flows and stochastic taylor series in ito calculus for general stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the ito flow map is given the computation relies on the lift to quasi shuffle algebras of formulas involving products of ito integrals of semimartingales whereas the chen strichartz formula computing the logarithm of the stratonovich flow map is classically expanded as a formal sum indexed by permutations the analogous formula in ito calculus is naturally indexed by surjections this reflects the change of algebraic background involved in the transition between the two integration theories lastly we extend our formula for the quasi shuffle chen strichartz series for the logarithm of the flow map to the non commutative case for linear matrix valued sdes driven by arbitrary semimartingales we obtain a similar formula

some applications of miller mocanu lemma on certain classes of meromorphic functions in this paper making use of a linear operator we introduce and study certain new classes of meromorphic functions we derive some inclusion results these classes contain many known classes as a special case

flots et series de taylor stochastiques we study the expansion of the solution of a stochastic differential equation as an infinite sum of iterated stochastic stratonovitch integrals this enables us to give a universal and explicit formula for any invariant diffusion on a lie group in terms of lie brackets as well as a universal and explicit formula for the brownian motion on a riemannian manifold in terms of derivatives of the curvature tensor the first of these formulae contains and extends to the non nilpotent case the results of doss 6 sussmann 17 yamato 18 fliess and normand cyrot 7 krener and lobry 19 and kunita 11 on the representation of solutions of stochastic differential equations

the noor integral and strongly starlike functions abstract let a denote the class of analytic functions f z defined in the unit disc satisfying the condition f 0 f 0 1 0 we introduce some new subclasses of strongly starlike functions defined by the noor integral and study their properties

convenient categories of smooth spaces a chen space is a set x equipped with a collection of plots i e maps from convex sets to x satisfying three simple axioms while an individual chen space can be much worse than a smooth manifold the category of all chen spaces is much better behaved than the category of smooth manifolds for example any subspace or quotient space of a chen space is a chen space and the space of smooth maps between chen spaces is again a chen space souriau s diffeological spaces share these convenient properties here we give a unified treatment of both formalisms following ideas of penon and dubuc we show that chen spaces diffeological spaces and even simplicial complexes are examples of concrete sheaves on a concrete site as a result the categories of such spaces are locally cartesian closed with all limits all colimits and a weak subobject classifier for the benefit of differential geometers our treatment explains most of the category theory we use 2011 john c baez and alexander e hoffnung

stochastic lie group integrators we present lie group integrators for nonlinear stochastic differential equations with noncommutative vector fields whose solution evolves on a smooth finite dimensional manifold given a lie group action that generates transport along the manifold we pull back the stochastic flow on the manifold to the lie group via the action and subsequently pull back the flow to the corresponding lie algebra via the exponential map we construct an approximation to the stochastic flow in the lie algebra via closed operations and then push back to the lie group and then to the manifold thus ensuring that our approximation lies in the manifold we call such schemes stochastic munthe kaas methods after their deterministic counterparts we also present stochastic lie group integration schemes based on castell gaines methods these involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential lie series they become stochastic lie group integrator schemes if we use munthe kaas methods as the underlying ordinary differential integrator further we show that some castell gaines methods are uniformly more accurate than the corresponding stochastic taylor schemes lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes

inclusion properties for certain classes of meromorphic functions associated with the choi saigo srivastava operator abstract the purpose of the present paper is to introduce several new classes of meromorphic functions defined by using a meromorphic analogue of the choi saigo srivastava operator for analytic functions and investigate various inclusion properties of these classes some interesting applications involving these and other classes of integral operators are also considered

casimir chaos in a boson fock space abstract we investigate casimir processes corresponding to central elements of the universal enveloping algebra associated with a family of representations of the unitary lie group u n which arises naturally in n dimensional quantum stochastic calculus in particular for a family of casimir elements g n m whose eigenvalues in the irreducible representation with highest weight l 1 n 1 l 2 n 2 l n are linear combinations of elementary symmetric polynomials in l 1 l n we obtain chaotic decompositions using iterated stochastic integrals the terms of the chaotic decomposition are themselves casimir processes corresponding to casimir elements z n m we find explicit formulae for the eigenvalues of z n m in an irreducible represenation finally by passing to a product of fock spaces we realise in the context of quantum stochastic calculus the random walk on the dual of u n first constructed by bi ne

the euler scheme for levy driven stochastic differential equations limit theorems we study the euler scheme for a stochastic differential equation driven by a levy process y more precisely we look at the asymptotic behavior of the normalized error process un xn x where x is the true solution and xn is its euler approximation with stepsize 1 n and un is an appropriate rate going to infinity if the normalized error processes converge or are at least tight we say that the sequence un is a rate which in addition is sharp when the limiting process or processes is not trivial we suppose that y has no gaussian part otherwise a rate is known to be u n sqrt n then rates are given in terms of the concentration of the levy measure of y around 0 and further we prove the convergence of the sequence un xn x to a nontrivial limit under some further assumptions which cover all stable processes and a lot of other levy processes whose levy measure behave like a stable levy measure near the origin for example when y is a symmetric stable process with index 0 2 a sharp rate is un n logn 1 when y is stable but not symmetric the rate is again un n logn 1 when 1 but it becomes un n logn 2 if 1 and un n if 1

semi localizations of semi abelian categories abstract a semi localization of a category is a full reflective subcategory with the property that the reflector is semi left exact there are many interesting examples of semi localizations as for instance any torsion free subcategory of a semi abelian category by specializing a result due to s mantovani we first characterize the categories which are semi localizations of exact mal tsev categories we then prove a new characterization of protomodular categories in terms of binary relations allowing us to obtain an abstract characterization of the semi localizations of exact protomodular categories this result is very useful to study the hereditarily torsion free subcategories of semi abelian categories some examples are considered in detail in the categories of groups crossed modules commutative rings and topological groups we finally explain how these results extend the corresponding ones obtained in the abelian context by w rump

some inclusion relationships and integral preserving properties of certain subclasses of meromorphic functions associated with a family of integral operators the object of the present paper is to investigate a family of integral operators defined on the space of normalized meromorphic functions by making use of these novel integral operators we introduce and investigate several new subclasses of starlike convex close to convex and quasi convex meromorphic functions in particular we establish some inclusion relationships associated with the aforementioned integral operators several interesting integral preserving properties are also considered

some inclusion properties of a certain family of integral operators abstract the authors introduce several new subclasses of analytic functions which are defined by means of a general integral operator i and investigate various inclusion properties of these subclasses many interesting applications involving these and other families of integral operators are also considered

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

asymptotically efficient runge kutta methods for a class of itoˆ and stratonovich equations in certain applications of stochastic differential equations approximate solutions must be found that depend only on samples of the driving process it is known that the order of convergence of such approximations is limited and that some are asymptotically efficient in the sense that they minimize the leading coefficient in the expansion of mean square errors as power series in the sample step size this article develops asymptotically efficient runge kt tta methods that involve evaluations either of the coefficients of an ito equation or of the coefficients of the corresponding stratonovich equation simpler approximations which converge with the maximum possible order but which are not asymptotically efficient are also defined the runge kutta methods for ito equations differ from those designed for ordinary differential equations in that they involve terms in the square root of the sample step size the approximations are tested along with the classical euler method on five examples the simulatio

pre control form of the generalized campbell baker hausdorff dynkin formula for affine nonholonomic systems abstract in this paper a pre control form of the generalized campbell baker hausdorff dynkin gcbhd formula is presented this form is dedicated to nonholonomic affine systems often encountered in robotics the simplest possible expression for the pre control gcbhd is obtained as control dependent functions pre multiply elements of the ph hall basis algorithmic aspects of deriving the formula are highlighted an application of the pre control form of the gcbhd formula in motion planning of nonholonomic systems is also reported

subclasses of meromorphic functions associated with the cho kwon srivastava operator abstract we consider a multiplier transformation and some subclasses of the class of meromorphic functions which was defined by means of the hadamard product by n e cho o s kwon and h m srivastava in n e cho o s kwon h m srivastava inclusion and argument properties for certain subclasses of meromorphic functions associated with a family of multiplier transformations j math anal appl 300 2004 505 520

rough evolution equations we generalize lyons rough paths theory in order to give a pathwise meaning to some nonlinear infinite dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise as an illustration we discuss a class of linear and nonlinear 1d spdes driven by a space time gaussian noise with singular space covariance and brownian time dependence

cubature on wiener space it is well known that there is a mathematical equivalence between solving parabolic partial differential equations pdes and the integration of certain functionals on wiener space monte carlo simulation of stochastic differential equations sdes is a naive approach based on this underlying principle in finite dimensions it is well known that cubature can be a very effective approach to integration we discuss the appropriate extension of this idea to wiener space in the process we develop high order numerical schemes valid for high dimensional sdes and semi elliptic pdes

ramification of rough paths the stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended chen s multiplicative property involves the durr connes kreimer coproduct on rooted trees this turns out to be the natural setting for a non geometric theory of rough paths

numerical methods for nonlinear stochastic differential equations with jumps we present and analyse two implicit methods for ito stochastic differential equations sdes with poisson driven jumps the first method ssbe is a split step extension of the backward euler method the second method cssbe arises from the introduction of a compensated martingale form of the poisson process we show that both methods are amenable to rigorous analysis when a one sided lipschitz condition rather than a more restrictive global lipschitz condition holds for the drift our analysis covers strong convergence and nonlinear stability we prove that both methods give strong convergence when the drift coefficient is one sided lipschitz and the diffusion and jump coefficients are globally lipschitz on the way to proving these results we show that a compensated form of the euler maruyama method converges strongly when the sde coefficients satisfy a local lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p 2 under our assumptions both ssbe and cssbe give well defined unique solutions for sufficiently small stepsizes and ssbe has the advantage that the restriction is independent of the jump intensity we also study the ability of the methods to reproduce exponential mean square stability in the case where the drift has a negative one sided lipschitz constant this work extends the deterministic nonlinear stability theory in numerical analysis we find that ssbe preserves stability under a stepsize constraint that is independent of the initial data cssbe satisfies an even stronger condition and gives a generalization of b stability finally we specialize to a linear test problem and show that cssbe has a natural extension of deterministic a stability the difference in stability properties of the ssbe and cssbe methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non jump literature

a theory of regularity structures we introduce a new notion of regularity structure that provides an algebraic framework allowing to describe functions and or distributions via a kind of jet or local taylor expansion around each point the main novel idea is to replace the classical polynomial model which is suitable for describing smooth functions by arbitrary models that are purpose built for the problem at hand in particular this allows to describe the local behaviour not only of functions but also of large classes of distributions we then build a calculus allowing to perform the various operations multiplication composition with smooth functions integration against singular kernels necessary to formulate fixed point equations for a very large class of semilinear pdes driven by some very singular typically random input this allows for the first time to give a mathematically rigorous meaning to many interesting stochastic pdes arising in physics the theory comes with convergence results that allow to interpret the solutions obtained in this way as limits of classical solutions to regularised problems possibly modified by the addition of diverging counterterms these counterterms arise naturally through the action of a renormalisation group which is defined canonically in terms of the regularity structure associated to the given class of pdes our theory also allows to easily recover many existing results on singular stochastic pdes kpz equation stochastic quantisation equations burgers type equations and to understand them as particular instances of a unified framework one surprising insight is that in all of these instances local solutions are actually smooth in the sense that they can be approximated locally to arbitrarily high degree as linear combinations of a fixed family of random functions distributions that play the role of polynomials in the theory as an example of a novel application we solve the long standing problem of building a natural markov process that is symmetric with respect to the finite volume measure describing the phi 4 3 euclidean quantum field theory it is natural to conjecture that the markov process built in this way describes the glauber dynamic of 3 dimensional ferromagnets near their critical temperature

high strong order methods for non commutative stochastic ordinary differential equation systems and the magnus formula in recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations sodes as sodes are often more appropriate than their deterministic counterparts in many modelling situations however unlike the deterministic case numerical methods for sodes are considerably less sophisticated due to the difficulty in representing the possibly large number of random variable approximations to the stochastic integrals although burrage and burrage high strong order explicit runge kutta methods for stochastic ordinary differential equations applied numerical mathematics 22 1996 81 101 were able to construct strong local order 1 5 stochastic runge kutta methods for certain cases it is known that all extant stochastic runge kutta methods suffer an order reduction down to strong order 0 5 if there is non commutativity between the functions associated with the multiple wiener processes this order reduction down to that of the euler maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques an additional difficulty in solving sodes arises even in the linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple wiener processes thus in this present paper first the work of magnus on the exponential solution of differential equations for a linear operator communications on pure and applied mathematics 7 1954 649 673 applied to deterministic non commutative linear problems will be applied to non commutative linear sodes and methods of strong order 1 5 for arbitrary linear non commutative sode systems will be constructed hence giving an accurate approximation to the general linear problem secondly for general nonlinear non commutative systems with an arbitrary number d of wiener processes it is shown that strong local order i runge kutta methods with d 1 stages can be constructed by evaluated a set of lie brackets as well as the standard function evaluations a method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of wiener processes finally some numerical results are presented which illustrate the efficacy of these approaches c 1999 elsevier science b v all rights reserved

a characterization of quasi toposes a locally presentable category 6 is the category of models of some theory f which can be described using a limits for a regular cardinal a our aim is to find conditions which ensure good properties of the category d of models when is 6 cartesian closed when is d a quasi topos when is d a topos and so on in this paper we focus our attention on some semantic conditions a locally a presentable category 8 can thus be presented as the category of cc left exact presheaves on a small category 9 with a colimits clearly each subpresheaf s of an e left exact presheaf f has an a left exact closure sc f this closure operation plays a key role in the theory of a left exact functors and we study first conditions on dz lex which ensure the universality of that closure operation we prove this universality to be equivalent to the universality of strongly epimorphic families in but also to the fact that a lex is an epireflective subcategory of a grothendieck topos this grothendieck topos will play an important role in the rest of the paper its construction depends heavily on the fact that under our assumptions a left exact subpresheaves are stable under arbitrary intersections from the point of view of the internal logic of the topos 4 of presheaves among the epireflective subcategories of a grothendieck topos we find the categories of separated objects for a given topology in this topos these

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

on integral operators abstract let fn z z 1 z n 1 n no and f 1 n be defined such that f n f 1 n z 1 z where denotes convolution hadamard product let f be analytic in the unit disc e we define a new operator inf f 1 n f which is analogous to one defined by ruscheweyh using this operator the classes m n are defined a function f analytic in e is in m n if and only if inf is close to convex the properties of f m n are discussed in some detail it is shown that m n m n 1 for n no and for n 0 1 m n consists entirely of univalent functions closure properties of some integral operators defined on m n are also given

the euler scheme for levy driven stochastic differential equations in relation with monte carlo methods to solve some integro differential equations we study the approximation problem of ee g x t by ee g bar x t n where x t 0 leq t leq t is the solution of a stochastic differential equation governed by a levy process z t bar x n t is defined by the euler discretization scheme with step fractn with appropriate assumptions we show that the error ee g x t ee g bar x t n can be expanded in powers of frac1n if the levy measure of z has finite moments of order high enough otherwise the rate of convergence is slower and its speed depends on the behavior of the tails of the levy measure the simulation of the increments of z t is also discussed

asymptotic expansion of stochastic flows we study the asymptotic expansion in small time of the solution of a stochastic differential equation we obtain a universal and explicit formula in terms of lie brackets and iterated stochastic stratonovich integrals this formula contains the results of doss 6 sussmann 15 fliess and normand cyrot 7 krener and lobry 10 yamato 17 and kunita 11 in the nilpotent case and extends to general diffusions the representation given by ben arous 3 for invariant diffusions on a lie group the main tool is an asymptotic expansion for deterministic ordinary differential equations given by strichartz 14

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

stochastic expansions and hopf algebras we study solutions to nonlinear stochastic differential systems driven by a multi dimensional wiener process a useful algorithm for strongly simulating such stochastic systems is the castell gaines method which is based on the exponential lie series when the diffusion vector fields commute it has been proved that at low orders this method is more accurate in the mean square error than corresponding stochastic taylor methods however it has also been shown that when the diffusion vector fields do not commute this is not true for strong order one methods here we prove that when there is no drift and the diffusion vector fields do not commute the exponential lie series is usurped by the sinh log series in other words the mean square error associated with a numerical method based on the sinh log series is always smaller than the corresponding stochastic taylor error in fact to all orders our proof uses the underlying hopf algebra structure of these series and a two alphabet associative algebra of shuffle and concatenation operations we illustrate the benefits of the proposed series in numerical studies

hopf monads on monoidal categories we define hopf monads on an arbitrary monoidal category extending the definition given in bruguieres and virelizier 2007 5 for monoidal categories with duals a hopf monad is a bimonad or opmonoidal monad whose fusion operators are invertible this definition can be formulated in terms of hopf adjunctions which are comonoidal adjunctions with an invertibility condition on a monoidal category with internal homs a hopf monad is a bimonad admitting a left and a right antipode hopf monads generalize hopf algebras to the non braided setting they also generalize hopf algebroids which are linear hopf monads on a category of bimodules admitting a right adjoint we show that any finite tensor category is the category of finite dimensional modules over a hopf algebroid any hopf algebra in the center of a monoidal category c gives rise to a hopf monad on c the hopf monads so obtained are exactly the augmented hopf monads more generally if a hopf monad t is a retract of a hopf monad p then p is a cross product of t by a hopf algebra of the center of the category of t modules generalizing the radford majid bosonization of hopf algebras we show that the comonoidal comonad of a hopf adjunction is canonically represented by a cocommutative central coalgebra as a corollary we obtain an extension of sweedler s hopf module decomposition theorem to hopf monads in fact to the weaker notion of pre hopf monad

efficient strong integrators for linear stochastic systems we present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of wiener processes these schemes are based on the neumann stochastic taylor and magnus expansions first we consider the case when the governing linear diffusion vector fields commute with each other but not with the linear drift vector field we prove that numerical methods based on the magnus expansion are more accurate in the mean square sense than corresponding stochastic taylor integration schemes second we derive the maximal rate of convergence for arbitrary multidimensional stochastic integrals approximated by their conditional expectations consequently for general nonlinear stochastic differential equations with noncommuting vector fields we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order third we consider the consequences in two numerical studies one of which is an application arising in stochastic linear quadratic optimal control

on certain subclass of meromorphic close to convex functions in the present paper we introduce and investigate a certain new subclass mk s l a b of meromorphic close to convex functions such results as inclusion relationships coefficient inequalities and convolution property are derived relevant connections of the results presented here with those obtained in earlier works are also pointed out

algebraic structures and stochastic differential equations driven by levy processes we construct an efficient integrator for stochastic differential systems driven by lvy processes an efficient integrator is a strong approximation that is more accurate than the corresponding stoc

the exponential lie series for continuous semimartingales we consider stochastic differential systems driven by continuous semimartingales and governed by non commuting vector fields we prove that the logarithm of the flowmap is an exponential lie series this relies on a natural change of basis to vector fields for the associated quadratic covariation processes analogous to stratonovich corrections the flowmap can then be expanded as a series in compositional powers of vector fields and the logaritm of the flowmap can thus be expanded in the lie algebra of vector fields further we give a direct self contained proof of the corresponding chen strichartz formula which provides an explicit formula for the lie series coefficients such exponential lie series are important in the development of strong lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves