conservation laws for integrable difference equations this paper deals with conservation laws for all integrable difference equations that belong to the famous adler bobenko suris classification all inequivalent three point conservation laws are found as are three five point conservation laws for each equation we also describe a method of generating conservation laws from known ones this method can be used to generate higher order conservation laws from those that are listed here

symmetries as integrability criteria for differential difference equations in this paper we review the results obtained by the generalized symmetry method in the case of differential difference equations during the last 20 years together with general theory of the method classification results are discussed for classes of equations which include the volterra toda and relativistic toda lattice equations

generalized symmetry integrability test for discrete equations on the square lattice we present an integrability test for discrete equations on the square lattice which is based on the existence of a generalized symmetry we apply this test to a number of equations obtained in different recent papers as a result we prove the integrability of seven equations which differ essentially from the qv equation introduced by viallet and thus from the adler bobenko suris list of equations contained therein

optimal transport and skorokhod embedding the skorokhod embedding problem is to represent a given probability as the distribution of brownian motion at a chosen stopping time over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties these constructions employ a variety of techniques ranging from excursion theory to potential and pde theory and have been used in many different branches of pure and applied probability we develop a new approach to skorokhod embedding based on ideas and concepts from optimal mass transport in analogy to the celebrated article of gangbo and mccann on the geometry of optimal transport we establish a geometric characterization of skorokhod embeddings with desired optimality properties this leads to a systematic method to construct optimal embeddings it allows us for the first time to derive all known optimal skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings while previous constructions typically used particular properties of brownian motion our approach applies to all sufficiently regular markov processes

the joint law of the maximum and terminal value of a martingale in this paper we characterise the possible joint laws of the maximum and terminal value of a uniformly integrable martingale we also characterise the joint laws of the maximum and terminal value of a convergent continuous local martingale vanishing at zero a number of earlier results on the possible laws of the maximum can be deduced quite easily

the root solution to the multi marginal embedding problem an optimal stopping and time reversal approach we provide a complete characterisation of the root solution to the skorokhod embedding problem sep by means of an optimal stopping formulation our methods are purely probabilistic and the analysis relies on a tailored time reversal argument this approach allows us to address the long standing question of a multiple marginals extension of the root solution of the sep our main result establishes a complete solution to the n marginal sep using first hitting times of barrier sets by the time space process the barriers are characterised by means of a recursive sequence of optimal stopping problems moreover we prove that our solution enjoys a global optimality property extending the one marginal root case our results hold for general one dimensional martingale diffusions

complete duality for martingale optimal transport on the line we study the optimal transport between two probability measures on the real line where the transport plans are laws of one step martingales a quasi sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward cost functions absence of a duality gap and existence of dual optimizers both properties are shown to fail in the classical formulation as a consequence of the duality result we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports

the generalized symmetry method for discrete equations the generalized symmetry method is applied to a class of completely discrete equations including the adler bobenko suris list assuming the existence of a generalized symmetry we derive a few integrability conditions suitable for testing and classifying the equations of this class those conditions are used at the end to test for integrability discretizations of some well known hyperbolic equations

an iterated azema yor type embedding for finitely many marginals we solve the nn marginal skorokhod embedding problem for a continuous local martingale and a sequence of probability measures 1 n 1 n which are in convex order and satisfy an additional technical assumption our construction is explicit and is a multiple marginal generalization of the azema and yor in seminaire de probabilites xiii univ strasbourg strasbourg 1977 78 1979 90 115 springer solution in particular we recover the stopping boundaries obtained by brown hobson and rogers probab theory related fields 119 2001 558 578 and madan and yor bernoulli 8 2002 509 536 our technical assumption is necessary for the explicit embedding as demonstrated with a counterexample we discuss extensions to the general case giving details when n 3n 3 in our analysis we compute the law of the maximum at each of the nn stopping times this is used in henry labordere et al ann appl probab 26 2016 1 44 to show that the construction maximizes the distribution of the maximum among all solutions to the nn marginal skorokhod embedding problem the result has direct implications for robust pricing and hedging of lookback options

deriving conservation laws for abs lattice equations from lax pairs in this paper we derive infinitely many conservation laws for abs lattice equations from their lax pairs these conservation laws can be algebraically expressed by means of some known polynomials for each equation the infinitely many conservation laws are not equivalent and are nontrivial we also show that the h1 h2 h3 q1 q2 q3 and a1 equations in the abs list share a generic discrete riccati equation

classification of integrable equations on quad graphs the consistency approach a classification of discrete integrable systems on quad graphs i e on surface cell decompositions with quadrilateral faces is given the notion of integrability laid in the basis of the classification is the three dimensional consistency this property yields among other features the existence of the discrete zero curvature representation with a spectral parameter for all integrable systems of the obtained exhaustive list the so called three leg forms are found this establishes lagrangian and symplectic structures for these systems and the connection to discrete systems of the toda type on arbitrary graphs generalizations of these ideas to the three dimensional integrable systems and to the quantum context are also discussed

symmetries and integrability of discrete equations defined on a black white lattice we study the deformations of the h equations presented recently by adler bobenko and suris which are naturally defined on a black white lattice for each one of these equations two different three leg forms are constructed leading to two different discrete toda type equations their multidimensional consistency leads to backlund transformations relating different members of this class as well as to lax pairs their symmetry analysis is presented yielding infinite hierarchies of generalized symmetries

infinitely many conservation laws for the discrete kdv equation rasin and hydon 2007 j phys a math theor 40 12763 73 suggested a way to construct an infinite number of conservation laws for the discrete kdv equation dkdv by repeated application of a certain symmetry to a known conservation law it was not decided however whether the resulting conservation laws were distinct and nontrivial in this paper we obtain the following results 1 we give an alternative method to construct an infinite number of conservation laws using a discrete version of the gardner transformation 2 we give a direct proof that the conservation laws obtained by the method of rasin and hydon are indeed distinct and nontrivial 3 we consider a continuum limit in which the dkdv equation becomes a first order eikonal equation in this limit the two sets of conservation laws become the same and are evidently distinct and nontrivial this proves the nontriviality of the conservation laws constructed by the gardner method and gives an alternative proof of the nontriviality of the conservation laws constructed by the method of rasin and hydon

the mapping class groups of reducible heegaard splittings of genus two the first named author was supported by the basic science research program through the national research foundation of korea nrf funded by the ministry of science ict and future planning nrf 2015r1a1a1a05001071 and by the ministry of education nrf 201800000001768 the second author was supported by jsps kakenhi grant numbers 15h03620 17k05254 17h06463 and jst crest grant number jpmjcr17j4

the minimum maximum of a continuous martingale with given initial and terminal laws let m t 0 t 1 be a continuous martingale with initial law m 0 0 and terminal law m 1 1 and let s sup 0 t 1 m t in this paper we prove that there exists a greatest lower bound with respect to stochastic ordering of probability measures on the law of s we give an explicit construction of this bound furthermore a martingale is constructed which attains this minimum by solving a skorokhod embedding problem the form of this martingale is motivated by a simple picture the result is applied to the robust hedging of a forward start digital option

conditions for the existence of higher symmetries of evolutionary equations on the lattice in this paper we construct a set of five conditions necessary for the existence of generalized symmetries for a class of differential difference equations depending only on nearest neighboring interaction these conditions are applied to prove the existence of new integrable equations belonging to this class

martingale optimal transport and robust hedging in continuous time the duality between the robust or equivalently model independent hedging of path dependent european options and a martingale optimal transport problem is proved the financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time the hedging problem is to construct a minimal super hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given fixed maturity the dual is a monge kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that have a given marginal at maturity in addition to duality a family of simple piecewise constant super replication portfolios that asymptotically achieve the minimal super replication cost is constructed

the geometry of multi marginal skorokhod embedding the skorokhod embedding problem is one of the classical problems in the theory of stochastic processes with applications in many different fields cf the surveys hobson in paris princeton lectures on mathematical finance 2010 volume 2003 of lecture notes in mathematics springer berlin 2011 obloj in probab surv 1 321 390 2004 many of these applications have natural multi marginal extensions leading to the optimal multi marginal skorokhod problem some of the first papers to consider this problem are brown et al probab theory relat fields 119 4 558 578 2001 hobson seminaire de probabilites xxxii volume 1686 of lecture notes in mathematics springer berlin 1998 madan and yor bernoulli 8 4 509 536 2002 however this turns out to be difficult using existing techniques only recently a complete solution was be obtained in cox et al probab theory relat fields 173 211 259 2018 establishing an extension of the root construction while other instances are only partially answered or remain wide open in this paper we extend the theory developed in beiglbock et al invent math 208 2 327 400 2017 to the multi marginal setup which is comparable to the extension of the optimal transport problem to the multi marginal optimal transport problem as for the one marginal case this viewpoint turns out to be very powerful in particular we are able to show that all classical optimal embeddings have natural multi marginal counterparts notably these different constructions are linked through a joint geometric structure and the classical solutions are recovered as particular cases moreover our results also have consequences for the study of the martingale transport problem as well as the peacock problem

affine linear and d4 symmetric lattice equations symmetry analysis and reductions we consider lattice equations on which are autonomous affine linear and possess the symmetries of the square some basic properties of equations of this type are derived as well as a sufficient linearization condition and a conservation law a systematic analysis of the lie point and the generalized three and five point symmetries is presented it leads to the generic form of the symmetry generators of all the equations in this class which satisfy a certain non degeneracy condition finally symmetry reductions of certain lattice equations to discrete analogs of the painleve equations are considered

symmetries and conservation laws of the abs equations and corresponding differential difference equations of volterra type a sequence of canonical conservation laws for all the adler bobenko suris abs equations is derived and is employed in the construction of a hierarchy of master symmetries for equations h1 h3 q1 q3 for the discrete potential and schwarzian kdv equations it is shown that their local generalized symmetries and nonlocal master symmetries in each lattice direction form centerless virasoro type algebras in particular for the discrete potential kdv the structure of its symmetry algebra is explicitly given interpreting the hierarchies of symmetries of equations h1 h3 q1 q3 as differential difference equations of yamilov s discretization of krichever novikov equation corresponding hierarchies of master symmetries along with isospectral and nonisospectral zero curvature representations are derived for all of them

wasserstein barycenters over riemannian manifolds we study barycenters in the space of probability measures on a riemannian manifold equipped with the wasserstein metric under reasonable assumptions we establish absolute continuity of the barycenter of general measures p p m p p m on wasserstein space extending on one hand results in the euclidean case for barycenters between finitely many measures of agueh and carlier 1 to the riemannian setting and on the other hand results in the riemannian case of cordero erausquin mccann schmuckenschlager 12 for barycenters between two measures to the multi marginal setting our work also extends these results to the case where is not finitely supported as applications we prove versions of jensen s inequality on wasserstein space and a generalized brunn minkowski inequality for a random measurable set on a riemannian manifold

asymptotic diagonalization of the discrete lax pair around singularities and conservation laws for dynamical systems a method of the formal diagonalization of the discrete linear operator with a parameter is studied in the case when the operator provides a lax operator for a nonlinear quad system the formal diagonalization method allows one to describe effectively conservation laws and generalized symmetries for this system asymptotic representation of the lax operators eigenfunctions are constructed and infinite series of conservation laws are described for the quad system connected with affine lie algebra for the modified discrete boussinesq system and for the discrete tzitzeica equation for a newly found multiquadratic discrete model conservation laws and several generalized symmetries are presented

algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations an algebraic structure related to discrete zero curvature equations is established it is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries the key of the theory is to generate nonisospectral flows t l l 0 from the discrete spectral problem associated with a given system of discrete evolution equations three examples are given

structure of optimal martingale transport plans in general dimensions given two probability measures mu and nu in convex order on r d we study the profile of one step martingale plans pi on r d x r d that optimize the expected value of the modulus of their increment among all martingales having mu and nu as marginals while there is a great deal of results for the real line i e when d 1 much less is known in the richer and more delicate higher dimensional case that we tackle in this paper we show that many structural results can be obtained provided the initial measure mu is absolutely continuous with respect to the lebesgue measure one such a property is that mu almost every x in r d is transported by the optimal martingale plan into a probability measure pi x concentrated on the extreme points of the closed convex hull of its support this will be established for the distance cost c x y vertical bar x y vertical bar in the two dimensional case and also for any d 3 as long as the marginals are in subharmonic order in some cases pi x is supported on the vertices of a k x dimensional polytope such as when the target measure is discrete duality plays a crucial role in our approach even though in contrast to standard optimal transports the dual extremal problem may not be attained in general we show however that martingale supporting borel subsets of r d x r d can be decomposed into a collection of mutually disjoint components by means of a convex paving of the source space in such a way that when the martingale is optimal for a general cost function each of the components then supports a restricted optimal martingale transport whose dual problem is attained this decomposition is used to obtain structural results in cases where global duality is not attained on the other hand it shows that certain optimal martingale supporting borel sets can be viewed as higher dimensional versions of nikodym type sets the paper focuses on the distance cost but much of the results hold for general lipschitz cost functions

on the lagrangian formulation of multidimensionally consistent systems multidimensional consistency has emerged as a key integrability property for partial difference equations p es defined on the space time lattice it has led among other major insights to a classification of scalar affine linear quadrilateral p es possessing this property leading to the so called adler bobenko suris abs list recently a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete lagrangian multi forms this description is based on a fundamental and highly non trivial property of lagrangians for those integrable lattice equations namely the fact that on the solutions of the corresponding p e the lagrange forms are closed i e they obey a closure relation here we extend those results to the continuous case it is known that associated with the integrable p es there exist systems of partial differential equations pdes in fact differential equations with regard to the parameters of the lattice as independent variables which equally possess the property of multidimensional consistency in this paper we establish a universal lagrange structure for affine linear quad lattices alongside a universal lagrange multi form structure for the corresponding continuous pdes and we show that the lagrange forms possess the closure property

superreplication of financial derivatives via convex programming we give a method based on convex programming to calculate the optimal superreplicating and subreplicating prices and corresponding hedging portfolios of a financial derivative in terms of other financial derivatives in a discrete time setting our method produces a model that matches the superreplicating or subreplicating price within an arbitrary precision and is consistent with the other financial derivatives prices applications include robust replication in terms of call prices with various strikes and maturities of forward start options volatility and variance swaps and derivatives cliquet calls barrier options and lookback and asian options numerical examples show that in some cases the best superreplicating and or subreplicating prices are within 10 of the price obtained by a standard model but considerably differ from it in other cases our method can incorporate bid ask spreads interest rates and dividends and various limitations to the diffusion model the supplementary material is a

on a problem of optimal transport under marginal martingale constraints the basic problem of optimal transportation consists in minimizing the expected costs e c x 1 x 2 by varying the joint distribution x 1 x 2 where the marginal distributions of the random variables x 1 and x 2 are fixed inspired by recent applications in mathematical finance and connections with the peacock problem we study this problem under the additional condition that x i i 1 2 is a martingale that is e x 2 x 1 x 1 we establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific cost functions in particular we identify a martingale coupling that resembles the classic monotone quantile coupling in several respects in analogy with the celebrated theorem of brenier the following behavior can be observed if the initial distribution is continuous then this monotone martingale is supported by the graphs of two functions t 1 t 2 r r

constrained optimal transport the classical duality theory of kantorovich c r doklady acad sci urss ns 37 199 201 1942 and kellerer z wahrsch verw gebiete 67 4 399 432 1984 for classical optimal transport is generalized to an abstract framework and a characterization of the dual elements is provided this abstract generalization is set in a banach lattice mathcal x with an order unit the problem is given as the supremum over a convex subset of the positive unit sphere of the topological dual of mathcal x and the dual problem is defined on the bi dual of mathcal x these results are then applied to several extensions of the classical optimal transport

a proof of the smale conjecture diff s3 0 4 the smale conjecture 9 is the assertion that the inclusion of the orthogonal group 0 4 into diff s3 the diffeomorphism group of the 3 sphere with the cw topology is a homotopy equivalence there are many equivalent forms of this conjecture some of which are listed in the appendix to this paper we shall prove

lattice equations hierarchies and hamiltonian structures by now many integrable systems with soliton solutions are known in two dimensions for fields u n1 n2 defined at the sites nln2 of a two dimensional 2d lattice for time dependent fields u n t defined at the sites n of a one dimensional chain and for fields u x t depending on two continuous variables x and t the relation between an integrable partial differential equation and its integrable discrete versions can be treated in the framework of the direct linearization method 1 2 which is based on a linear integral equation with arbitrary measure and contour 3 4 in the treatment use is made of backlund transformations bt s which are generated by scalar multiplications of the free wave function and or measure in the integral equation 5 the integrable lattice equations are obtained in the form of bianchi identities expressing the commutativity of bt s the integrable equations with one or more continuous variables and their direct linearizations are obtained applying suitable continuum limits to the lattice equation and the integral equation at the same time

how to detect the integrability of discrete systems several integrability tests for discrete equations will be reviewed all tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as lax pairs specifically singularity confinement algebraic entropy nevanlinna theory diophantine integrability and discrete systems over finite fields will be described

the maximum maximum of a martingale constrained by an intermediate law let m t be any martingale with m0 0 an intermediate law m1 1 and terminal law m2 2 and let m 2 sup0 t 2m t in this paper we prove that there exists an upper bound with respect to stochastic ordering of probability measures on the law of m 2 we construct using excursion theory a martingale which attains this maximum finally we apply this result to the robust hedging of a lookback option

the two sided stefan problem with a spatially dependent latent heat we prove existence and uniqueness of solutions to a problem which generalizes the two sided stefan problem the initial temperature distribution and variable latent heat may be given by positive measures rather than point functions and the free boundaries which result are essentially arbitrary increasing functions which need not exhibit any degree of smoothness in general nevertheless the solutions are classical in the sense that all derivatives and boundary values have the classical interpretation we also study connections with the skorohod embedding problem of probability theory and with a general class of optimal stoppmg problems

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159