algebraic methods for interactive proof systems a new algebraic technique for the construction of interactive proof systems is presented our technique is used to prove that every language in the polynomial time hierarchy has an interactive proof system this technique played a pivotal role in the recent proofs that ip pspace 28 and that mip nexp 4

on dinur s proof of the pcp theorem probabilistically checkable proofs are proofs that can checked probabilistically by reading very few bits of the proof in the early 1990 s it was shown that proofs could be transformed into probabilistically checkable ones with only a modest increase in their size the initial transformations though elementary were a little too complex a recent work due to irit dinur gives a dramatically simple and radically new construction of probabilistically checkable proofs this article explains the notion of a probabilistically checkable proof presents the formal definition and then introduces the reader to dinur s work along with some of the context

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

mimicking the one dimensional marginal distributions of processes having an ito differential let t be a stochastic process starting from 0 with ito differential d xi left t right delta left t omega right dw left t right beta left t omega right dt where left w left t right mathfrak f t right is a wiener process and are bounded mathfrak f processes such that t is uniformly positive definite then it is proved that there exists a stochastic differential equation dx left t right sigma left t x left t right right dw left t right b left t x left t right right dt with non random coefficients which admits a weak solutionx t having the same one dimensional probability distribution as t for everyt the coefficients andb have a simple interpretation sigma left t x right left e left delta delta t left t right xi left t right x right right tfrac 1 2 b left t x right e left beta left t right xi left t right x right

multi parameter tikhonov regularization and model function approach to the damped morozov principle for choosing regularization parameters in this paper we study the multi parameter tikhonov regularization method which adds multiple different penalties to exhibit multi scale features of the solution an optimal error bound of the regularization solution is obtained by a priori choice of multiple regularization parameters some theoretical results of the regularization solution about the dependence on regularization parameters are presented then an a posteriori parameter choice i e the damped morozov discrepancy principle is introduced to determine multiple regularization parameters five model functions i e two hyperbolic model functions a linear model function an exponential model function and a logarithmic model function are proposed to solve the damped morozov discrepancy principle furthermore four efficient model function algorithms are developed for finding reasonable multiple regularization parameters and their convergence properties are also studied numerical results of several examples show that the damped discrepancy principle is competitive with the standard one and the model function algorithms are efficient for choosing regularization parameters

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

probabilistic checking of proofs a new characterization of np we give a new characterization of np the class np contains exactly those languages l for which membership proofs a proof that an input x is in l can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof we discuss implications of this characterization specifically we show that approximating clique and independent set even in a very weak sense is np hard

on the martingale problem for degenerate parabolic partial differential operators with unbounded coefficients and a mimicking theorem for ito processes using results from our companion article 26 on a schauder approach to existence of solutions to a degenerate parabolic partial differential equation we solve three intertwined problems motivated by probability theory and mathematical finance concerning degenerate diffusion processes we show that the martingale problem associated with a degenerate elliptic differential operator with unbounded locally holder continuous coefficients on a half space is wellposed in the sense of stroock and varadhan second we prove existence uniqueness and the strong markov property for weak solutions to a stochastic differential equation with degenerate diffusion and unbounded coefficients with suitable holder continuity properties third for an ito process with degenerate diffusion and unbounded but appropriately regular coefficients we prove existence of a strong markov process unique in the sense of probability law whose onedimensional marginal probability distributions match those of the given ito process

a schauder approach to degenerate parabolic partial differential equations with unbounded coefficients abstract motivated by applications to probability and mathematical finance we consider a parabolic partial differential equation on a half space whose coefficients are suitably holder continuous and allowed to grow linearly in the spatial variable and which become degenerate along the boundary of the half space we establish existence and uniqueness of solutions in weighted holder spaces which incorporate both the degeneracy at the boundary and the unboundedness of the coefficients in our companion article feehan and pop 12 we apply the main result of this article to show that the martingale problem associated with a degenerate elliptic partial differential operator is well posed in the sense of stroock and varadhan

ip pspace in this paper it is proven that when both randomization and interaction are allowed the proofs that can be verified in polynomial time are exactly those proofs that can be generated with polynomial space

a supermartingale characterization of sets of stochastic integrals and applications a supermartingale characterization of sets of stochastic integrals is given along with its applications to control and diffusion approximation the characterization is convenient for passing to the limit under natural conditions it is proved that the set of distributions of controlled diffusion processes is convex and compact

pspace has constant round quantum interactive proof systems in this paper we introduce quantum interactive proof systems which are interactive proof systems in which the prover and verifier may perform quantum computations and exchange quantum messages it is proved that every language in pspace has a quantum interactive proof system that requires a total of only three messages to be sent between the prover and verifier and has exponentially small one sided probability of error it follows that quantum interactive proof systems are strictly more powerful than classical interactive proof systems in the constant round case unless the polynomial time hierarchy collapses to the second level

bounding queries in the analytic polynomial time hierarchy abstract in a previous paper the present authors baier and wagner 1996 investigated an hierarchy over p using word quantifiers as well as two types of set quantifiers the so called analytic polynomial time hierarchy the fact that some constructions there result in a bounded number of oracle queries and the recent pcp results which can be expressed by set quantifiers with a bounded number of queries motivated us to examine a hierarchy which extends the analytic polynomial time hierarchy by considering restrictions on the number of oracle queries this hierarchy is called bounded analytic polynomial time hierarchy we show that every class from this hierarchy having a certain normal form coincides with one of the classes np conp pspace kexp or kexp k 1 all these characterizations remain valid if the queries are asked in a nonadaptive form i e in parallel

convergence tests on constant dirichlet series by means of the theory of mathematical analysis this paper derives the convergence tests for the following two kinds of constant dirichlet series and an improper integral n 1 p n f j n d n n 1 1 n 1 f j n d n and 1 p t f j t d t dt as applications we give two assertions to demonstrate the effectiveness of these convergence tests

on the drawdowns and drawups in diffusion type models with running maxima and minima we obtain closed form expressions for the values of joint laplace transforms of the running maximum and minimum of a diffusion type process stopped at the first time at which the associated drawdown and drawup processes hit constant levels it is assumed that the coefficients of the diffusion type process are regular functions of the running values of the process itself its maximum and minimum as well as its maximum drawdown and maximum drawup processes the proof is based on the solution to the equivalent boundary value problems and application of the normal reflection conditions for the value functions at the edges of the state space of the resulting five dimensional markov process we show that the joint laplace transforms represent linear combinations of solutions to the systems of first order partial differential equations arising from the application of the normal reflection conditions

convex regularization of local volatility models from option prices convergence analysis and rates abstract we study a convex regularization of the local volatility surface identification problem for the black scholes partial differential equation from prices of european call options this is a highly nonlinear ill posed problem which in practice is subject to different noise levels associated to bid ask spreads and sampling errors we analyze in appropriate function spaces different properties of the parameter to solution map that assigns to a given volatility surface the corresponding option prices using such properties we show stability and convergence of the regularized solutions in terms of the bregman distance with respect to a class of convex regularization functionals when the noise level goes to zero we improve convergence rates available in the literature for the volatility identification problem furthermore in the present context we relate convex regularization with the notion of exponential families in statistics finally we connect convex regularization functionals with convex risk measures through fenchel conjugation we do this by showing that if the source condition for the regularization functional is satisfied then convex risk measures can be constructed

multi parameter regularization techniques for ill conditioned linear systems when a system of linear equations is ill conditioned regularization techniques provide a quite useful tool for trying to overcome the numerical inherent difficulties the ill conditioned system is replaced by another one whose solution depends on a regularization term formed by a scalar and a matrix which are to be chosen in this paper we consider the case of several regularizations terms added simultaneously thus overcoming the problem of the best choice of the regularization matrix the error of this procedure is analyzed and numerical results prove its efficiency

multi parameter regularization and its numerical realization in this paper we propose and analyse a choice of parameters in the multi parameter regularization of tikhonov type a modified discrepancy principle is presented within the multi parameter regularization framework an order optimal error bound is obtained under the standard smoothness assumptions we also propose a numerical realization of the multi parameter discrepancy principle based on the model function approximation numerical experiments on a series of test problems support theoretical results finally we show how the proposed approach can be successfully implemented in laplacian regularized least squares for learning from labeled and unlabeled examples

marginal density expansions for diffusions and stochastic volatility ii applications in 17 we discussed density expansions for multidimensional diffusions x 1 x d at fixed time t and projected to their first l coordinates in the small noise regime global conditions were found which replace the well known not in cutlocus condition known from heat kernel asymptotics in the present paper we discuss financial applications these include tail and implied volatility asymptotics in some correlated stochastic volatility models in particular we solve a problem left open by a gulisashvili and e m stein 2009

the mean field schrodinger problem ergodic behavior entropy estimates and functional inequalities we study the mean field schrodinger problem mfsp that is the problem of finding the most likely evolution of a cloud of interacting brownian particles conditionally on the observation of their initial and final configuration its rigorous formulation is in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent brownian particles we undertake a fine study of the dynamics of its solutions including quantitative energy dissipation estimates yielding the exponential convergence to equilibrium as the time between observations grows larger and larger as well as a novel class of functional inequalities involving the mean field entropic cost i e the optimal value in mfsp our strategy unveils an interesting connection between forward backward stochastic differential equations and the riemannian calculus on the space of probability measures introduced by otto which is of independent interest

a general heuristic for choosing the regularization parameter in ill posed problems for a variety of regularization methods including tikhonov regularization landweber iteration nu method iteration and the method of conjugate gradients we develop and illustrate a heuristi

marginal density expansions for diffusions and stochastic volatility i theoretical foundations density expansions for hypoelliptic diffusions x1 xd are revisited we are particularly interested in density expansions of the projection at time t 0 with l d global conditions are found that replace the well known not in cut locus condition known from heat kernel asymptotics our small noise expansion allows for a second order exponential factor as an application new light is shed on the takanobu watanabe expansion of brownian motion and levy s stochastic area further applications include tail and implied volatility asymptotics in some stochastic volatility models discussed in the companion paper 12 2013 wiley periodicals inc

convergence of best entropy estimates given a finite number of moments of an unknown density on a finite measure space the best entropy estimate that nonnegative density x with the given moments which minimizes the boltzmann shannon entropy i x x log x is considered a direct proof is given that i has the kadec property in l if yn converges weakly to 37 and i yn converges to i 37 then yn converges to 37 in norm as a corollary it is obtained that as the number of given moments increases the best entropy estimates converge in li norm to the best entropy estimate of the limiting problem which is simply in the determined case furthermore for classical moment problems on intervals with strictly positive and sufficiently smooth error bounds and uniform convergence are actually obtained

on absolute summability factors of infinite series in this paper a general theorem on a d k summability methods has been proved this theorem includes as a special case a known result in e savas factors for a k summability of infinite series comput math appl 53 2007 1045 1049

on the power of multi prover interactive protocols abstract we look at complexity issues of interactive proof systems with multiple provers separated from each other this model developed by ben or et al 1988 allows the verifier to play the provers off each other we show this model equivalent to an alternative interactive proof system model using oracles as provers we also show that every language accepted by these models lies in nondeterministic exponential time we exhibit a relativized world where a co np language does not have multiple prover interactive proofs finally we show a simple example that one cannot parallelize multiple prover protocols as easily as the single prover model

the pcp theorem by gap amplification the pcp theorem arora and safra 1998 arora et al 1998 says that every language in np has a witness format that can be checked probabilistically by reading only a constant number of bits from the proof the celebrated equivalence of this theorem and inapproximability of certain optimization problems due to feige et al 1996 has placed the pcp theorem at the heart of the area of inapproximability in this work we present a new proof of the pcp theorem that draws on this equivalence we give a combinatorial proof for the np hardness of approximating a certain constraint satisfaction problem which can then be reinterpreted to yield the pcp theorem our approach is to consider the unsat value of a constraint system which is the smallest fraction of unsatisfied constraints ranging over all possible assignments for the underlying variables we describe a new combinatorial amplification transformation that doubles the unsat value of a constraint system with only a linear blowup in the size of the system the amplification step causes an increase in alphabet size that is corrected by a standard pcp composition step iterative application of these two steps yields a proof for the pcp theorem the amplification lemma relies on a new notion of graph powering that can be applied to systems of binary constraints this powering amplifies the unsat value of a constraint system provided that the underlying graph structure is an expander we also extend our amplification lemma towards construction of assignment testers alternatively pcps of proximity which are slightly stronger objects than pcps we then construct pcps and locally testable codes whose length is linear up to a polylog factor and whose correctness can be probabilistically verified by making a constant number of queries namely we prove sat pcp 1 2 1 log2 n poly log n o 1

the calibration of stochastic local volatility models an inverse problem perspective abstract we tackle the calibration of the stochastic local volatility slv model this is the class of financial models that combines the local volatility and stochastic volatility features and has been subject of the attention by many researchers and practitioners recently the corresponding inverse problem consists in finding certain functional coefficients in a class of parabolic partial differential equations from observed values of the solutions more precisely given a calibrated local volatility surface and a choice of stochastic volatility parameters we calibrate the corresponding leverage function our approach makes use of regularization techniques from inverse problem theory respecting the integrity of the data and thus avoiding data interpolation the result is a stable and efficient algorithm which is resilient to instabilities in the regions of low probability density of the spot price and of the instantaneous variance we substantiate our claims with numerical experiments using synthetic and real data