaneously our construction is quite intuitive and leads to a generalized stochastic calculus it is also shown that for each fixed t our integral coincides with that defined by skorohod in 18

positivity and stability of coupled differential difference equations with time varying delays this paper studies the asymptotic stability of a special class of coupled delay differential difference equations with internal positive property an explicit characterization on the positivity of coupled differential difference equations is firstly given then based on the positivity of coupled differential difference equations with constant delays we investigate the entrywise monotonicity and asymptotic property of their state trajectories starting from appropriately chosen initial conditions furthermore the time varying delay system is analyzed through comparing with the corresponding constant delay system it turns out that an internally positive coupled differential difference equation with bounded time varying delays is asymptotically stable as long as the corresponding delay free system is asymptotically stable

integration by parts on wiener space and the existence of occupation densities we present a general criterion for the existence of an occupation density which is based on the integration by parts formula on wiener space this criterion is applied to two particular examples of anticipating processes first we discuss the case of brownian motion plus a nonadapted absolutely continuous drift and second we consider the case of a skorohod integral process finally a version of tanaka s formula is proved for the skorohod integral process 1 introduction an extension of the ito stochastic integral for processes which are not necessarily adapted to the brownian motion was introduced by skorohod in 12 the skorohod integral shares some of the probabilistic properties of the ito integral and on the other hand it has a more analytic flavor because it can be regarded as the adjoint of the derivative operator recently a stochastic calculus for skorohod integral processes has been developed in 9 and a version of the change of variable formula ito s formula for skorohod integral processes has been established a crucial point in the derivation of this formula is the fact that skorohod integral processes possess quadratic variations which are given in the same way as for their it6 counterparts and consequently are nontrivial that is the trajectories of these processes are continuous but highly irregular this observation leads us to investigate their occupation densities or local times different methods have been used so far to study the occupation densities of anticipating processes berman s idea to employ fourier analysis see 2 has been applied by imkeller in 6 7 to find integral criteria for the existence of a square integrable local time for a skorohod integral process which belongs to the second wiener chaos a more stochastic approach has been presented in 8 again for processes living in the second wiener chaos although this method can be generalized to deal with more general processes its main idea is to represent the skorohod integral process we are dealing with as the composition of a gaussian semimartingale depending on an infinite dimensional parameter with a gaussian vector and to use an infinite dimensional generalization of kolmogorov s continuity criterion in order to find the local time of this composition

stochastic evolution equations with random generators we prove the existence of a unique mild solution for a stochastic evolution equation on a hilbert space driven by a cylindrical wiener process the generator of the corresponding evolution system is supposed to be random and adapted to the filtration generated by the wiener process the proof is based on a maximal inequality for the skorohod integral deduced from the ito s formula for this anticipating stochastic integral

anticipated backward stochastic differential equations in this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations anticipated bsdes in these equations the generator includes not only the values of solutions of the present but also the future we show that these anticipated bsdes have unique solutions a comparison theorem for their solutions and a duality between them and stochastic differential delay equations

estimating stable delay intervals with a discretized lyapunov krasovskii functional formulation abstract in general a system with time delay may have multiple stable delay intervals especially a stable delay interval does not always contain zero asymptotically accurate stability conditions such as discretized lyapunov krasovskii functional dlf method and sum of square sos method are especially effective for such systems in this article a dlf based method is proposed to estimate the maximal stable delay interval accurately without using bisection when one point in this interval is given the method is formulated as a generalized eigenvalue problem gevp of linear matrix inequalities lmis and an accurate estimate may be reached by iteration either in a finite number of steps or asymptotically the coupled differential difference equation formulation is used to illustrate the method however the idea can be easily adapted to the traditional differential difference equation setting

calcul stochastique non adapte pour des processus a deux parametres formules de changement de variables de type stratonovitch et de type skorohod nous etablissons une formule de changement de variables de type stratonovitch pour des processus non adaptes indexes par 0 1 2 qui est analogue a celle existant en calcul deterministe pour cela nous definissons une integrale de stratonovitch simple resp double nous en deduisons une formule de changement de variables de type skorohod ne contenant pas d integrale curviligne nous utilisons des regularisations du processus de wiener obtenues par convolution avec une approximation de l unite

decroissance non exponentielle du noyau de la chaleur in this paper we give estimates on the heat kernel on the diagonal in terms of exp logt 1 when1 vanishes moreover we compute by means of the stochastic calculus of variations the decay of the flatness of some heat kernels on the other hand we show that the computations in both cases are very similar

diagonal versions and quantum stochastic integrals on the symmetric fock space with nonadapted integrands we develop a quantum stochastic calculus for nonadapted processes and the three standard integrators on the symmetric fock space over an arbitrary polish space the integrable processes are restricted to a class of operators which possess some kernel description our approach is mainly based on the concept of diagonalized versions 16 in the case of the real line we consider connections to the usual stochastic and quantum stochastic calculi and derive ito formulae besides the standard integrals with two integrands and one integrator these formulae lead to a new type of quantum stochastic integrals which has two integrators and three integrands

evolution equation of a stochastic semigroup with white noise drift we study the existence and uniqueness of the solution of a functionvalued stochastic evolution equation based on a stochastic semigroup whose kernel p s t y x is brownian in s and t the kernel p is supposed to be measurable with respect to the increments of an underlying wiener process in the interval s t the evolution equation is then anticipative and choosing the skorohod formulation we establish existence and uniqueness of a continuous solution with values in l r as an application we prove the existence of a mild solution of the stochastic parabolic equation dut xudt v dt x ru f t x u w dt x where v and w are brownian in time with respect to a common ltration in this case p is the formal backward heat kernel of x v dt x rx supported by the dgicyt grant no pb96 0087 ysupported by nsf international opportunities postdoctoral fellowship no int9600278 and nsf nato postdoctoral fellowship no dge 9633937 ams subject classi cation primary 60h15 secondary 60h07

convergence of rational bernstein operators abstract in this paper we discuss convergence properties and error estimates of rational bernstein operators introduced by pi ul and sablonniere it is shown that the rational bernstein operators converge to the identity operator if and only if the maximal difference between two consecutive nodes is converging to zero further a voronovskaja theorem is given based on the explicit computation of higher order moments for the rational bernstein operator

frame bundle of riemannian path space and ricci tensor in adapted differential geometry abstract the vanishing of the renormalized ricci tensor of the path space above a ricci flat riemannian manifold is discussed

viability property for a backward stochastic differential equation and applications to partial differential equations in the present paper we study conditions under which the solutions of a backward stochastic differential equation remains in a given set of constraints this property is the so called viability property in a separate section this condition is translated to a class of partial differential equations

quantum and non causal stochastic calculus the quantum stochastic calculus initiated by hudson and parthasarathy and the non causal stochastic calculus originating with the papers of hitsuda and skorohod are two potent extensions of the ito calculus currently enjoying intensive development the former provides a quantum probabilistic extension of schrodinger s equation enabling the construction of a markov process for a quantum dynamical semigroup the latter allows the treatment of stochastic differential equations which involve terms which anticipate the future in this paper the close relationship between these theories is displayed and a noncausal quantum stochastic calculus already in demand from physics is described

stability of coupled functional differential difference equations general coupled functional differential difference equations are considered a new approach to exponential stability of such equations is presented consequently some novel criteria for exponentia