small and strong formulations for unions of convex sets from the cayley embedding there is often a significant trade off between formulation strength and size in mixed integer programming when modeling convex disjunctive constraints e g unions of convex sets adding auxiliary continuous variables can sometimes help resolve this trade off however standard formulations that use such auxiliary continuous variables can have a worse than expected computational effectiveness which is often attributed precisely to these auxiliary continuous variables for this reason there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables we introduce a technique to construct formulations without these detrimental continuous auxiliary variables to develop this technique we introduce a natural non polyhedral generalization of the cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding we then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints in particular we show it can recover and generalize all known strong formulations without continuous auxiliary variables

approximate constraint satisfaction requires large lp relaxations we prove super polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems we show that for these problems polynomial sized linear programs are no more powerful than programs arising from a constant number of rounds of the sherali adams hierarchy in particular any polynomial sized linear program for m ax c ut has an integrality gap of and any such linear program for m ax 3 s at has an integrality gap of f

markov snakes and superprocesses we suggest the name markov snakes for a class of path valued markov processes introduced recently by j f le gall in connection with the theory of branching measure valued processes le gall applied this class to investigate path properties of superdiffusions and to approach probabilistically partial differential equations involving a nonlinear operator v v2 we establish an isomorphism theorem which allows to translate results on continuous superprocesses into the language of markov snakes and vice versa by using this theorem we get limit theorems for discrete markov snakes

embedding formulations and complexity for unions of polyhedra it is well known that selecting a good mixed integer programming mip formulation is crucial for effectively obtaining a solution with state of the art solvers although best practices and guidelines for constructing good formulations abound there is rarely a single systematic construction approach that leads to the best possible formulation here we introduce embedding formulations and complexity as a new mip formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size more specifically they yield the smallest possible ideal formulation i e one whose lp relaxation has integral extreme points among all formulations that only use 0 1 auxiliary variables we use the paradigm to characterize optimal formulations for special ordered sets of type 2 and certain piecewise linear functions of two variables we also show that the resultant formulations can provide a significant computational advantage over all known formulations for pi

high order compact finite difference scheme for option pricing in stochastic volatility models we derive a new high order compact finite difference scheme for option pricing in stochastic volatility models the scheme is fourth order accurate in space and second order accurate in time under some restrictions theoretical results like unconditional stability in the sense of von neumann are presented where the analysis becomes too involved we validate our findings by a numerical study numerical experiments for the european option pricing problem are presented we observe fourth order convergence for non smooth payoff

pricing and hedging european style options in levy based stochastic volatility models considering the leverage effect abstract this paper develops a semi analytical exact solution for the discrete time variance optimal hedging strategy based on two dimensional fourier cosine series expansions this approach is novel and is more efficient than those previously suggested in the literature numerical studies for different models also suggest that the algorithm has better accuracy and stability for the pricing and hedging problems a levy based stochastic volatility model considering the leverage effect is also proposed we have employed data on apple stock prices and the corresponding option contracts to test this model to evaluate its ability to perform pricing and hedging in summary our model design is relatively reliable in the context of the known empirical patterns such as the existence of jumps the volatility smile and the negative correlation between asset returns and volatilities

kolmogorov s test for super brownian motion we prove a kolmogorov test for super brownian motion started at the dirac mass at the origin more precisely we determine the functions g such that for all t small enough the support of the process at time t will be contained in the ball of radius g t centered at 0 as a consequence we get a necessary and sufficient condition for the existence in certain space time domains of a solution of the associated semilinear partial differential equation that blows up at the origin

a class of path valued markov processes and its applications to superprocesses let s be a continuous markov process satisfying certain regularity assumptions we introduce a path valued strong markov process associated with s which is closely related to the so called superprocess with spatial motion s in particular a subseth of the state space of s intersects the range of the superprocess if and only if the set of paths that hith is not polar for the path valued process the latter property can be investigated using the tools of the potential theory of symmetric markov processes a set is not polar if and only if it supports a measure of finite energy the same approach can be applied to study sets that are polar for the graph of the superprocess in the special case when s is a diffusion process we recover certain results recently obtained by dynkin

a novel pricing method for european options based on fourier cosine series expansions here we develop an option pricing method for european options based on the fourier cosine series and call it the cos method the key insight is in the close relation of the characteristic function with the series coefficients of the fourier cosine expansion of the density function in most cases the convergence rate of the cos method is exponential and the computational complexity is linear its range of application covers underlying asset processes for which the characteristic function is known and various types of option contracts we will present the method and its applications in two separate parts the first one is this paper where we deal with european options in particular in a follow up paper we will present its application to options with early exercise features

on the distributional complexity of disjointness we prove that the distributional communication complexity of the predicate disjointness with respect to a very simple measure on inputs is n

efficient protocols for generating bipartite classical distributions and quantum states we investigate the fundamental problem of generating bipartite classical distributions or quantum states by designing efficient communication protocols and proving their optimality we establish a number of intriguing connections to fundamental measures in optimization convex geometry and information theory 1 to generate a classical distribution p x y we tightly characterize the minimum amount of quantum communication needed by the psd rank of p as a matrix a measure recently proposed by fiorini et al proc 44th acm symp theory comput pp 95 106 2012 in studies of the minimum size of extended formulations of optimization problems such as tsp this echos the previous characterization for the optimal classical communication cost by the nonnegative rank of p the result is obtained via investigating the more general case of bipartite quantum state generation and designing an optimal protocol for it 2 when an approximation is allowed to generate a distribution x y p we present a classical protocol of the communication cost o c x y 1 where c x y is common information a well studied measure in information theory introduced by wyner ieee trans inf theory 21 2 163 179 1975 this also links nonnegative rank and common information two seemingly unrelated quantities in different fields 3 for approximately generating a quantum pure state we completely characterize the minimum cost by a corresponding approximate rank closing a possibly exponential gap left in ambainis etal siam j comput 32 6 1570 1585 2003

exponential lower bounds for polytopes in combinatorial optimization we solve a 20 year old problem posed by yannakakis and prove that no polynomial size linear program lp exists whose associated polytope projects to the traveling salesman polytope even if the lp is not required to be symmetric moreover we prove that this holds also for the cut polytope and the stable set polytope these results were discovered through a new connection that we make between one way quantum communication protocols and semidefinite programming reformulations of lps

high order adi scheme for option pricing in stochastic volatility models we propose a new high order alternating direction implicit adi finite difference scheme for the solution of initial boundary value problems of convection diffusion type with mixed derivatives and non constant coefficients as they arise from stochastic volatility models in option pricing our approach combines different high order spatial discretisations with hundsdorfer and verwer s adi time stepping method to obtain an efficient method which is fourth order accurate in space and second order accurate in time numerical experiments for the european put option pricing problem using heston s stochastic volatility model confirm the high order convergence

pricing early exercise and discrete barrier options by fourier cosine series expansions we present a pricing method based on fourier cosine expansions for early exercise and discretely monitored barrier options the method works well for exponential levy asset price models the error convergence is exponential for processes characterized by very smooth rm c infty a b in mathbb r transitional probability density functions the computational complexity is o m 1 n log n with n a small number of terms from the series expansion and m the number of early exercise monitoring dates this paper is the follow up of fang and oosterlee in siam j sci comput 31 2 826 848 2008 in which we presented the impressive performance of the fourier cosine series method for european options

a fourier cosine method for an efficient computation of solutions to bsdes we develop a fourier method to solve backward stochastic differential equations bsdes a general theta discretization of the time integrands leads to an induction scheme with conditional expectations these are approximated by using fourier cosine series expansions relying on the availability of a characteristic function the method is applied to bsdes with jumps numerical experiments demonstrate the applicability of bsdes in financial and economic problems and show fast convergence of our efficient probabilistic numerical method

integrality gaps of 2 o 1 for vertex cover sdps in the lovasz schrijver hierarchy linear and semidefinite programming are highly successful approaches for obtaining good approximations for np hard optimization problems for example breakthrough approximation algorithms for max cut and sparsest cut use semidefinite programming perhaps the most prominent np hard problem whose exact approximation factor is still unresolved is vertex cover probabilistically checkable proof pcp based techniques of dinur and safra ann of math 2 162 2005 pp 439 486 show that it is not possible to achieve a factor better than 1 36 on the other hand no known algorithm does better than the factor of 2 achieved by the simple greedy algorithm there is a widespread belief that semidefinite programming sdp techniques are the most promising methods available for improving upon this factor of 2 following a line of study initiated by arora et al theory comput 2 2006 pp 19 51 our aim is to show that a large family of linear programming lp and sdp based algorithms fail to produce an approximation for vertex cover better than 2 lovasz and schrijver siam j optim 1 1991 pp 166 190 introduced the systems ls and ls for systematically tightening lp and sdp relaxations respectively over many rounds these systems naturally capture large classes of lp and sdp relaxations indeed ls captures the celebrated sdp based algorithms for max cut and sparsest cut mentioned above we rule out polynomial time sdp based 2 omega 1 approximations for vertex cover using ls in particular for every epsilon 0 we prove an integrality gap of 2 epsilon for vertex cover sdps obtained by tightening the standard lp relaxation with omega sqrt log n log log n rounds of ls while tight integrality gaps were known for vertex cover in the weaker ls system g schoenebeck l trevisan and m tulsiani proceedings of the 39th annual acm symposium on theory of computing acm press new york 2007 pp 302 310 previous results did not rule out a 2 omega 1 approximation after even two rounds of ls

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

intersection equivalence of brownian paths and certain branching processes we show that sample paths of brownian motion and other stable processes intersect the same sets as certain random cantor sets constructed by a branching process with this approach the classical result that two independent brownian paths in four dimensions do not intersect reduces to the dying out of a critical branching process and estimates due to lawler 1982 for the long range intersection probability of several random walk paths reduce to kolmogrov s 1938 law for the lifetime of a critical branching process extensions to random walks with long jumps and applications to hausdorff dimension are also derived

efficient pricing of european options on two underlying assets by frame duality abstract the proj method for pricing european options on one underlying asset was proposed by j lars kirkby and then was applied to price bermudan and asian options in this paper we extend the method to higher dimensions especially two dimensions in which some exotic options can be priced our method does not rely on a priori truncation of the integration range and exhibits excellent performance compared with other state of the art methods particularly for fatter tailed short maturity models we also discuss the errors introduced in each approximation and give corresponding error bounds numerical results on implementation of this method to price for popular two assets options under both the geometric brownian motion and variance gamma dynamics demonstrate remarkable accuracy and robustness

nondeterministic quantum query and communication complexities we study nondeterministic quantum algorithms for boolean functions f such algorithms have positive acceptance probability on input x iff f x 1 in the setting of query complexity we show that the nondeterministic quantum complexity of a boolean function is equal to its nondeterministic polynomial degree we also prove a quantum vs classical gap of 1 vs n for nondeterministic query complexity for a total function in the setting of communication complexity we show that the nondeterministic quantum complexity of a two party function is equal to the logarithm of the rank of a nondeterministic version of the communication matrix this implies that the quantum communication complexities of the equality and disjointness functions are n 1 if we do not allow any error probability we also exhibit a total function in which the nondeterministic quantum communication complexity is exponentially smaller than its classical counterpart

super brownian motion path properties and hitting probabilities sample path properties of super brownian motion including a one sided modulus of continuity and exact hausdorff measure function of the range and closed support are obtained analytic estimates for the probability of hitting balls lead to upper bounds on the hausdorff measure of the set of k multiple points and a sufficient condition for a set to be polar

recombined multinomial tree based on saddle point approximation and its application to levy models options pricing abstract this paper studies the constructing methods of a recombined multinomial tree based on saddle point approximation and its application to levy models options pricing firstly the levy process and the european option pricing are introduced then we used the characteristic function of levy process to generate density function value at discrete point in a certain range based on the saddle point approximation method then we provide the method to construct recombined multinomial tree and give the pricing formula of european option and path dependent option pricing based on the backward iteration finally we used the cgmy process to demonstrate its application to european option american option and american barrier option pricing it proves that saddle point approximation turns the inverse fourier integral transform problem to several function value calculation comparing to ifft the speed of calculation using this method is faster and it avoid the negative probability density function value based on ifft because of the linear growth of the number of nodes we can extend the saddle point approximation method to the calculation of path dependent option

on coordinate transformation and grid stretching for sparse grid pricing of basket options we evaluate two coordinate transformation techniques in combination with grid stretching for pricing basket options in a sparse grid setting the sparse grid technique is a basic technique for solving a high dimensional partial differential equation by creating a small hypercube sub grid in the composite sparse grid we can also determine hedge parameters accurately we evaluate these techniques for multi asset examples with up to five underlying assets in the basket

theta bodies for polynomial ideals inspired by a question of lovasz we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal called theta bodies of the ideal these relaxations generalize lovasz s construction of the theta body of a graph we establish a relationship between theta bodies and lasserre s relaxations for real varieties which allows in many cases for theta bodies to be expressed as feasible regions of semidefinite programs examples from combinatorial optimization are given lovasz asked to characterize ideals for which the first theta body equals the closure of the convex hull of its real variety we answer this question for vanishing ideals of finite point sets via several equivalent characterizations we also give a geometric description of the first theta body for all ideals

characterization of g regularity for super brownian motion and consequences for parabolic partial differential equations we give a characterization of g regularity for super brownian motion and the brownian snake more precisely we define a capacity on e 0 x r d which is not invariant by translation we then prove that the measure of hitting a borel set a c e for the graph of the brownian snake excursion starting at 0 0 is comparable up to multiplicative constants to its capacity this implies that super brownian motion started at time 0 at the dirac mass 0 hits immediately a i e 0 0 is g regular for a c if and only if its capacity is infinite as a direct consequence if q e is a domain such that 0 0 q we give a necessary and sufficient condition for the existence on q of a positive solution of t u 1 2 u 2u 2 which blows up at 0 0 we also give an estimate of the hitting probabilities for the support of super brownian motion at fixed time we prove that if d 2 the support of super brownian motion is intersection equivalent to the range of brownian motion

a highly efficient shannon wavelet inverse fourier technique for pricing european options in the search for robust accurate and highly efficient financial option valuation techniques we here present the swift method shannon wavelets inverse fourier technique based on shannon wavelets swift comes with control over approximation errors made by means of sharp quantitative error bounds the nature of the local shannon wavelets basis enables us to adaptively determine the proper size of the computational interval numerical experiments on european style options show exponential convergence and confirm the bounds robustness and efficiency

balas formulation for the union of polytopes is optimal a celebrated theorem of balas gives a linear mixed integer formulation for the union of two nonempty polytopes whose relaxation gives the convex hull of this union the number of inequalities in balas formulation is linear in the number of inequalities that describe the two polytopes and the number of variables is doubled in this paper we show that this is best possible in every dimension there exist two nonempty polytopes such that if a formulation for the convex hull of their union has a number of inequalities that is polynomial in the number of inequalities that describe the two polytopes then the number of additional variables is at least linear in the dimension of the polytopes we then show that this result essentially carries over if one wants to approximate the convex hull of the union of two polytopes and also in the more restrictive setting of lift and project

two dimensional fourier cosine series expansion method for pricing financial options the cos method for pricing european and bermudan options with one underlying asset was developed in f fang and c w oosterlee siam j sci comput 31 2008 pp 826 848 and f fang and c w oosterlee numer math 114 2009 pp 27 62 in this paper we extend the method to higher dimensions with a multidimensional asset price process the algorithm can be applied to for example pricing two color rainbow options but also to pricing under the popular heston stochastic volatility model for smooth density functions the resulting method converges exponentially in the number of terms in the fourier cosine series summations otherwise we achieve algebraic convergence the use of an fft algorithm for asset prices modeled by levy processes makes the algorithm highly efficient we perform extensive numerical experiments

a fast preconditioned iterative method for two dimensional options pricing under fractional differential models abstract in recent years fractional partial differential equation fpde has been widely applied in options pricing problems which better explains many important empirical facts of financial markets however the vast majority of the literatures focus on pricing the single asset option under the fpde framework in this paper a two dimensional fpde governing the valuation of rainbow options is established where two underlying assets are assumed to follow independent exponential levy processes and its boundary conditions are determined by solving one dimensional fpdes a second order accurate finite difference scheme is proposed to discretize the two dimensional fpde given the block toeplitz with toeplitz block structure of the coefficient matrix a fast preconditioned krylov subspace method is developed for solving the resulting linear system with o n log n computational complexity per iteration where n is the matrix size the proposed preconditioner accelerates the convergence of the iterative method with theoretical analysis numerical examples are given to demonstrate the accuracy and efficiency of our proposed numerical methods

quantum communication and complexity in the setting of communication complexity two distributed parties want to compute a function depending on both their inputs using as little communication as possible the required communication can sometimes be significantly lowered if we allow the parties the use of quantum communication we survey the main results of the young area of quantum communication complexity its relation to teleportation and dense coding the main examples of fast quantum communication protocols lower bounds and some applications

wiener s test for super brownian motion and the brownian snake we prove a wiener type criterion for super brownian motion and the brownian snake if f is a borel subset of d and x d we provide a necessary and sufficientcondition for super brownian motion started at x to immediately hit the set f equivalently this condition is necessary and sufficient for the hitting time of f by thebrownian snake with initial point x to be 0 a key ingredient of the proof isan estimate showing that the hitting probability of f is comparable up to multiplicative constants to the relevant capacity of f this estimate which is of independent interest refines previous results due to perkins and dynkin an important role is played by additivefunctionals of the brownian snake which are investigated here via the potentialtheory of symmetric markov processes as a direct application of our probabilisticresults we obtain a necessary and sufficient condition for the existence in a domain d of a positivesolution of the equation u u 2 which explodes at a given point of d

on the potential theory of symmetric markov processes on reprend des resultats de dynkin et on considere un processus de markov fort symetrique continu a droite on etudie la relation existant entre le cone de mesures positives d energie finie et la capacite

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

efficient numerical fourier methods for coupled forward backward sdes we develop three numerical methods to solve coupled forward backward stochastic differential equations we propose three different discretization techniques for the forward stochastic differential equation a theta discretization of the time integrands is used to arrive at schemes with conditional expectations these conditional expectations are approximated by using the cos method which relies on the availability of the conditional characteristic function of the discrete forward process the numerical methods are applied to different problems including a financial problem richardson extrapolation is used to obtain more accurate results resulting in the observation of second order convergence in the number of time steps advantages and disadvantages of each method are compared against each other

green s and dirichlet spaces associated with fine markov processes this is the second paper in a series devoted to green s and dirichlet spaces in the first paper we have investigated green s space k and the dirichlet space h associated with a symmetric markov transition function pt x b now we assume that p is a transition function of a fine markov process x and we prove that a the space h can be built from functions which are right continuous along almost all paths b the positive cone k in k can be identified with a cone m of measures on the state space c the positive cone h in h can be interpreted as the cone of green s potentials of measures m to every measurable set b in the state space e there correspond a subspace k b of k and a subspace h b of h the orthogonal projections of k onto k and of h onto h b can be expressed in terms of the hitting probabilities of b by the markov process x as the main tool we use additive functionals of x corresponding to measures m

wiener criteria for existence of large solutions of nonlinear parabolic equations with absorption in a non cylindrical domain abstract we obtain a necessary condition and a sufficient condition both expressed in terms of wiener type tests involving the parabolic w q 2 1 capacity where q q q 1 and q 1 for the existence of large solutions to equation t u u u q 0 in a non cylindrical domain we provide also a sufficient condition for the existence of such solutions to equation t u u e u 1 0 besides we apply our results to equation t u u a u p b u q 0 for a b 0 1 p 2 and q 1

mixed integer linear programming formulation techniques a wide range of problems can be modeled as mixed integer linear programming mip problems using standard formulation techniques however in some cases the resulting mip can be either too weak or

approximate extended formulations mixed integer programming mip formulations are typically tightened through the use of a separation algorithm and the addition of violated cuts using extended formulations involving new variables is a possible alternative but this often results in prohibitively large mips where even the linear programming relaxations are hard or impossible to solve in this paper we demonstrate how in certain cases it is possible and interesting to define approximate extended formulations in all the examples considered our description involves a single control parameter k large values of k result in strong but large formulations in particular when k takes its maximum value the approximate formulation is identical to the complete extended formulation through this approximation parameter the user has control over the tradeoff between the strength and the size of the formulation

lower bounds on the sizes of integer programs without additional variables let x x be the set of integer points in some polyhedron we investigate the smallest number of facets of any polyhedron whose set of integer points is x x this quantity which we call the relaxation complexity of x x corresponds to the smallest number of linear inequalities of any integer program having x x as the set of feasible solutions that does not use auxiliary variables we show that the use of auxiliary variables is essential for constructing polynomial size integer programming formulations in many relevant cases in particular we provide asymptotically tight exponential lower bounds on the relaxation complexity of the integer points of several well known combinatorial polytopes including the traveling salesman polytope and the spanning tree polytope in addition to the material in the extended abstract kaibel and weltge 2014 we include omitted proofs supporting figures discussions about properties of coefficients in such formulations and facts about the complexity of formulations in more general settings

mixed integer reformulations of integer programs and the affine tu dimension of a matrix we study the reformulation of integer linear programs by means of a mixed integer linear program with fewer integer variables such reformulations can be solved efficiently with mixed integer linear programming techniques we exhibit examples that demonstrate how integer programs can be reformulated using far fewer integer variables to this end we introduce a generalization of total unimodularity called the affine tu dimension of a matrix and study related theory and algorithms for determining the affine tu dimension of a matrix we also present bounds on the number of integer variables needed to represent certain integer hulls