hecke algebra correspondences for the metaplectic group over a p adic field of odd residual characteristic gan and savin proved a correspondence between the bernstein components of the even and odd weil representations of the metaplectic group and the components of the trivial representation of the equal rank odd orthogonal groups in this paper we extend their result to the case of even residual characteristic

forced oscillations in quadratically damped systems bayliss 1975 has studied the question whether in the case of linear differential equations the relationship between the stability of the homogeneous equations and the existence of almost periodic solutions to the inhomogeneous equation is preserved by finite difference approximations in the current investigation analogous properties are considered for the case in which the damping is quadratic rather than linear the properties of the considered equation for arbitrary forcing terms are examined and the validity is proved of a theorem concerning the characteristics of the unique solution by using the lipschitz continuity of the mapping and the contracting mapping principle almost periodic solutions can be found for perturbations of the considered equation attention is also given to the lipschitz continuity of the solution operator and the results of numerical tests which have been conducted to test the discussed theory

adaptive weak approximation of diffusions with jumps this work develops adaptive time stepping algorithms for the approximation of a functional of a diffusion with jumps based on a jump augmented monte carlo euler maruyama method which achieve a prescribed precision the main result is the derivation of new expansions for the time discretization error with computable leading order term in a posteriori form which are based on stochastic flows and discrete dual backward functions combined with proper estimation of the statistical error they lead to efficient and accurate computation of global error estimates extending the results by a szepessy r tempone and g e zouraris comm pure appl math 54 2001 pp 1169 1214 adaptive algorithms for either deterministic or trajectory dependent time stepping are proposed numerical examples show the performance of the proposed error approximations and the adaptive schemes

variance reduced weak monte carlo simulations of stochastically driven oscillators of engineering interest abstract analytical solutions of nonlinear and higher dimensional stochastically driven oscillators are rarely possible and this leaves the direct monte carlo simulation of the governing stochastic differential equations sdes as the only tool to obtain the required numerical solution engineers in particular are mostly interested in weak numerical solutions which provide a faster and simpler computational framework to obtain the statistical expectations moments of the response functions the numerical integration tools considered in this study are weak versions of stochastic euler and stochastic newmark methods a well known limitation of a monte carlo approach is however the requirement of a large ensemble size in order to arrive at convergent estimates of the statistical quantities of interest presently a simple form of a variance reduction strategy is proposed such that the ensemble size may be significantly reduced without affecting the accuracy of the predicted expectations of any function of the response vector process provided that the function can be adequately represented through a power series approximation the basis of the variance reduction strategy is to appropriately augment the governing system equations and then weakly replace the stochastic forcing function which is typically a filtered white noise process through a set of variance reduced functions in the process the additional computational cost due to system augmentation is far smaller than the accrued advantages due to a drastically reduced ensemble size indeed we show that the proposed method appears to work satisfactorily even in the special case of the ensemble size being just 1 the variance reduction scheme is first illustrated through applications to a nonlinear duffing equation driven by additive and multiplicative white noise processes a problem for which exact stationary solutions are known this is followed up with applications of the strategy to a few higher dimensional systems i e 2 and 3 dof nonlinear oscillators under additive white noises

on the approximations of solutions to neutral sdes with markovian switching and jumps under non lipschitz conditions abstract in this paper we investigate the existence and uniqueness of solutions to neutral stochastic differential equations with markovian switching and jumps nsdewmsjs under non lipschitz conditions on the other hand we present the euler approximate solutions for nsdewmsjs and show that the convergence of the euler approximate solutions to the true solutions by applying it o formula bihari s lemma and burkholder davis gundy s lemma some examples are provided to illustrate the main results

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

weak approximations and extrapolations of stochastic differential equations with jumps numerical discretization schemes are developed to approximate functionals of stochastic differential equations with jumps and the convergence is shown to have an appropriate order for the euler scheme and the second order weak scheme the leading coefficient of their global errors are determined by the stochastic taylor expansion based on the error expression the extrapolation technique can be applied to get a higher order convergence numerical examples are provided to compare various weak schemes and extrapolations

convergence and stability of the backward euler method for jump diffusion sdes with super linearly growing diffusion and jump coefficients abstract this paper firstly investigates convergence of the backward euler method for stochastic differential equations sdes driven by brownian motion and compound poisson process different from existing works we do not impose a globally lipschitz condition on the jump coefficient but formulate a more relaxed condition to allow for its super linear growth it is shown that the mean square convergence order of this method can be arbitrarily close to 1 2 under mild assumptions imposed on sdes allowing for possibly super linearly growing drift diffusion and jump coefficients an exact order 1 2 is recovered when further differentiability assumption is put on the coefficients furthermore the considered method is able to inherit the mean square stability of a wider class of levy noise driven sdes for all stepsizes these results are finally supported by some numerical experiments

caracterisation de la correspondance de langlands locale par les facteurs e de paires soitf un corps local non archimedien a corps residuel fini la conjecture de langlands locale prouvee par laumon rapoport et stuhler quandf est de caracteristique non nulle associe a une classe d isomorphisme de representations irreductibles du groupe de weil def une classe d isomorphisme de representations admissibles irreductibles supercuspidales d un groupe lineaire surf le groupe gl n f si est de dimensionn nous prouvons dans cet appendice qu il existe au plus une telle correspondance qui preserve les facteurs e de paires et coincide en dimension 1 avec l application donnee par la theorie du corps de classes

comparison of semimartingales and levy processes in this paper we derive comparison results for terminal values of d dimensional special semimartingales and also for finite dimensional distributions of multivariate levy processes the comparison is with respect to nondecreasing increasing convex increasing directionally convex and increasing supermodular functions we use three different approaches in the first approach we give sufficient conditions on the local predictable characteristics that imply ordering of terminal values of semimartingales this generalizes some recent convex comparison results of exponential models in math finance 8 1998 93 126 finance stoch 4 2000 209 222 proc steklov inst math 237 2002 73 113 finance stoch 10 2006 222 249 in the second part we give comparison results for finite dimensional distributions of levy processes with infinite levy measure in the first step we derive a comparison result for markov processes based on a monotone separating transition kernel by a coupling argument we get an application to the comparison of compound poisson processes these comparisons are then extended by an approximation argument to the ordering of levy processes with infinite levy measure the third approach is based on mixing representations which are known for several relevant distribution classes we discuss this approach in detail for the comparison of generalized hyperbolic distributions and for normal inverse gaussian processes

strong approximations of stochastic differential equations with jumps this paper is a survey of strong discrete time approximations of jump diffusion processes described by stochastic differential equations sdes it also presents new results on strong discrete time approximations for the specific case of pure jump sdes strong approximations based on jump adapted time discretizations which produce no discretization error in the case of pure jump processes are analyzed the computational complexity of these approximations is proportional to the jump intensity by exploiting a stochastic expansion for pure jump processes higher order discrete time approximations whose computational complexity is not dependent on the jump intensity are proposed for the specific case of pure jump sdes the strong order of convergence of strong taylor schemes is established under weaker conditions than those currently known in the literature

a proof of langland s conjecture on plancherel measures complementary series of p adic groups analysis of p adic reductive groups our first result theorem 7 9 proves a conjecture of langlands on normalization of intertwining operators by means of local langlands root numbers and l functions at least when the group is quasi split and the inducing representation is generic assuming two natural conjectures in harmonic analysis of p adic groups we also prove the validity of the conjecture in general theorem 9 5 as our second result we obtain all the complementary series and special representations of quasi split p adic groups coming from rank one parabolic subgroups and generic supercuspidal represen

langlands parameters for epipelagic representations of hbox gl _ n let f be a non archimedean local field an irreducible cuspidal representation of mathrm gl n f is epipelagic if its swan conductor equals 1 we give a full and explicit description of the langlands parameters of such representations in terms of elementary representation theory and local class field theory

product expansion for stochastic jump diffusions and its application to numerical approximation abstract we derive a product expansion of the exponential lie series in terms of a philip hall basis for the chen series corresponding to the stochastic jump diffusion as in sussmann in c i byrnes and a lindquist eds theory and applications of nonlinear control systems north holland amsterdam 1986 pp 323 335 for the deterministic case based on the expansion we establish the stratonovich taylor hall sth schemes such that each scheme involves only the minimum number of multiple stochastic integrals which can be regarded as systems of stochastic differential equations and approximated by a lower order scheme with an appropriate step size to ensure the necessary accuracy mean square convergence of the sth schemes is shown and numerical examples are provided to illustrate the results

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

extrapolation methods for the weak approximation of ito diffusions higher order weak extrapolation methods for the approximation of functionals of ito diffusions are considered under appropriate regularity conditions it is shown that extrapolations allow a considerable increase in the weak order of convergence of a discrete time one step approximation method numerical experiments indicate the efficiency of extrapolations based on higher order weak schemes for stochastic differential equations with additive noise

equidistribution theorems for holomorphic siegel modular forms for gsp_4 hecke fields and n level density this paper is a continuation of kim et al j inst math jussieu 2018 we supplement four results on a family of holomorphic siegel cusp forms for gsp 4 mathbb q first we improve the result on hecke fields namely we prove that the degree of hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of gsp 4 second we prove simultaneous vertical sato tate theorem namely we prove simultaneous equidistribution of hecke eigenvalues at finitely many primes third we compute the n level density of degree 4 spinor l functions and thus we can distinguish the symmetry type depending on the root numbers this is conditional on certain conjecture on root numbers fourth we consider equidistribution of paramodular forms in this case we can prove the conjecture on root numbers main tools are the equidistribution theorem in our previous work and shin templier s compos math 150 12 2003 2053 2014 work

a family of weak stochastic newmark methods for simplified and efficient monte carlo simulations of oscillators engineers are often more concerned with the computation of statistical moments or mathematical expectations of the response of stochastically driven dynamical systems than with the determination of path wise response histories with this in perspective weak stochastic solutions of dynamical systems modelled as n degrees of freedom dof mechanical oscillators and driven by additive and or multiplicative white noise or filtered white noise processes are considered in this study while weak stochastic solutions are simpler and quicker to compute than strong sample path wise solutions it must be emphasized that sample realizations of weak solutions have no path wise similarity with strong solutions however the statistical moment of any continuous and sufficiently differentiable deterministic function of the weak stochastic response is close to that of the true response if it exists within a certain order of a given time step size computation of weak response therefore assumes great significance in the context of simulations of stochastically driven dynamical systems oscillators of engineering interest to efficiently generate such weak responses a novel class of weak stochastic newmark methods wsnms based on implicit ito taylor expansions of displacement and velocity fields is proposed the resulting multiple stochastic integrals msis in these expansions are replaced by a set of random variables with considerably simpler and discrete probability densities in fact yet another simplifying feature of the present strategy is that there is no need to model and compute some of the higher level msis estimates of error orders of these methods in terms of a given time step size are derived and a proof of global convergence provided numerical illustrations are provided and compared with exact solutions whenever available to demonstrate the accuracy simplicity and higher computational speed of wsnms vis a vis a few other popularly used stochastic integration schemes as well as the path wise versions of the stochastic newmark scheme copyright 2006 john wiley sons ltd

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

on the asymptotic stability and numerical analysis of solutions to nonlinear stochastic differential equations with jumps this paper is concerned with the stability and numerical analysis of solution to highly nonlinear stochastic differential equations with jumps by the ito formula stochastic inequality and semi martingale convergence theorem we study the asymptotic stability in the p th moment and almost sure exponential stability of solutions under the local lipschitz condition and nonlinear growth condition on the other hand we also show the convergence in probability of numerical schemes under nonlinear growth condition finally an example is provided to illustrate the theory

runge kutta methods for jump diffusion differential equations in this paper we consider runge kutta methods for jump diffusion differential equations we present a study of their mean square convergence properties for problems with multiplicative noise we are concerned with two classes of runge kutta methods first we analyse schemes where the drift is approximated by a runge kutta ansatz and the diffusion and jump part by a maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the runge kutta ansatz for the drift the second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise we present results showing when the implicit stochastic equations defining the stage values of the runge kutta methods are uniquely solvable finally simulation results illustrate the theoretical findings