time reversal for infinite dimensional diffusions in this paper we analize the reversibility of the diffusion property for the solution of certain infinite dimensional systems of stochastic differential equations necessary and sufficient conditions ensuring this reversibility are given the proofs use the techniques of the stochastic calculus of variations

white noise analysis for levy processes we construct a white noise theory for levy processes the starting point of this theory is a chaos expansion for square integrable random variables we use this approach to malliavin calculus to prove the following white noise generalization of the clark haussmann ocone formula for levy processes f e f m 1 0te dt m f ft y t m dt here e f is the generalized expectation the operators dt m f m 1 are generalized malliavin derivatives is the wick product and for all m 1y t m is the white noise of power jump processes yt m in particular y t 1 is the white noise of the levy process the formula holds for all f g l2 where g is a space of stochastic distributions and is a white noise probability measure finally we give an application of this formula to partial observation minimal variance hedging problems in financial markets driven by levy processes

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

the malliavin calculus a functional analytic approach in his groundbreaking articles s 91 malliavin introduced a technique for obtaining elliptic regularity results using function space calculus in his formulation the function space calculus on which his theory rests is the martingale calculus coming from the ornstein uhlenbeck process on wiener space cf 121 for a detailed discussion of malliavin s theory from this point of view although this formulation has a great deal to recommend it and is particularly pleasing to afficionados of the so called brownian sheet most analysts including those who are reasonably facile with the machinery of probability theory are unlikely to find the technical difficulties inherent in this approach outweighed by the eventual rewards shigekawa 111 discovered that there is an alternative formulation of malliavin s theory in shigekawa s formulation no mention need be made of the ornstein uhlenbeck process instead what he relies on is a sobolev type extension of the frechet dervative with wiener measure playing the role played by lebesgue measure in the finite dimensional context a third formulation of malliavin s theory was recently provided by bismut l bismut s idea is to exploit the quasi invariance of wiener measure as it is manifested in a beautiful relation due to haussmann 4 the present article has two aims in the first place section 1 2 and 3 are devoted to yet another formulation of malliavin s calculus although the end results are precisely the same as those obtained in 121 no reference is made to the ornstein uhlenbeck process on which the original theory depended instead the machinery is entirely that of elementary functional analysis and is devoid of any sophisticated martingale calculus it has been my intent that these three sections should be readily accessible to any analyst who has had some acquintance with wiener measure

a convex log concave correlation inequality for gaussian measure and an application to abstract wiener spaces this paper deals with a generalization of a result due to brascamp and lieb which states that in the space of probabilities with log concave density with respect to a gaussian measure on open image in new window this gaussian measure is the one which has strongest moments we show that this theorem remains true if we replace x by a general convex function then we deduce a correlation inequality for convex functions quite better than the one already known finally we prove results concerning stochastic analysis on abstract wiener spaces through the notion of approximate limit

multiple wiener ito integrals possessing a continuous extension letf w be a wiener functional defined byf w in f wherein f denotes the multiple wiener ito integral of ordern of the symmetricl2 0 1 n kernelf we show that a necessary and sufficient condition for the existence of a continuous extension off i e the existence of a function o from the continuous functions on 0 1 which are zero at zero to which is continuous in the supremum norms and for which o w f w a s is that there exists a multimeasure dt1 dt n on 0 1 n such thatf t1 t n t1 1 t n 1 a e lebesgue on 0 1 n recall that a multimeasure a1 a n is for every fixedi and every fixedai ai 1 ai 1 an a signed measure ina i and there exists multimeasures which are not measures it is furthermore shown that iff t1 t2 t n t1 1 t n 1 then all the tracesf k k leqq left frac n 2 right off exist eachf k induces ann 2k multimeasure denoted by k the following relation holds i n f sum limits k 0 n 2 left frac 1 2 right k frac n k n 2k int limits 0 1 n 2k w t 1 cdot cdot cdot w t n 2k cdot mu k dt 1 dt n 2k and each of the integrals in the above expression equals the multiple stratonovich or ogawa type integral of the tracef k namely int limits 0 1 n 2k w t 1 cdot cdot cdot w t n 2k mu k dt 1 dt n 2k i n 2k circ f k

onsager machlup functionals for non trace class spde s an onsager machlup functional limit is derived for a class of spde s whose principal part is not trace class both nondegenerate and degenerate limits are obtained and are illustrated by examples the proof uses fkg type inequalities

integration by parts and time reversal for diffusion processes in this paper we obtain necessary and sufficient conditions for the reversibility of the diffusion property assuming the existence of a density at every time t the proofs are based on techniques of the stochastic calculus of variations

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

integration par parties dans l espace de wiener et approximation du temps local let be a real valued stochastic process having a continuous local timel u t u 0 t t andxe t e x t t 0 the regularization ofx by means of the convolution with the approximation of unity e the main theorem in this paper theorem 3 5 is a generalization of various results about the approximation for fixedu of the local timel u by means of a convenient normalization of the numbernxe u of crossings of the processxe with the levelu especially this theorem extends to a class of not necessarily markovian continuous martingales a result of this type for one dimensional diffusions due to azais a2 the methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic processes with stochastic analysis techniques based upon integration by parts in the wiener space

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

valuation of boundary linked assets by stochastic boundary value problems solved with a wavelet collocation algorithm this article studies the valuation of boundary linked assets and their derivatives in continuous time markets valuing boundary linked assets requires the solution of a stochastic differential equation with boundary conditions which often is not markovian we propose a wavelet collocation algorithm for solving a milstein approximation to the stochastic boundary problem its convergence properties are studied furthermore we value boundary linked derivatives using malliavin calculus and monte carlo methods we apply these ideas to value european call options of boundary linked assets

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

geometric analysis of conditional independence on wiener space in this paper we study the conditional independence of sigma fields on the wiener space using the tools of the stochastic calculus of variations particular emphasis is given to the relation between the splitting of the random tangent spaces associated to the sigma fields and the conditional independence

transformation of wiener measure under anticipative flows lett f be a transformation from the wiener space to itself with the range off assumed to be in the cameron martin space the absolute continuity and the density function associated witht is considered t is assumed to be embedded in or defined through a parameterizationt t f t andf t is assumed to be differentiable int the paper deals first with the case where the range of thet derivative off t is also in the cameron martin space and new representations for the radon nikodym derivative and the carleman fredholm determinant are derived the case where thet derivative off t is not in the cameron martin space is considered next and results on the absolute continuity and the density function under conditions which are considerably weaker than previously known conditions are presented

absolute continuity of the law of an infinite dimensional wiener functional with respect to the wiener probability in this paper we study conditions ensuring that the law of ac 0 1 valued functional defined on an abstract wiener space is absolutely continuous with respect to the wiener measure onc 0 1 these conditions extend those established byp malliavin 12 13 for finite dimensional wiener functionals and those of 15 for hilbert valued functionals

ito s lemma without non anticipatory conditions a stochastic integral with respect to brownian motion which extends ito s integral to anticipatory integrands is constructed and investigated this stochastic integral is different from the skorokhod integral the ito lemma is proved for this integral

representation of the distributions on wiener space and stochastic calculus of variations abstract in this work we construct projections and dual projections with respect to the past of the wiener process for the cameron martin space valued distributions on the wiener space with their help we then prove on the one hand an extension of clark s formula to the representation of the scalar distributions with the extended ito integral and on the other hand we give a decomposition of the vector valued distributions by exactly calculating the kernel of the divergence operator

finite dimensional approximation of riemannian path space geometry abstract in this paper we construct a finite dimensional approximation for the geometry on the path space over a compact riemannian manifold this approximation allows to construct the horizontal lift of the ornstein uhlenbeck process on the path space through the markovian connection we also prove a representation formula for the heat semigroup on adapted vector fields as well as a commutation formula for its derivative

dynamic topologies and their applications to crisp topologies fuzzifications of crisp topologies and fuzzy topologies on the crisp real line abstract we introduce and develop topologies whose members are dynamic sets a la zeleny this development spawns the notions of phase and co phase spaces fuzzy dual dynamic dual and co fuzzy dual with the following results being obtained result a the notion of fuzzy dual provides a unifying framework for the diverse fuzzification schemes of hutton gantner steinlage warren klein and lowen result b the dynamic duals of the fuzzy real lines are both homeostatic and coheterostatic result c the fuzzy real lines re interpreted as fuzzy duals of r with the usual euclidean topology t induce on r via the notion of co fuzzy dual a class of canonical stratified fuzzy topologies co r l e g if l is a demorgan algebra and lc 1 then the compactness of a r is equivalent to the ordinary compactness of a w r t t and if l is a hutton algebra r l co r l is metrizable in the sense of hutton and erceg with erceg metric d l such that d l p r 1 p s 1 r s where pr is the crisp point or impulse with support r i e d l extends the euclidean metric on r

applications of localic separation axioms compactness axioms representations and compactifications to poslat topological spaces fuzzy topology first appeared about twenty seven years ago with chang s initial paper 1968 yet the fundamental issues of compactness and low order separation axioms remain unresolved this paper resolves these issues with respect to topological compactification categorical compactification and stone representation theorems using the theory of locales and previous work of the author in fuzzy sobriety our results give the first compactification reflectors for all traditional fuzzy topology construct classes of generalized stone representations of boolean algebras and distributive lattices and show categorically that the fuzzy analogue of classical compact hausdorff spaces is the categories of chang compact localic regular sober fuzzy topological spaces

application of malliavin calculus to a class of stochastic differential equations in this article we deal with stochastic differential equations driven by an infinite dimensional brownian motion under some non degeneracy conditions the existence and smoothness of the density for the law of the solution is proved

quantum martingale measures and stochastic partial differential equations in fock space a concept of quantum martingale measure is introduced and examples are constructed as quantum stochastic spectral integrals in fock space these are then utilized as space time noise to drive a parabolic stochastic partial differential equation spde we establish the existence and uniqueness of the solutions as families of densely defined closable operators in fock space that are jointly continuous in time and space variables and satisfy a markov property

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

occupation densities for stochastic integral processes in the second wiener chaos the two point distributions of skorohod integral processes in the second wiener chaos are mainly described by a hilbert schmidt operatort giving the mutual interaction of infinitely many gaussian components and by simple multiplication operators so are the fourier transforms of their occupation measures this enables us to use the well known fourier analytic criterion discovered and elaborated by berman to derive integral conditions for the existence of their occupation densities in terms of associated hilbert schmidt operators ift is a trace class operator we get a necessary and sufficient criterion if it is not still a sufficient one in a case in which the interaction is particularly simple we verify the appropriate integral condition and show that the results are essentially beyond the reach of enlargement of filtrations techniques of semimartingale theory

malliavin derivatives and derivatives of functionals of the wiener process with respect to a scale parameter let f c 0w be a functional of the wiener process with variance parameter c 2 0 and let f cw be an extension of f c 0w to f cw c in 0 c 0 relations are derived between the malliavin derivatives between the derivatives with respect to the scale parameter partial f rho cw partial rho p 1 and noncoherent derivatives such as de f cw sqrt varepsilon c tilde w mid w d varepsilon varepsilon 0 where tilde w is another wiener process independent of w and between the generator of the nontime homogeneous ornstein uhlenbeck process

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

a conditional approach to the anticipating girsanov transformation we study the law of a stochastic differential equation d xi t d omega t k 1 xi omega dt where the drift anticipates the future behavior of the brownian path for example the endpoint we first investigate anticipation of the endpoint using a conditional girsanov transformation and methods of malliavin calculus a combination with results of buckdahn 2 leads to new versions of the anticipating girsanov transformation of ramer and kusuoka and in particular to explicit formulas for the carleman fredholm determinant

the ito formula for anticipative processes with nonmonotonous time scale via the malliavin calculus this work is devoted to derive ito type formulae for anticipative stochastic processes with nonmonotonous time using the malliavin calculus techniques and the fundamental theorem of the differential calculus the same method is applied also to give an ito ventcell type formula in the anticipative case

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

theito formula for quantum semimartingales we construct a sequence of concordance invariants for classical links which depend on the peripheral isomorphism type of the nilpotent quotients of the link fundamental group the terminology stems from the fact that we replace the magnus expansion in the definition of milnor s bar mu invariants by the similar campbell hausdorff expansion the main point is that we introduce a new universal indeterminacy which depends only on the number of components of the link the campbell hausdorff invariants are new effectively computable and can efficiently distinguish unordered and unoriented isotopy types of links as we indicate on several families of closed braid examples they also satisfy certain natural dependence relations which generalize well known symmetries of the bar mu invariants 1991 mathematics subject classification 81s25 46l10 46l50 47a60

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 structure of independence on wiener space motives par des resultats de travaux anterieurs sur la caracterisation de l independance de variables aleatoires dans l espace de wiener les auteurs etudient ici la relation entre l independance de certaines classes de sous algebres et l orthogonalite des h derivees faibles des variables aleatoires qui engendrent ces sous algebres ou c h est l espace de wiener classique

stochastic calculus of variations for stochastic partial differential equations abstract this paper develops the stochastic calculus of variations for hilbert space valued solutions to stochastic evolution equations whose operators satisfy a coercivity condition an application is made to the solutions of a class of stochastic pde s which includes the zakai equation of nonlinear filtering in particular a lie algebraic criterion is presented that implies that all finite dimensional projections of the solution define random variables which admit a density this criterion generalizes hypoellipticity type conditions for existence and regularity of densities for finite dimensional stochastic differential equations

applications of the degree theorem to absolute continuity on wiener space let h p be an abstract wiener space and define a shift on byt f wheref is anh valued random variable we study the absolute continuity of the measurespot 1and f p ot 1 with respect top using the techniques of the degree theory of wiener maps where f det2 1 f exp f 1 2 f 2

anticipative girsanov transformations the transformations of measures induced by left w int limits 0 bullet k s ds right with ks possibly anticipating the wiener process ws is discussed and a girsanovtype theorem under rather weak assumptions on ks is derived

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

l topological meet theorems kindler gave a purely topological condition for an intersection of subsets of some set to be nonempty which plays an important role in topology this paper generalizes kindler s theorems to the fuzzy lattice based setting due to a proper choice of compactness and connectedness axioms it shows that almost all basic results of kindler survive in the extended setting

a quantum nonadapted ito formula and stochastic analysis in fock scale abstract a generalized definition of quantum stochastic qs integrals and differentials is given in the free of adaptiveness and dimensionality form in terms of malliavin derivative on a projective fock space and their uniform continuity with respect to the inductive limite convergence is proved a new form of qs calculus based on an inductive algebraic structure in an indefinite space is developed and a nonadaptive generalization of the qs ito formula for its representation in fock space is derived 1 the problem of solution of general qs evolution equations in a hilbert space is solved in terms of the constructed operator representation of chronological products defined in the indefinite space and the isometry and homomorphism property respectively for operators and maps of these solutions corresponding to the peseudounitary and homomorphism property of the qs integrable generators is proved

boundary value processes estimation and identification recent results obtained for boundary value processes and the associated smoothing and identification problems are presented in this paper both lumped and distributed parameter models are considered some open problems are discussed and the fundamental mathematical difficulties that arise in studying nonlinear extensions of the proposed models are mentioned

on the relation between the stratonovich and ogawa integrals l auteur montre que l existence de l integrale d ogawa implique l existence de l integrale de stratonovich et que les deux sont egales sous condition d une definition convenable de l integrale stochastique de stratonovich non adaptee

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

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

lowen para lowen and α level functors and fuzzy topologies on the crisp real line on etudie les relations entre des foncteurs et l de lowen definis pour des cadres l et les foncteurs para lowen g l et la classe cor l l des topologies floues sur r

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

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

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

some relations among classes of σ fields on wiener space in this work we study sigma fields and their tangent spaces on the wiener space which are invariant in some sense with respect to the basic operators of the malliavin calculus

fuzzy path and fuzzy connectedness abstract this paper discusses fuzzy connectedness in fuzzy topology zadeh s classical paper 1 has introduced the fundamental concept of a fuzzy set upon which the foundation of fuzzy mathematics rests the theory of fuzzy topology is a branch of such mathematics our purpose is to introduce some basic concepts as fuzzy path fuzzy path connected set locally fuzzy path connected set in the fuzzy topological spaces and to prove some fundamental properties related to fuzzy connectedness

a white noise approach to a class of non linear stochastic heat equations abstract cauchy problems for a class of non linear stochastic evolution equations are studied they are formulated as integral equations for generalized random fields by methods of white noise analysis s transformation characterization theorem etc these problems are reduced to fixed point problems in appropriatly constructed banach spaces this technique provides a systematic treatment of existence and uniqueness questions for a variety of equations the method is applied to non linear heat equations non linear volterra equations and non linear ordinary differential equations which may all be anticipating for each case examples are given such as the stochastic burgers equation and stochastic reaction diffusion equations

convergence to spdes in stratonovich form we consider the perturbation of parabolic operators of the form t p x d by large amplitude highly oscillatory spatially dependent potentials modeled as gaussian random fields the amplitude of the potential is chosen so that the solution to the random equation is affected by the randomness at the leading order we show that when the dimension is smaller than the order of the elliptic pseudo differential operator p x d the perturbed parabolic equation admits a solution given by a duhamel expansion moreover as the correlation length of the potential vanishes we show that the latter solution converges in distribution to the solution of a stochastic parabolic equation with multiplicative noise that should be interpreted in the stratonovich sense the theory of mild solutions for such stochastic partial differential equations is developed

spectral families of quantum stochastic integrals we develop a theory of spectral integration for quantum stochastic integrals of certain families of processes driven by creation conservation and annihilation processes in fock space these give a non commutative generalisation of classical stochastic integrals driven by poisson random measures a stochastic calculus for these processes is developed and used to obtain unitary operator valued solutions of stochastic differential equations as an application we construct stochastic flows on operator algebras driven by levy processes with finite levy measure

approximation solution of nonlinear stratonovich volterra integral equations by applying modification of hat functions in this paper we studied the hyers ulam rassias stability for nonlinear stratonovich volterra integral equations also we obtained stochastic operational matrix of modification of hat functions mhfs to numerically solve the nonlinear stratonovich volterra integral equations the main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved finally numerical examples are presented to certify convergence and accuracy of the method

transformation of the wiener measure under non invertible shifts in this paper we consider the transformation of measure induced by a not necessarily invertible perturbation of the identity the radon nikodym density for the image of the wiener measure and the associated girsanov type density are derived an application of these results yields an extension of the degree theorem

covariance inequalities induced by biorthogonal operators abstract the logarithmic derivative of a weakly second order probability measure on a general vector space x is an example of an operator from x into l 2 which preserves the pairing of x and x under the inner product of l 2 this biorthogonality property is exploited to obtain by means of comparison a lower bound on s covariance operator the logarithmic derivative is then explicitly evaluated for probability measures induced by the solutions to stochastic ordinary and partial differential equations

random non linear wave equations smoothness of the solutions we show existence and uniqueness for the solution of a onedimensional wave equation with non linear random forcing then we give sufficient conditions for the solution at a given time and a given point to have a density and for this density to be smooth the proof uses the extension of the malliavin calculus to the two parameters wiener functionals