hermite polynomial based expansion of european option prices abstract we seek a closed form series approximation of european option prices under a variety of diffusion models the proposed convergent series are derived using the hermite polynomial approach departing from the usual option pricing routine in the literature our model assumptions have no requirements for affine dynamics or explicit characteristic functions moreover convergent expansions provide a distinct insight into how and on which order the model parameters affect option prices in contrast with small time asymptotic expansions in the literature with closed form expansions we explicitly translate model features into option prices such as mean reverting drift and self exciting or skewed jumps numerical examples illustrate the accuracy of this approach and its advantage over alternative expansion methods

analysing images of curved surfaces a reflectance map makes the relationship between image intensity and surface orientation explicit trade offs between image intensity and surface orientation emerge which cannot be resolved locally in a single view existing methods for determining surface orientation from a single view embody assumptions about surface curvature the hessian matrix is introduced to represent surface curvature properties of surface curvature are expressed as properties of the hessian matrix for several classes of surface image analysis simplifies this result has already been established for planar surfaces forming trihedral corners similar simplification is demonstrated for developable surfaces and for the subclass of surfaces known as generalized cones these studies help to delineate shape information that can be determined from geometric measurements at object boundaries and shape information that can be determined from intensity measurements over sections of smoothly curved surface a novel technique called photometric stereo is discussed the idea of stereo is to obtain multiple images in order to determine the underlying scene precisely in photometric stereo the viewing direction is constant multiple images are obtained by varying the incident illumination it is shown that this provides sufficient information to determine surface orientation at each image point

saddlepoint approximations for continuous time markov processes this paper proposes saddlepoint expansions as a means to generate closed form approximations to the transition densities and cumulative distribution functions of markov processes this method is applicable to a large class of models considered in finance for which a laplace or characteristic functions but not the transition density can be found in closed form but even when such a computation is not possible explicitly we go one step further by showing how useful approximations can be obtained by replacing the laplace or characteristic functions by an expansion in small time

recovering surface shape and orientation from texture texture provides an important source of information about the three dimensional structure of visible surfaces particularly for stationary monocular views to recover 3d structure the distorting effects of projection must be distinguished from properties of the texture on which the distortion acts this requires that assumptions must be made about the texture yet the unpredictability of natural textures precludes the use of highly restrictive assumptions the recovery method reported in this paper exploits the minimal assumption that textures do not mimic projective effects this assumption determines the strategy of attributing as much as possible of the variation observed in the image to projection equivalently the interpretation is chosen for which the texture prior to projection is made as uniform as possible this strategy was implemented using statistical methods first for the restricted case of planar surfaces and then by extension for curved surfaces the technique was applied successfully to natural images

the effects of random and discrete sampling when estimating continuous time diffusions high frequency financial data are not only discretely sampled in time but the time separating successive observations is often random we analyze the consequences of this dual feature of the data when estimating a continuous time model in particular we measure the additional effects of the randomness of the sampling intervals over and beyond those due to the discreteness of the data we also examine the effect of simply ignoring the sampling randomness we find that in many situations the randomness of the sampling has a larger impact than the discreteness of the data

wu s method and its application to perspective viewing abstract an algebraic method for proving a class of geometry theorems recently proposed by wu is informally discussed key concepts relevant to the method are explained an application of this method to perspective viewing in image understanding is discussed finally it is outlined how the method can be helpful to prove geometry theorems involving inequalities to capture the relation of a point being in between two given points this is also illustrated using an example from image understanding

estimators of diffusions with randomly spaced discrete observations a general theory we provide a general method to analyze the asymptotic properties of a variety of estimators of continuous time diffusion processes when the data are not only discretely sampled in time but the time separating successive observations may possibly be random we introduce a new operator the generalized infinitesimal generator to obtain taylor expansions of the asymptotic moments of the estimators as a special case our results apply to the situation where the data are discretely sampled at a fixed nonrandom time interval we include as specific examples estimators based on maximum likelihood and discrete approximations such as the euler scheme

analysis of wiener functionals malliavin calculus and its applications to heat kernels on donne un expose facilement accessible du calcul de malliavin du calcul differentiel de dimension infinie sur l espace de wiener

asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of malliavin watanabe the asymptotic expansions of the probability distributions of statistics for the small diffusion are derived by means of the malliavin calculus from this the second order efficiency of the maximum likelihood estimator is proved

malliavin calculus and asymptotic expansion for martingales we present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale the emphasis is put on the use of the malliavin calculus the uniform nondegeneracy of the malliavin covariance under certain truncation plays an essential role as the cramer condition did in the case of independent observations applications to statistics are presented

a new delta expansion for multivariate diffusions via the ito taylor expansion abstract in this paper we develop a new delta expansion approach to deriving analytical approximation to the transition densities of multivariate diffusions using the ito taylor expansion of the conditional expectation of the dirac delta function our approach yields an explicit recursive formulas for the expansion coefficients and is universally applicable for a wide spectrum of models particularly the time inhomogeneous non affine irreducible multivariate diffusions we show that this new approach can be viewed as an extension of ait sahalia 2002 and lee et al 2014 to the case of multivariate models the derived expansions are proved to converge to the true probability density as the observational time interval shrinks the obtained approximations can thereby be used to carry out the maximum likelihood estimation for the diffusions with discretely observed data extensive numerical experiments demonstrate the accuracy and effectiveness of our approach

explicit form of approximate transition probability density functions of diffusion processes abstract a continuous time diffusion process is very popular in modeling and provides useful tools to analyze particularly but not restricted to a variety of economic and financial variables the transition probability density function tpdf of a diffusion process plays an important role in understanding and explaining the dynamics of the process a new way to find closed form approximate tpdfs for multivariate diffusions is proposed in this paper this method can be applied to general multivariate time inhomogeneous diffusion processes as long as roughly speaking they have smooth drift and volatility functions a diffusion process is said to be reducible if it can be converted into a unit diffusion process where the volatility is the identity matrix we have established how to obtain the approximate tpdf of a reducible diffusion explicitly when a diffusion process is not reducible an explicit form of approximate tpdf can be obtained by using the results in ait sahalia 2008 and choi 2013 the tpdf expansion suggested here enables us to obtain a recursive formula for the coefficient of the approximate tpdf for a multivariate jump diffusion monte carlo simulation studies of conducting maximum likelihood estimation mle using our approximations provide convincing evidence that our tpdf expansion can be used for the mle when the true tpdf is unavailable we also applied our approximate tpdfs to option pricing the differences between our option prices and those from the extended black scholes formula are shown to be quite small this implies that our methods can be employed to price assets whose underlying state variables follow general diffusion models

parameter estimation and bias correction for diffusion processes this paper considers parameter estimation for continuous time diffusion processes which are commonly used to model dynamics of financial securities including interest rates to understand why the drift parameters are more difficult to estimate than the diffusion parameter as observed in previous studies we first develop expansions for the bias and variance of parameter estimators for two of the most employed interest rate processes vasicek and cir processes then we study the first order approximate maximum likelihood estimator for linear drift processes a parametric bootstrap procedure is proposed to correct bias for general diffusion processes with a theoretical justification simulation studies confirm the theoretical findings and show that the bootstrap proposal can effectively reduce both the bias and the mean square error of parameter estimates for both univariate and multivariate processes the advantages of using more accurate parameter estimators when calculating various option prices in finance are demonstrated by an empirical study

asymptotic expansions and bootstrapping distributions for dependent variables a martingale approach the paper develops a one step triangular array asymptotic expansion for continuous martingales which are asymptotically normal mixing conditions are not required but the quadratic variations of the martingales must satisfy a law of large numbers and a central limit type condition from this result we derive expansions for the distributions of estimators in asymptotically ergodic differential equation models and also for the bootstrapping estimators of these distributions

computation of surface orientation and structure of objects using grid coding in this correspondence algorithms are introduced to infer surface orientation and structure of visible object surfaces using grid coding we adopt the active lighting technique to spatially encode the scene for analysis the observed objects which can have surfaces of arbitrary shape are assumed to rest on a plane base plane in a scene which is encoded with light cast through a grid plane two orthogonal grid patterns are used where each pattern is obtained with a set of equally spaced stripes marked on a glass pane the scene is observed through a camera and the object surface orientation is determined using the projected patterns on the object surface if the surfaces under consideration obey certain smoothness constraints a dense orientation map can be obtained through proper interpolation the surface structure can then be recovered given this dense orientation map both planar and curved surfaces can be handled in a uniform manner the algorithms we propose yield reasonably accurate results and are relatively tolerant to noise especially when compared to shape from shading techniques in contrast to other grid coding techniques reported which match the grid junctions for depth reconstruction under the stereopsis principle our techniques use the direction of the projected stripes to infer local surface orientation and do not require any correspondence relationship between either the grid lines or the grid junctions to be specified the algorithm has the ability to register images and can therefore be embedded in a system which integrates knowledge from multiple views

cylindrical algebraic decomposition i the basic algorithm given a set of r variate integral polynomials a cylindrical algebraic decomposition cad of euclidean r space e r partitions e r into connected subsets compatible with the zeros of the polynomials by compatible with the zeros of the polynomials we mean that on each subset of e r each of the polynomials either vanishes everywhere or nowhere for example consider the bivariate polynomial y 4 2 y 3 y 2 3 x 2 y 2 x 4

on the approximate maximum likelihood estimation for diffusion processes the transition density of a diffusion process does not admit an explicit expression in general which prevents the full maximum likelihood estimation mle based on discretely observed sample paths ait sahalia j finance 54 1999 1361 1395 econometrica 70 2002 223 262 proposed asymptotic expansions to the transition densities of diffusion processes which lead to an approximate maximum likelihood estimation amle for parameters built on ait sahalia s econometrica 70 2002 223 262 ann statist 36 2008 906 937 proposal and analysis on the amle we establish the consistency and convergence rate of the amle which reveal the roles played by the number of terms used in the asymptotic density expansions and the sampling interval between successive observations we find conditions under which the amle has the same asymptotic distribution as that of the full mle a first order approximation to the fisher information matrix is proposed

martingale expansions and second order inference the paper develops a one step triangular array edgeworth expansion for multivariate martingales that are essentially asymptotically ergodic both discrete and continuous time are covered the expansion is in a test function topology we investigate when the expansion has the usual edgeworth form looking in particular at likelihood inference including cox regression and at inference for stationary time series the triangular array nature of the results make them useful for bootstrapping and a result pointing in that direction is shown for cox regression

closed form likelihood approximation and estimation of jump diffusions with an application to the realignment risk of the chinese yuan this paper provides closed form likelihood approximations for multivariate jump diffusion processes widely used in finance for a fixed order of approximation the maximum likelihood estimator mle computed from this approximate likelihood achieves the asymptotic efficiency of the true yet uncomputable mle as the sampling interval shrinks this method is used to uncover the realignment probability of the chinese yuan since february 2002 the market implied realignment intensity has increased fivefold the term structure of the forward realignment rate which completely characterizes future realignment probabilities is hump shaped and peaks at mid 2004 the realignment probability responds quickly to economic news releases and government interventions r 2007 elsevier b v all rights reserved

estimating jump diffusions using closed form likelihood expansions the indispensable role of likelihood expansions in financial econometrics for continuous time models has been established since the ground breaking work of ait sahalia 1999 2002a 2008 jump diffusions play an important role in modeling a variety of economic and financial variables as a significant generalization of li 2013 we propose a new closed form expansion for transition density of poisson driven jump diffusion models and its application in maximum likelihood estimation based on discretely sampled data technically speaking our expansion is obtained by perturbing paths of the underlying model correction terms can be calculated explicitly using any symbolic software numerical examples and monte carlo evidence for illustrating the performance of density expansion and the resulting approximate mle are provided in order to demonstrate the practical applicability of the method using the theoretical results in hayashi and ishikawa 2012 some convergence properties related to the density expansion and the approximate mle method can be justified under some standard sufficient but not necessary conditions

recovering three dimensional shape from a single image of curved objects an algorithm to recover three dimensional shape i e surface orientation and relative depth from a single segmented image is presented it is assumed that the scene is composed of opaque regular solid objects bounded by piecewise smooth surfaces with no markings or textures it is also assumed that the reflectance map r n is known for the canonical case of lambertian surfaces illuminated by a point light source this implies knowing the light source direction a variational formulation of line drawing and shading constraints in a common framework is developed the global constraints are partitioned into constraint sets corresponding to the faces edges and vertices in the scene for a face the constraints are given by horn s image irradiance equation a variational formulation of the constraints at an edge both from the known direction of the image curve corresponding to the edge and shading is developed at a vertex the constraints are modeled by a system of nonlinear equations an algorithm is presented to solve this system of constraints

solutions to the seepage face model for dual porosity flows with hysteresis abstract this paper is concerned with theoretical analysis of the initial boundary value problem for the dual porosity model of darcian flows with the preisach hysteresis operator due to the presence of the so called unilateral boundary condition the problem comes as an evolution variational inequality we apply the penalty method to prove the global in time existence of the variational solution by constructing approximate solutions satisfying suitable energy type estimates

numerical shape from shading and occluding boundaries an iterative method for computing shape from shading using occluding boundary information is proposed some applications of this method are shown we employ the stereographic plane to express the orientations of surface patches rather than the more commonly used gradient space use of the stereographic plane makes it possible to incorporate occluding boundary information but forces us to employ a smoothness constraint different from the one previously proposed the new constraint follows directly from a particular definition of surface smoothness we solve the set of equations arising from the smoothness constraints and the image irradiance equation iteratively using occluding boundary information to supply boundary conditions good initial values are found at certain points to help reduce the number of iterations required to reach a reasonable solution numerical experiments show that the method is effective and robust finally we analyze scanning electron microscope sem pictures using this method other applications are also proposed

empirically derived estimates of the complexity of labeling line drawings of polyhedral scenes abstract several results have been obtained in the past about the complexity of understanding line drawings of polyhedral scenes kirousis and papadimitriou 1988 have shown that the problem of labeling line drawings of trihedral scenes is np complete the human brain however seems to grasp at a glance the 3d structure associated with a line drawing a possible explanation of this discrepancy offered by kirousis and papadimitriou themselves is that the worst case complexity does not reflect the real difficulty of labeling line drawings which might be far less in the average or in typical cases however no statistical analysis has ever been carried out to test this conjecture the core of this paper is an algorithm for the generation of random instances of polyhedral scenes random instances of line drawings are then obtained as perspective projections of these scenes and can be used as an input to standard labeling algorithms so as to derive experimental estimates of the complexity of these algorithms the results indicate that the median case complexity is linear in the number of junctions this substantiates the conjecture that typical instances of line drawings are easy to label and may help explain the ease by which the brain is able to solve the problem

an adaptive time discretization of the classical and the dual porosity model of richards equation this paper presents a numerical solution to the equations describing darcian flow in a variably saturated porous medium a classical richards equation model richards 1931 1 and an extension of it that approximates the flow in media with preferential paths a dual porosity model gerke and van genuchten 1993 8 a numerical solver to this problem the drutes computer program was developed and released during our investigation a new technique which maintains an adaptive time step defined here as the retention curve zone approach was constructed and tested the aim was to limit the error of a linear approximation to the time derivative part finally parameter identification was performed in order to compare the behavior of the dual porosity model with data obtained from a non homogenized fracture and matrix flow simulation experiment

the complexity of understanding line drawings of origami scenes this paper deals with the interpretation of line drawings of origami scenes kanade 1980 that is scenes obtained by assembling planar panels of negligible thickness and it addresses the computational complexity of the problem of consistently assigning suitable labels to the segments describing 3d properties as convexity concavity and occlusion labeling problem the main results of the paper are the following a the labeling problem for line drawings of origami scenes is np as for the case of trihedral scenes b the problem remains np even if the location of the vanishing points in the image plane is given whereas for trihedral scenes the problem was polynomially solvable c in case the vanishing points are known the labeling problem can be subdivided into two subproblems the paneling problem and the labeling a paneled line drawing problem which are both polynomially solvable there may be however exponentially many legal panelings to check against labelability

interpreting line drawings of curved objects in this thesis we study the problem of interpreting line drawings of scenes composed or opaque regular solid objects bounded by piecewise smooth surfaces with no markings or texture on them it is assumed that the line drawing has been formed by orthographic projection of such a scene under general viewpoint that the line drawing is error free and that there are no lines due to shadows or specularities our definition implicitly excludes laminae wires and the apices of cones a major component of the interpretation of line drawings is line labelling by line labelling we mean a classification of each image curve as corresponding to either a depth or orientation discontinuity in the scene and b further subclassification of each kind of discontinuity for a depth discontinuity we determine whether it is a limb a locus of points on the surface where the line of sight is tangent to the surface or an occluding edge a tangent plane discontinuity of the surface for an orientation discontinuity we determine whether it corresponds to a convex or concave edge this thesis presents the first mathematically rigorous scheme for labelling line drawings of the class of scenes described by analysing the projection of the neighborhoods of different kinds of points on a piecewise smooth surface we are able to catalog all local labelling possibilities for the different types of junctions in a line drawing an algorithm is developed which utilizes this catalog to determine all legal labellings of the line drawing a local minimum complexity rule at each vertex select those labellings which correspond to the minimum number of faces meeting at the vertex is used in order to prune highly counter intuitive interpretations the labelling scheme was implemented and tested on a number of line drawings the labellings obtained are few and by and large in accordance with human interpretations

geometric reasoning for constructing 3d scene descriptions from images abstract there are many applications for a vision system which derives a three dimensional model of a scene from one or more images and stores the model for easy retrieval and matching the derivation of a 3d model of a scene involves transformations between four levels of representation images 2d features 3d structures and 3d geometric models geometric reasoning is used to perform these transformations as well as for the eventual model matching since the image formation process is many to one the problem of deriving 3d features from 2d features is ill constrained additional constraints may be derived from knowledge of the domain from which the images were taken the 3d mosaic system has successfully used domain specific knowledge to drive the geometric reasoning necessary to acquire 3d models for complex real world urban scenes to generalize this approach a framework for the representation and use of domain knowledge for geometric reasoning for vision is proposed

a multi level geometric reasoning system for vision abstract geometry is known to play an important role in image understanding and machine vision geometric reasoning is employed in different forms in model formation and model matching problems in model based vision research geometric representations and reasoning methods are often used implicitly in the design of vision systems and algorithms used there an approach towards model based vision in which geometric and algebraic reasoning is explicit is discussed a multi level reasoning system for machine vision based on this approach is being designed and developed at general electric corporate research and development three key components of this system a hierarchical organization of geometric knowledge for its systematic and efficient use as well as to control the search space labeling algorithms and algebraic reasoning algorithms are presented the use of such a system for deriving a three dimensional model from two dimensional images as well as for using a two dimensional image for matching against another two dimensional image called view consistency problems is discussed

recovery of the three dimensional shape of an object from a single view given a single picture which is a projection of a three dimensional scene onto the two dimensional picture plane we usually have definite ideas about the 3 d shapes of objects to do this we need to use assumptions about the world and the image formation process since there exist a large number of shapes which can produce the same picture the purpose of this paper is to identify some of these assumptions mostly geometrical ones by demonstrating how the theory and techniques which exploit such assumptions can provide a systematic shape recovery method the method consists of two parts the first is the application of the origami theory which models the world as a collection of plane surfaces and recovers the possible shapes qualitatively the second is the technique of mapping image regularities into shape constraints for recovering the probable shapes quantitatively actual shape recovery from a single view is demonstrated for the scenes of an object such as a box and a chair given a single image the method recovers the 3 d shapes of an object in it and generates images of the same object as we would see it from other directions

an algebraic approach to shape from image problems abstract this paper presents a new method for recovering three dimensional shapes of polyhedral objects from their single view images the problem of recovery is formulated in a constrained optimization problem in which the constraints reflect the assumption that the scene is composed of polyhedral objects and the objective function to be minimized is a weighted sum of quadric errors of surface information such as shading and texture for practical purpose it is decomposed into the two more tractable problems a linear programming problem and an unconstrained optimization problem in the present method the global constraints placed by the polyhedron assumption are represented in terms of linear algebra whereas similar constraints have usually been represented in terms of a gradient space moreover superstrictness of the constraints can be circumvented by a new concept position free incidence structure for this reason the present method has several advantages it can recover the polyhedral shape even if image data are incorrect due to vertex position errors it can deal with perspective projection as well as orthographic projection the number of variables in the optimization problem is very small three or a little greater than three and any kinds of surface information can be incorporated in a unifying manner