on the mean field and classical limits of quantum mechanics the main result in this paper is a new inequality bearing on solutions of the n body linear schrodinger equation and of the mean field hartree equation this inequality implies that the mean field limit of the quantum mechanics of n identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation this result applies to the case of c1 1 interaction potentials the quantity measuring the approximation of the n body quantum dynamics by its mean field limit is analogous to the monge kantorovich or wasserstein distance with exponent 2 the inequality satisfied by this quantity is reminiscent of the work of dobrushin on the mean field limit in classical mechanics func anal appl 13 115 123 1979 our approach to this problem is based on a direct analysis of the n particle liouville equation and avoids using techniques based on the bbgky hierarchy or on second quantization

a penalty function method based on smoothing lower order penalty function the paper introduces a smoothing technique for a lower order penalty function for constrained optimization problems cop it is proved that the optimal solution to the smoothed penalty optimization problem is a e2 approximate optimal solution to the original optimization problem under some mild assumptions based on the smoothed penalty function an algorithm for solving cop is proposed and some numerical examples are given

ricci curvature of finite markov chains via convexity of the entropy we study a new notion of ricci curvature that applies to markov chains on discrete spaces this notion relies on geodesic convexity of the entropy and is analogous to the one introduced by lott sturm and villani for geodesic measure spaces in order to apply to the discrete setting the role of the wasserstein metric is taken over by a different metric having the property that continuous time markov chains are gradient flows of the entropy using this notion of ricci curvature we prove discrete analogues of fundamental results by bakry emery and otto villani further we show that ricci curvature bounds are preserved under tensorisation as a special case we obtain the sharp ricci curvature lower bound for the discrete hypercube

convexity of quasi entropy type functions lieb s and ando s convexity theorems revisited given a positive function f on 0 and a non zero real parameter we consider a function if a b x tr x f larb 1 rb x in three matrices a b 0 and x this generalizes the notion of monotone metrics on positive definite matrices and in the literature 1 has been typical we investigate how operator monotony of f is sufficient and or necessary for joint convexity concavity of if a b x similar discussions are given for quasi entropies and quantum skew informations

a robust and accurate finite difference method for a generalized black scholes equation in this paper we present a numerical method for a generalized black scholes equation which is used for option pricing the method is based on a central difference spatial discretization on a piecewise uniform mesh and an implicit time stepping technique our scheme is stable for arbitrary volatility and arbitrary interest rate and is second order convergent with respect to the spatial variable furthermore the present paper efficiently treats the singularities of the non smooth payoff function numerical results support the theoretical results

a power penalty method for solving a nonlinear parabolic complementarity problem abstract in this paper we present a penalty method for solving a complementarity problem involving a second order nonlinear parabolic differential operator in this work we first rewrite the complementarity problem as a nonlinear variational inequality then we define a nonlinear parabolic partial differential equation pde approximating the variational inequality using a power penalty term with a penalty constant 1 a power parameter k 0 and a smoothing parameter e we prove that the solution to the penalized pde converges to that of the variational inequality in an appropriate norm at an arbitrary exponential rate of the form o k e 1 e 1 k 1 2 numerical experiments performed to verify the theoretical results show that the computed rates of convergence in both and k are close to the theoretical ones

an upwind numerical approach for an american and european option pricing model the numerical solution of several mathematical models arising in financial economics for the valuation of both european and american call options on different types of assets is considered all the models are based on the black scholes partial differential equation in the case of european options a numerical upwind scheme for solving the boundary backward parabolic partial differential equation problem is presented when treating with american options an additional inequality constraint leads to a discretized linear complementarity problem in each case the numerical approximation of option values is computed by means of optimization algorithms in particular uzawa s method allows to compute the optimal exercise boundary which corresponds to the classical concept of moving boundary in continuum mechanics

numerical approximation of a time fractional black scholes equation abstract in this paper a time fractional black scholes equation is examined we transform the initial value problem into an equivalent integral differential equation with a weakly singular kernel and use an integral discretization scheme on an adapted mesh for the time discretization a rigorous analysis about the convergence of the time discretization scheme is given by taking account of the possibly singular behavior of the exact solution and first order convergence with respect to the time variable is proved for overcoming the possibly nonphysical oscillation in the computed solution caused by the degeneracy of the black scholes differential operator we employ a central difference scheme on a piecewise uniform mesh for the spatial discretization it is proved that the scheme is stable and second order convergent with respect to the spatial variable numerical experiments support these theoretical results

concentration of quantum states from quantum functional and transportation cost inequalities quantum functional inequalities e g the logarithmic sobolev and poincare inequalities have found widespread application in the study of the behavior of primitive quantum markov semigroups the classical counterparts of these inequalities are related to each other via a so called transportation cost inequality of order 2 tc2 the latter inequality relies on the notion of a metric on the set of probability distributions called the wasserstein distance of order 2 tc2 in turn implies a transportation cost inequality of order 1 tc1 in this paper we introduce quantum generalizations of the inequalities tc1 and tc2 making use of appropriate quantum versions of the wasserstein distances one recently defined by carlen and maas and the other defined by us we establish that these inequalities are related to each other and to the quantum modified logarithmic sobolev and poincare inequalities as in the classical case we also show that these inequalities imply certain concentration type results

non commutative calculus optimal transport and functional inequalities in dissipative quantum systems we study dynamical optimal transport metrics between density matrices associated to symmetric dirichlet forms on finite dimensional c algebras our setting covers arbitrary skew derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for markov chains lindblad equations and the fermi ornstein uhlenbeck semigroup we develop a non nommutative differential calculus that allows us to obtain non commutative ricci curvature bounds logarithmic sobolev inequalities transport entropy inequalities and spectral gap estimates

global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities the smoothing newton method for solving a system of nonsmooth equations f x 0 which may arise from the nonlinear complementarity problem the variational inequality problem or other problems can be regarded as a variant of the smoothing method at the kth step the nonsmooth function f is approximated by a smooth function f and the derivative of f at x k is used as the newton iterative matrix the merits of smoothing methods and smoothing newton methods are global convergence and convenience in handling in this paper we show that the smoothing newton method is also superlinearly convergent if f is semismooth at the solution and f satisfies a jacobian consistency property we show that most common smooth functions such as the gabriel more function have this property as an application we show that for box constrained variational inequalities if the involved function is p uniform the iteration sequence generated by the smoothing newton method will converge to the unique solution of the problem globally and superlinearly quadratically

black box complexity of local minimization the complexity of local minimization in the black box model that is the model in which the objective function and its gradient are available as external subroutines is studied the black box model is used for example in all the optimization algorithms in dennis and schnabel numerical methods for unconstrained optimization and nonlinear equations prentice hall englewood cliffs nj 1983 the first main result is that the complexity grows polynomially with the number of variables n in contrast to other related black box problems global minimization and brouwer fixed points for which the worst case complexity is exponential in n the second contribution is the construction of a family of functions that are bad cases for all possible black box local optimization algorithms

contractivity of positive and trace preserving maps under lp norms we provide a complete picture of contractivity of trace preserving positive maps with respect to p norms we show that for p 1 contractivity holds in general if and only if the map is unital when the domain is restricted to the traceless subspace of hermitian matrices then contractivity is shown to hold in the case of qubits for arbitrary p 1 and in the case of qutrits if and only if p 1 in all noncontractive cases best possible bounds on the p norms are derived

optimal convergence rate of the explicit finite difference scheme for american option valuation an optimal convergence rate o dx for an explicit finite difference scheme for a variational inequality problem is obtained under the stability condition s email protected dx 2 1 using completely pde methods as a corollary a binomial tree scheme of an american put option where s email protected dx 2 1 is convergent unconditionally with the rate o dt 1 2

a nonlinear lagrangian approach to constrained optimization problems in this paper we study nonlinear lagrangian functions for constrained optimization problems which are in general nonlinear with respect to the objective function we establish an equivalence between two types of zero duality gap properties which are described using augmented lagrangian dual functions and nonlinear lagrangian dual functions respectively furthermore we show the existence of a path of optimal solutions generated by nonlinear lagrangian problems and show its convergence toward the optimal set of the original problem we analyze the convergence of several classes of nonlinear lagrangian problems in terms of their first and second order necessary optimality conditions

option pricing using a computational method based on reproducing kernel abstract one of the most important subject in financial mathematics is the option pricing the most famous result in this area is black scholes formula for pricing european options this paper is concerned with a method for solving a generalized black scholes equation in a reproducing kernel hilbert space subsequently the convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method furthermore the error estimates for obtained approximation in reproducing kernel hilbert space are presented finally a numerical example is considered to illustrate the computation efficiency and accuracy of the proposed method

necessary and sufficient optimality conditions for a class of nonsmooth minimization problems the purpose of this paper is to derive in a unified way second order necessary and sufficient optimality criteria for four types of nonsmooth minimization problems thediscrete minimax problem thediscrete l1 approximation the minimization of theexact penalty function and the minimization of theclassical exterior penalty function our results correct and supplement conditions obtained by various authors in recent papers

an augmented lagrangian approach with a variable transformation in nonlinear programming abstract tangent cone and regular normal cone of a closed set under an invertible variable transformation around a given point are investigated which lead to the concepts of 1 tangent cone of a set and 1 subderivative of a function when the notion of 1 subderivative is applied to perturbation functions a class of augmented lagrangians involving an invertible mapping of perturbation variables are obtained in which dualizing parameterization and augmenting functions are not necessarily convex in perturbation variables a necessary and sufficient condition for the exact penalty representation under the proposed augmented lagrangian scheme is obtained for an augmenting function with an euclidean norm a sufficient condition resp a sufficient and necessary condition for an arbitrary vector resp 0 to support an exact penalty representation is given in terms of 1 subderivatives an example of the variable transformation applied to constrained optimization problems is given which yields several exact penalization results in the literature

quantum logarithmic sobolev inequalities and rapid mixing a family of logarithmic sobolev inequalities on finite dimensional quantum state spaces is introduced the framework of non commutative lp spaces is reviewed and the relationship between quantum logarithmic sobolev inequalities and the hypercontractivity of quantum semigroups is discussed this relationship is central for the derivation of lower bounds for the logarithmic sobolev ls constants essential results for the family of inequalities are proved and we show an upper bound to the generalized ls constant in terms of the spectral gap of the generator of the semigroup these inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups when the ls constant and the spectral gap are of the same order convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory we provide a number of examples where improved bounds on the mixing time of several semigroups are obtained incl

tau functions for the dirac operator on the cylinder the goal of the present paper is to calculate the determinant of the dirac operator with a mass in the cylindrical geometry the domain of this operator consists of functions that realize a unitary one dimensional representation of the fundamental group of the cylinder with n marked points the determinant represents a version of the isomonodromic function introduced by m sato t miwa and m jimbo it is calculated by comparison of two sections of the det bundle over an infinite dimensional grassmannian the latter is composed of the spaces of boundary values of some local solutions to the dirac equation the principal ingredients of the computation are the formulae for the green function of the singular dirac operator and for the so called canonical basis of global solutions on the 1 punctured cylinder we also derive a set of deformation equations satisfied by the expansion coefficients of the canonical basis in the general case and find a more explicit expression for the function in the simplest case n 2

short distance asymptotics of ising correlations we prove that the short distance asymptotics for the even ising model scaling functions from below tc is given by the luther peschel formula generalizations to the odd scaling functions and holonomic fields are given

matrix optimal mass transport a quantum mechanical approach in this paper we describe a possible generalization of the wasserstein 2 metric originally defined on the space of scalar probability densities to the space of hermitian matrices with trace one and to the space of matrix valued probability densities our approach follows a control theoretic optimization formulation of the wasserstein 2 metric having its roots in fluid dynamics and utilizes certain results from the quantum mechanics of open systems in particular the lindblad equation it allows determining the gradient flow for the quantum entropy relative to this matricial wasserstein metric

an inverse problem for a parabolic variational inequality arising in volatility calibration with american options in finance the price of an american option is obtained from the price of the underlying asset by solving a parabolic variational inequality the free boundary associated with this variational inequality can be interpreted as the price for which the option should be exercised the calibration of volatility from the observations of the prices of an american option yields an inverse problem for the previously mentioned parabolic variational inequality after studying the variational inequality and the exercise price we give results concerning the sensitivity of the option price and of the exercise price with respect to the variations of the volatility the inverse problem is addressed by a least square method with suitable regularization terms we give necessary optimality conditions involving an adjoint state for a simplified inverse problem and we study the differentiability of the cost function optimality conditions are also given for the genuine calibration problem

the zero duality gap property and lower semicontinuity of the perturbation function we examine the validity of the zero duality gap properties for two important dual schemes a generalized augmented lagrangian dual scheme and a nonlinear lagrange type dual scheme the necessary and sufficient conditions for the zero duality gap property to hold are established in terms of the lower semicontinuity of the perturbation functions

finite element error estimates for a nonlocal problem in american option valuation based on new exact formulations of american option problems on bounded domains error estimates are established for finite element approximations of american option prices under admissible regularity some numerical results are also presented

numerical methods for reaction diffusion problems with non differentiable kinetics we consider a class of steady state semilinear reaction diffusion problems with non differentiable kinetics the analytical properties of these problems have received considerable attention in the literature we take a first step in analyzing their numerical approximation we present a finite element method and establish error bounds which are optimal for some of the problems in addition we also discuss a finite difference approach numerical experiments for one and two dimensional problems are reported

a unified augmented lagrangian approach to duality and exact penalization in this paper the existence of an optimal path and its convergence to the optimal set of a primal problem of minimizing an extended real valued function are established via a generalized augmented lagrangian and corresponding generalized augmented lagrangian problems in which no convexity is imposed on the augmenting function these results further imply a zero duality gap property between the primal problem and the generalized augmented lagrangian dual problem a necessary and sufficient condition for the exact penalty representation in the framework of a generalized augmented lagrangian is obtained in the context of constrained programs we show that generalized augmented lagrangians present a unified approach to several classes of exact penalization results some equivalences among exact penalization results are obtained

convergence of a fitted finite volume method for the penalized black scholes equation governing european and american option pricing in this paper we present an analysis of a numerical method for a degenerate partial differential equation called the black scholes equation governing american and european option pricing the method is based on a fitted finite volume spatial discretization and an implicit time stepping technique the analysis is performed within the framework of the vertical method of lines where the spatial discretization is formulated as a petrov galerkin finite element method with each basis function of the trial space being determined by a set of two point boundary value problems we establish the stability and an error bound for the solutions of the fully discretized system numerical results are presented to validate the theoretical results

a super convergent unsymmetric finite volume method for convection diffusion equations abstract the design of the dual mesh of a conventional fitted finite volume method is revisited with the aim to obtain better convergence rates a new choice of dual mesh points is proposed and it is proved that at these points the approximate flux density and the solution have a second order accuracy the results of numerical experiments substantiate the theoretical findings

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

finite difference scheme with a moving mesh for pricing asian options in this paper we propose a stable numerical method for pricing asian call options which is based on a central difference scheme with a moving mesh in the spatial discretization and the rannacher time stepping scheme in the time discretization at each time mesh point we make a piecewise uniform mesh to discretise the space interval which ensures that the matrix associated with the discrete operator is an m matrix hence the spatial discretization scheme is maximum norm stable for arbitrary volatility and arbitrary interest rate we show that the scheme is second order convergent with respect to both time and spatial variables numerical results support the theoretical results

a smoothing type algorithm for solving system of inequalities in this paper we consider system of inequalities by constructing a new smoothing function the problem is approximated via a family of parameterized smooth equations a newton type algorithm is applied to solve iteratively the smooth equations so that a solution of the problem concerned is found we show that the algorithm is globally and locally quadratically convergent under suitable assumptions preliminary numerical results are reported

a superlinearly convergent method for a class of complementarity problems with non lipschitzian functions we consider a class of complementarity problems involving functions which are not lipschitz continuous in this paper we reformulate this class of non lipschitzian complementarity problems into a lipschitzian complementarity problem then we propose an inexact smoothing newton method to solve this lipschitzian complementarity problem we prove that our proposed method converges quadratically and globally under a mild condition numerical results show that this method is promising this method can solve these kinds of complementarity problems with one million variables in reasonable time on a pc with 1 gb of ram

a fast numerical method for the black scholes equation of american options this paper introduces a fast numerical method for computing american option pricing problems governed by the black scholes equation the treatment of the free boundary is based on some properties of the solution of the black scholes equation an artificial boundary condition is also used at the other end of the domain the finite difference method is used to solve the resulting problem computational results are given for some american call option problems the results show that the new treatment is very efficient and gives better accuracy than the normal finite difference method

asymptotics of a class of solutions to the cylindrical toda equations the small t asymptotics of a class of solutions to the 2d cylindrical toda equations is computed the solutions qk t have the representation qk t log det i k k log det i k k 1 where kk are integral operators this class includes the n periodic cylindrical toda equations for n 2 our results reduce to the previously computed asymptotics of the 2d radial sinh gordon equation and for n 3 and with an additional symmetry constraint they reduce to earlier results for the radial bullough dodd equation both of these special cases are examples of painleviii and have arisen in various applications the asymptotics of qk t are derived by computing the small t asymptotics det i k k b k t n a k

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

an upwind finite difference method for a nonlinear black scholes equation governing european option valuation under transaction costs in this paper we develop a numerical method for a nonlinear parabolic partial differential equation arising from pricing european options under transaction costs the method is based on an upwind finite difference scheme for the spatial discretization and a fully implicit time stepping scheme we prove that the system matrix from this scheme is an m matrix and that the approximate solution converges unconditionally to the viscosity solution to the equation by showing that the scheme is consistent monotone and unconditionally stable a newton iterative algorithm is proposed for solving the discretized nonlinear system of which the jacobian matrix is shown to be also an m matrix numerical experiments are performed to demonstrate the accuracy and robustness of the method

a fixed point method for a class of super large scale nonlinear complementarity problems we consider a class of complementarity problems involving functions which are nonlinear in this paper we reformulate this nonlinear complementarity problem as a system of absolute value equations which is nonsmooth then we propose a fixed point method to solve this nonsmooth system we prove that the proposed method is globally linearly convergent under a mild condition the proposed method is greatly effective not only for small and medium size problems but also for large and super large scale problems especially our method can efficiently solve super large scale problems with a million variables in a few tens of minutes on a pc

decreasing functions with applications to penalization the theory of increasing positively homogeneous functions defined on the positive orthant is applied to the class of decreasing functions a multiplicative version of the inf convolution operation is studied for decreasing functions modified penalty functions for some constrained optimization problems are introduced that are in general nonlinear with respect to the objective function of the original problem as the perturbation function of a constrained optimization problem is decreasing the theory of decreasing functions is subsequently applied to the study of modified penalty functions the zero duality gap property and the exact penalization

exact augmented lagrangian duality for mixed integer linear programming we investigate the augmented lagrangian dual ald for mixed integer linear programming mip problems ald modifies the classical lagrangian dual by appending a nonlinear penalty function on the violation of the dualized constraints in order to reduce the duality gap we first provide a primal characterization for ald for mips and prove that ald is able to asymptotically achieve zero duality gap when the weight on the penalty function is allowed to go to infinity this provides an alternative characterization and proof of a recent result in boland and eberhard math program 150 2 491 509 2015 proposition 3 we further show that under some mild conditions ald using any norm as the augmenting function is able to close the duality gap of an mip with a finite penalty coefficient this generalizes the result in boland and eberhard 2015 corollary 1 from pure integer programming problems with bounded feasible region to general mips we also present an example where ald with a quadratic augmenting function is not able to close the duality gap for any finite penalty coefficient

operator lipschitz functions in schatten von neumann classes this paper resolves a number of problems in the perturbation theory of linear operators linked with the 45 year old conjecure of m g kre n in particular we prove that every lipschitz function is operator lipschitz in the schatten von neumann ideals s 1 0 such that left f a f b right alpha leqslant c alpha left f right text lip 1 left a b right alpha where f is a lipschitz function with left f right text lip 1 mathop sup limits lambda ne mu lambda mu in mathbb r left frac f left lambda right f left mu right lambda mu right infty left cdot right alpha is the norm is s and a and b are self adjoint linear operators such that a b in s alpha

a penalty method for a mixed nonlinear complementarity problem abstract we propose a power penalty method for a mixed nonlinear complementarity problem mncp and show that the solution to the penalty equation converges to that of the mncp exponentially as the penalty parameter approaches infinity provided that the mapping involved in the mncp is both continuous and monotone furthermore a convergence theorem is established when the monotonicity assumption on the mapping is removed to demonstrate the usefulness and the convergence rates of this method we design a non trivial test mncp problem arising in shape preserving bi harmonic interpolation and apply our method to this test problem the numerical results confirm our theoretical findings

quintic b spline collocation approach for solving generalized black scholes equation governing option pricing we develop a numerical algorithm for solving generalized black scholes partial differential equation which arises in european option pricing the method comprises the horizontal method of lines for time integration and method 1 2 1 1 corresponds to the back ward euler method and 1 2 corresponds to the crank nicolson method to discretize in temporal direction and the quintic b spline collocation method in uniform spatial direction the convergence analysis and stability of proposed method are discussed in detail it is justifying that the approximate solution converges to the exact solution of orders o k h 3 for the back ward euler method and o k 2 h 3 for the crank nicolson method where k and h are mesh sizes in the time and space directions respectively the proposed method is also shown to be unconditionally stable this scheme applied on some test examples the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence results shown by this method are found to be in good agreement with the known exact solutions the produced results are also seen to be more accurate than some available results given in the literature

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

b spline based monotone multigrid methods for the efficient numerical solution of elliptic variational inequalities on closed convex sets multigrid methods based on piecewise linear finite elements have been investigated over the past decades essential to their success is the appropriate approximation of the constraint set on coarser grids which is based on function values for piecewise linear finite elements on the other hand there are a number of problems which profit from higher order approximations among these are the problem of pricing american options formulated as a parabolic boundary value problem involving black scholes equation with a free boundary in addition to computing the free boundary the optimal exercise price of the option of particular importance are accurate pointwise derivatives of the value of the stock option up to order two the so called greek letters in this paper we propose a monotone multigrid method for discretizations in terms of b splines of arbitrary order to solve elliptic variational inequalities on a closed convex set in order to maintain monotonicity upper bound and quasi optimality lower bound of the coarse grid corrections we propose an optimized coarse grid correction ocgc algorithm which is based on b spline expansion coefficients we prove that the ocgc algorithm is of optimal complexity of the degrees of freedom of the coarse grid and therefore the resulting monotone multigrid method is of asymptotically optimal multigrid complexity finally the method is applied to a standard model for the valuation of american options in particular it is shown that a discretization based on b splines of order four enables us to compute the second derivative of the value of the stock option to high precision

gradient flows of the entropy for finite markov chains let k be an irreducible and reversible markov kernel on a finite set x we construct a metric w on the set of probability measures on x and show that with respect to this metric the law of the continuous time markov chain evolves as the gradient flow of the entropy this result is a discrete counterpart of the wasserstein gradient flow interpretation of the heat flow in rn by jordan kinderlehrer and otto 1998 the metric w is similar to but different from the l2 wasserstein metric and is defined via a discrete variant of the benamou brenier formula

asymptotics of a τ function arising in the two dimensional ising model the short distance assymptotics of the function associated to the 2 point function of the two dimensional ising model is computed as a function of the integration constant defined from the long distance behavior of the function the result is expressible in terms of the barnes double gamma function equivalently the barnesg function

derivations as square roots of dirichlet forms abstract the aim of this work is to analyze the structure of a tracially symmetric dirichlet form on a c algebra in terms of a killing weight and a closable derivation taking values in a hilbert space with a bimodule structure it is shown that the generator of the associate markovian semigroup always appears in a natural way as the divergence of a closable derivation applications are shown to the decomposition of dirichlet forms and to the construction of differential calculus on metric spaces

poincare modified logarithmic sobolev and isoperimetric inequalities for markov chains with non negative ricci curvature abstract we study functional inequalities for markov chains on discrete spaces with entropic ricci curvature bounded from below our main results are that when curvature is non negative but not necessarily positive the spectral gap the cheeger isoperimetric constant and the modified logarithmic sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space with respect to a suitable metric these estimates are discrete analogues of classical results of riemannian geometry obtained by li and yau buser and wang

an exponentially fitted finite volume method for the numerical solution of 2d unsteady incompressible flow problems abstract in this paper we develop and test an exponentially fitted finite volume method for the numerical solution of the navier stokes equations describing 2d incompressible flows the method is based on an unstructured delaunay mesh and its dual dirichlet tessellation combined with a locally constant approximation to the flux this yields a piecewise exponential approximation to the exact solution numerical tests are presented for a linear advection diffusion problem with boundary layers the method is then applied to the driven cavity problem with reynolds numbers up to 10 4 the numerical results indicate that the method is robust for a wide range of values of the reynolds number in the case re 10 4 unsteady solutions are captured if the mesh is sufficiently fine

asymptotic analysis for penalty and barrier methods in convex and linear programming we consider a wide class of penalty and barrier methods for convex programming which includes a number of specific functions proposed in the literature we provide a systematic way to generate penalty and barrier functions in this class and we analyze the existence of primal and dual optimal paths generated by these penalty methods as well as their convergence to the primal and dual optimal sets for linear programming we prove that these optimal paths converge to single points

optimal hypercontractivity for fermi fields and related non commutative integration inequalities optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained these are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by nelson in the process several results of independent interest in the theory of non commutative integration are established

an analog of the 2 wasserstein metric in non commutative probability under which the fermionic fokker planck equation is gradient flow for the entropy let mathfrak c denote the clifford algebra over mathbb r n which is the von neumann algebra generated by n self adjoint operators qj j 1 n satisfying the canonical anticommutation relations qiqj qjqi 2 iji and let denote the normalized trace on mathfrak c this algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom let mathfrak p denote the set of all positive operators rho in mathfrak c such that 1 these are the non commutative analogs of probability densities in the non commutative probability space mathfrak c tau the fermionic fokker planck equation is a quantum mechanical analog of the classical fokker planck equation with which it has much in common such as the same optimal hypercontractivity properties in this paper we construct a riemannian metric on mathfrak p that we show to be a natural analog of the classical 2 wasserstein metric and we show that in analogy with the classical case the fermionic fokker planck equation is gradient flow in this metric for the relative entropy with respect to the ground state we derive a number of consequences of this such as a sharp talagrand inequality for this metric and we prove a number of results pertaining to this metric several open problems are raised

pricing options under jump diffusion processes with fitted finite volume method this paper develops a numerical method for a partial integro differential equation and a partial integro differential complementarity problem arising from european and american options valuations respectively when the underlying assets are driven by a jump diffusion process the method is based on a fitted finite volume scheme for the spatial discretization and the crank nicolson scheme for the time discretization the fully discretized system is solved by an iterative method coupled with an fft for the evaluation of the discretized integral term while the constraint in the american option model is imposed by adding a penalty term to the original partial integro differential complementarity problem we show that the system matrix of the discretized system is an m matrix and propose an algorithm for solving the discretized system numerical experiments are implemented to show the efficiency and robustness of this method

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence