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

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

transformation de fourier et temps d occupation browniens in this paper we study oscillatory stochastic integrals of the form gamma lambda int limits 0 infty exp i lambda b s g s d s where is a non zero parameter andg a square integrable function we study integrability properties of and its behavior as a function of using stochastic calculus techniques martingale theory representation of ito for a random variable of the wiener space lemma of garsia rodemich rumsey we also obtain limit theorems in law related to the variables based upon an asymptotic version of a theorem of knight on orthogonal continuous martingales

quasi sure analysis and stratonovich anticipative stochastic differential equations consider a stochastic differential equation on d with smooth and bounded coefficients we apply the techniques of the quasi sure analysis to show that this equation can be solved pathwise out of a slim set furthermore we can restrict the equation to the level sets of a nondegenerate and smooth random variable and this provides a method to construct the solution to an anticipating stochastic differential equation with smooth and nondegenerate initial condition

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 with functional boundary conditions we consider linear nth order stochastic differential equations on 0 1 with linear boundary conditions supported by a finite subset of 0 1 we study some features of the solution to these problems and especially its conditional independence properties of markovian type

stochastic heat equation with random coefficients we prove the existence of a unique solution for a one dimensional stochastic parabolic partial differential equation with random and adapted coefficients perturbed by a two parameter white noise the proof is based on a maximal inequality for the skorohod integral deduced from ito s formula for this anticipating stochastic integral

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

the substitution theorem for semilinear stochastic partial differential equations abstract in this article we establish a substitution theorem for semilinear stochastic evolution equations see s depending on the initial condition as an infinite dimensional parameter due to the infinite dimensionality of the initial conditions and of the stochastic dynamics existing finite dimensional results do not apply the substitution theorem is proved using malliavin calculus techniques together with new estimates on the underlying stochastic semiflow applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations spde s with anticipating initial conditions and non ergodic stationary solutions in particular our result gives a new existence theorem for solutions of semilinear stratonovich spde s with anticipating initial conditions

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

the robustness optimization of a multivariable feedback system in hankel norm space this paper is concerned with the optimization of robustness in a multi input multi output linear time invariant finite dimensional feedback system the objective is to derive a closed form solution to the problems of the synthesis of optimal controllers that either minimize the weighted sensitivity function or maximize the excess robust ness with respect to hankel norm criterion an all pass solution for the minimum hankel norm is employed to treat our problems finally the infinite norm case is also discussed

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

weak approximations a malliavin calculus approach we introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that may be anticipating our main assumption is that the process and the initial random variable have to be smooth in the malliavin sense furthermore if the inverse of the malliavin covariance matrix associated with the process under consideration is sufficiently integrable then approximations for densities and distributions can also be achieved we apply these ideas to the case of stochastic differential equations with boundary conditions and the composition of two diffusions

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

a relation algebraic model of robust correctness we propose a new and uniform abstract relational approach to demonic nondeterminism and robust correctness similar to hoare s chaos semantics it is based on a specific set of relations on flat lattices this set forms a complete lattice furthermore we deal with the refinement of programs among other things we show the correctness of the unfold fold method for demonic nondeterminism and robust correctness as refinement relation and investigate relationships to dijkstra s wp calculus and morgan s specification statement authors abstract

the malliavin calculus and stochastic delay equations we consider stochastic delay systems dx t g x t r dw t driven by multi dimensional brownian motion w the diffusion coefficient g is smooth with a possible degeneracy at 0 for a large class of deterministic initial paths we show that the solution x t admits a smooth density with respect to lebesgue measure the proof is based on malliavin calculus together with new probabilistic lower bounds on the solution x

a different quantum stochastic calculus for the poisson process we show that a gradient operator defined by perturbations of the poisson process jump times can be used with its adjoint operator instead of the annihilation and creation operators on the poisson charlier chaotic decomposition to represent the poisson process the quantum stochastic integration and the ito formula are developed accordingly leading to commutation relations which are different from the ccr an analog of the weyl representation is defined for a subgroup ofsl 2 showing that the exponential and geometric distributions are closely related in this approach

a characterization of reciprocal processes via an integration by parts formula on the path space we characterize in this paper the class of reciprocal processes associated to a brownian diffusion therefore not necessarily gaussian as the set of probability measures under which a certain integration by parts formula holds on the path space this functional equation can be interpreted as a perturbed duality equation between malliavin derivative operator and stochastic integration an application to periodic ornstein uhlenbeck process is presented we also deduce from our integration by parts formula the existence of nelson derivatives for general reciprocal processes

linear skorohod stochastic differential equations let andb be bounded processes on the wiener space p c 0 1 which are possibly anticipating the brownian motionw t t and let be a bounded random variable we deduce the existence and uniqueness of a solutionx for the linear equation with skorohod integral x t eta int limits 0 t sigma s x s dw s int limits 0 t b s x s ds t in 0 1 1 under rather weak assumptions on and no additional requirement onb and the description of the solutionx requires to study the family t t t 0 1 of transformationt t of into itself associated to 1 by the equation t t omega int limits 0 t wedge sigma s t s omega ds omega in omega t in 0 1

boundary value problems for stochastic differential equations in this paper we study stochastic differential equations with boundary conditions at the endpoints of a time interval instead of the customary initial condition we present existence and uniqueness results and study the markov property of the solution in the one dimensional case we prove that the solution is a markov field operatorname iff the drift is affine

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

linear estimation of boundary value stochastic processes part i the role and construction of complementary models this paper presents a substantial extension of the method of complementary models for minimum variance linear estimation introduced by weinert and desai in their important paper 1 specifically the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions a major contribution of this paper is an application of green s identity in deriving a differential operator representation of the estimator to clarify the development and to illustrate the range of applications of our approach two brief examples are provided one is a 1 d discrete two point boundary value process and the other is a 2 d process governed by poisson s equation on the unit disk

a new approach to stochastic evolution equations with adapted drift abstract in this paper we develop a new approach to stochastic evolution equations with an unbounded drift a which is dependent on time and the underlying probability space in an adapted way it is well known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term in this paper we give a new representation formula for the stochastic convolution which avoids integration of non adapted processes here we mainly consider the parabolic setting we establish connections with other solution concepts such as weak solutions the usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift at the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift

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

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

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

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

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

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

the riemann approach to stochastic integration using non uniform meshes in this note we shall prove that the stochastic integral with respect to a semimartingale can be defined by riemann s approach however in this approach we use non uniform meshes instead of the usual uniform meshes

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 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

on the quasi linear reflected backward stochastic partial differential equations abstract this paper is concerned with the quasi linear reflected backward stochastic partial differential equation rbspde for short basing on the theory of backward stochastic partial differential equation and the parabolic capacity and potential we first associate rbspde to a variational problem and via the penalization method we prove the existence and uniqueness of the solution for linear rbspde with laplacian leading coefficients with the continuity approach we further obtain the well posedness of general quasi linear rbspdes related results including ito formulas for backward stochastic partial differential equations with stochastic regular measures the comparison principle for solutions of rbspdes and the connections with reflected backward stochastic differential equations and optimal stopping problems are addressed as well

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

programs as partial graphs i flow equivalence and correctness abstract a common feature of most theoretical investigations on semantics correctness and termination is the strict distinction between the descriptional tool used for the flow of program control and the completely different tool describing single program steps since these different methods are difficult to handle when it is necessary that they be used together this paper presents a unified approach to the presentation of these concepts in terms of tarski s and riguet s relational algebra partial graphs and programs are introduced to formally manipulate relational notions of semantics correctness and termination the second part of this paper will extend these notions to systems of recursive programs

absolute continuity of distributions of solutions of anticipating stochastic differential equations abstract we consider a stochastic differential equation with anticipating initial value and drift and give two types of sufficient conditions for the absolute continuity of the distribution of the solution with respect to the lebesgue measure using the malliavin calculus

robust controllers for multivariable model matching systems abstract the robustness properties of multivariable perfect model matching systems are presented in order to stabilize the nominal system a general parametrized 2 d controller is introduced a necessary and sufficient condition of robust stabilization is derived for the perfect model matching system under external disturbances and non linear time varying perturbations the results presented could be viewed as the extension of the work by astrom 1980

2 d exact model matching the exact model matching problem in single input single output 2 d systems is solved by applying polyaomial techniques all solutions are described in a parametric form natural solvability conditions for causal and or stable solutions are given and a simple design procedure is proposed it consists in solving a linear 2 d polynomial equation which directly yields a suitable compensator

an inverse random source problem for the helmholtz equation this paper is concerned with an inverse random source problem for the one dimensional stochastic helmholtz equation which is to reconstruct the statistical properties of the random source function from boundary measurements of the radiating random electric field although the emphasis of the paper is on the inverse problem we adapt a computationally more efficient approach to study the solution of the direct problem in the context of the scattering model specifically the direct model problem is equivalently formulated into a two point spatially stochastic boundary value problem for which the existence and uniqueness of the pathwise solution is proved in particular an explicit formula is deduced for the solution from an integral representation by solving the two point boundary value problem based on this formula a novel and efficient strategy which is entirely done by using the fast fourier transform is proposed to reconstruct the mean and the variance of the random source function from measurements at one boundary point where the measurements are assumed to be available for many realizations of the source term numerical examples are presented to demonstrate the validity and effectiveness of the proposed method

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

embedding a demonic semilattice in a relation algebra we present a refinement ordering between binary relations viewed as programs or specifications this ordering induces a complete join semilattice that can be embedded in a relation algebra this embedding then allows an easy proof of many properties of the refinement semilattice by making use of the well known corresponding properties of relation algebras the operations of the refinement semilattice corresponding to join and composition in the embedding algebra are respectively demonic join and demonic composition the weakest prespecification and postspecification operators of hoare and he defined over a relation algebra also have corresponding operators in the semilattice

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

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

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

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

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

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

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

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

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

on the non commutative fractional wishart process abstract we investigate the process of eigenvalues of a fractional wishart process defined by n b b where b is the matrix fractional brownian motion recently studied in 18 using stochastic calculus with respect to the young integral we show that with probability one the eigenvalues do not collide at any time when the matrix process b has entries given by independent fractional brownian motions with hurst parameter h 1 2 1 we derive a stochastic differential equation in the malliavin calculus sense for the eigenvalues of the corresponding fractional wishart process finally a functional limit theorem for the empirical measure valued process of eigenvalues of a fractional wishart process is obtained the limit is characterized and referred to as the non commutative fractional wishart process which constitutes the family of fractional dilations of the free poisson distribution

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

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

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

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

parallel composition and decomposition of specifications in the discipline of derivation of programs from specifications by refinement the demonic calculus of relations is a well established tool which allows the use of demonic operators on relations considered as specifications of a program s input output behaviour together with the standard relation algebraic operators to reason about specifications ordered by the refinement ordering the central contribution of this paper is the introduction of a demonic product operator that embodies a natural concept of parallel composition of specifications of course this raises the question of parallel decomposition which in turn motivates the introduction of demonic implication and totalisation operators finally we also present first results concerning parallel decomposition

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

optimal control of mean field backward doubly stochastic systems driven by ito levy processes abstractin this paper we introduce a new class of backward doubly stochastic differential equations in short bdsde called mean field backward doubly stochastic differential equations in short m

integration with respect to fractal functions and stochastic calculus i the classical lebesgue stieltjes integral b a fdg of real or complex valued functions on a finite interval a b is extended to a large class of integrands f and integrators g of unbounded variation the key is to use composition formulas and integration by part rules for fractional integrals and weyl derivatives in the special case of holder continuous functions f and g of summed order greater than 1 convergence of the corresponding riemann stieltjes sums is proved the results are applied to stochastic integrals where g is replaced by the wiener process and f by adapted as well as anticipating random functions in the anticipating case we work within slobodeckij spaces and introduce a stochastic integral for which the classical ito formula remains valid moreover this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional brownian motion

various implementations of implementable transfer matrices an overall transfer matrix g f s is said to be implementable for a given plant g s if the resulting system uses only proper compensators has all forward paths passing through the plant is well posed and totally stable a necessary condition for an overall transfer matrix to be implementable is established the sufficiency of the condition is established by constructing several implementations the open loop unity feedback two input one output compensator and plant input output feedback configurations are used the implementations either extend or improve the existing results

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

on improving control loop robustness of model matching controllers this note first characterizes the class of all stabilizing controllers for a two degree of freedom control system which achieve a prescribed achievable transfer function the char acterization is in terms of an arbitrary proper stable transfer function with this characterization robust model matching is formulated as a standard hm optimization problem this means that standard controller designs for a nominal plant such as linear quadratic gaussian ones can be enhanced to give improved robustness properties using hm design tech niques

skorohod integration and stochastic calculus beyond the fractional brownian scale abstract we extend the skorohod integral allowing integration with respect to gaussian processes that can be more irregular than any fractional brownian motion this is done by restricting the class of test random variables used to define skorohod integrability a detailed analysis of the size of this class is given it is proved to be non empty even for gaussian processes which are not continuous on any closed interval despite the extreme irregularity of these stochastic integrators the skorohod integral is shown to be uniquely defined and to be useful an ito formula is established it is employed to derive a tanaka formula for a corresponding local time linear additive and multiplicative stochastic differential equations are solved an analysis of existence for the stochastic heat equation is given

skorohod stochastic differential equations of diffusion type leta b bec2 r1 functions with bounded derivatives of first and second order we study stochastic differential equations dx t a x t dw t b x t dt 0 leqq t leqq 1 whose initial valuex0 is a frechet differentiable random variable which may depend on the whole path of the driving brownian motion w t this anticipation requires to pass from the ito integral to the skorohod integral we show that the equation has a unique local solution x t 0 t t0 for sufficiently smallt0 0 and we provide conditions for the existence of a global solution x t 0 t 1

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

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

homogenization of a singular random one dimensional pde with time varying coefficients in this paper we study the homogenization of a nonautonomous parabolic equation with a large random rapidly oscillating potential in the case of one dimensional spatial variable we show that if the potential is a statistically homogeneous rapidly oscillating function of both temporal and spatial variables then under proper mixing assumptions the limit equation is deterministic and convergence in probability holds to the contrary for the potential having a microstructure only in one of these variables the limit problem is stochastic and we only have convergence in law

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

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

relational algebraic semantics of deterministic and nondeterministic programs abstract abstract relational algebra is proposed as a practical means to describe the denotational semantics of programming languages we apply this method of semantics description to a functional language and demonstrate the usefulness of this approach by some examples in particular we prove the correctness of a program transformation rule within our calculus then the deterministic language dfp is extended to a nondeterministic functional language nfp in order to investigate three kinds of nondeterminism viz erratic angelic and demonic nondeterminism we provide a relational algebraic semantics for each kind of nondeterminism these three different semantics are not obtained by using three different orderings on certain domains but by using three different interpretations on the same ordered domain the alternative kinds of nondeterminism are then compared and an illustrative example is given finally we characterize the natural numbers with relational algebraic means the uniform description of domains and programs makes it possible to prove the termination of concrete deterministic and nondeterministic programs by induction

girsanov theorem for anticipative shifts on poisson space we study the absolute continuity of the image measure of the canonical poisson probability measure under nonlinear shifts the radon nykodim density function is expressed using a carleman fredholm determinant and a divergence operator results are obtained for non necessarily invertible transformations under almost sure differentiability hypothesis

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

on a class of backward doubly stochastic differential equations in this paper a new class of backward doubly stochastic differential equations is studied this type of equations has a more general form of the forward ito integrals compared to the ones which have been studied until now we conclude that unique solutions of these equations can be represented with the help of solutions of the corresponding backward doubly stochastic differential equations considered earlier in paper 5 by pardoux and peng some comparison theorems are also given as well as a probabilistic interpretation for solutions of the corresponding quasilinear stochastic partial differential equations

stochastic equations with delay optimal control via bsdes and regular solutions of hamilton jacobi bellman equations we consider an ito stochastic differential equation with delay driven by brownian motion whose solution by an appropriate reformulation defines a markov process x with values in a space of continuous functions mathbf c with generator mathcal l we then consider a backward stochastic differential equation depending on x with unknown processes y z and we study properties of the resulting system in particular we identify the process z as a deterministic functional of x we next prove that the forward backward system provides a suitable solution to a class of parabolic partial differential equations on the space mathbf c driven by mathcal l and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects we also include applications to optimal stochastic control of differential equation with delay in particular we characterize optimal controls as feedback laws in terms of the process x