on the geometry of the term structure of interest rates we present recently developed geometric methods for the analysis of finite dimensional term structure models of the interest rates this includes an extension of the frobenius theorem for fr echet spaces in particular this approach puts new light on many of the classical models such as the hull white extended vasicek and cox ingersoll ross short rate models the notion of a finite dimensional realization fdr is central for this analysis we motivate it classify all generic fdrs and provide some new results for the corresponding factor processes such as hypoellipticity of its generators and the existence of smooth densities furthermore we include finite dimensional external factors thus admitting a stochastic volatility structure

a variation embedding theorem and applications fractional sobolev spaces also known as besov or slobodetski spaces arise in many areas of analysis stochastic analysis in particular we prove an embedding into certain q variation spaces applications include a new route to a regularity result by kusuoka for stochastic differential equations integration against besov paths a regularity criterion for rough paths and a new regularity result for cameron martin paths associated to fractional brownian motion

a nonparametric estimation procedure for bivariate extreme value copulas summary a bivariate extreme value distribution with fixed marginals is generated by a onedimensional map called a dependence function this paper proposes a new nonparametric estimator of this function its asymptotic properties are examined and its small sample behaviour is compared to that of other rank based and likelihood based procedures the new estimator is shown to be uniformly strongly convergent and asymptotically unbiased through simulations it is also seen to perform reasonably well against the maximum likelihood estimator based on the correct model and to have smaller l1 l2 and l errors than any existing nonparametric alternative the n consistency of the proposed estimator leads to nonparametric estimation of tawn s 1988 dependence measure that may be used to test independence in small samples

non degeneracy of wiener functionals arising from rough differential equations malliavin calculus is about sobolev type regularity of functionals on wiener space the main example being the ito map obtained by solving stochastic differential equations rough path analysis is about strong regularity of the solution to possibly stochastic differential equations we combine arguments of both theories and discuss the existence of a density for solutions to stochastic differential equations driven by a general class of non degenerate gaussian processes including processes with sample path regularity worse than brownian motion

a tree approach to p variation and to integration we consider a real valued path it is possible to associate a tree to this path and we explore the relations between the tree the properties of p variation of the path and integration with respect to the path in particular the fractal dimension of the tree is estimated from the variations of the path and young integrals with respect to the path as well as integrals from the rough paths theory are written as integrals on the tree examples include some stochastic paths such as martingales levy processes and fractional brownian motions for which an estimator of the hurst parameter is given

regularity of laws and ergodicity of hypoelliptic sdes driven by rough paths we consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior in particular we focus on the case when the driving noise is a rough path valued fractional brownian motion with hurst parameter h 2 1 1 our contribution in this work is twofold first when the driving vector fields satisfy hormander s c elebrated lie bracket condition we derive explicit quantitative bounds on the i nverse of the malliavin ma trix en route to this we provide a novel deterministic version of norris s lemma for differential equations driven by rough paths this result with the added assumption that the linearized equation has moments will then yield that the transition laws have a smooth density with respect to lebesgue measure our second main result states that under hormander s condi tion the solutions to rough differential equations driven by fractional brownian motion with h 2 1 1 enjoy a suitable version of the strong feller property under a standard controllability condition this implies that they admit a unique stationary solution that is physical in the sense that it does not look into the future

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

modelling extreme multivariate events summary the classical treatment of multivariate extreme values is through componentwise ordering though in practice most interest is in actual extreme events here the point process of observations which are extreme in at least one component is considered parametric models for the dependence between components must satisfy certain constraints two new techniques for generating such models are presented aspects of the statistical estimation of the resulting models are discussed and are illustrated with an application to oceanographic data

modelling dependence within joint tail regions standard approaches for modelling dependence within joint tail regions are based on extreme value methods which assume max stability a particular form of joint tail dependence we develop joint tail models based on a broader class of dependence structure which provides a natural link between max stable models and weaker forms of dependence including independence and negative association this approach overcomes many of the problems that are encountered with standard methods and is the basis for a poisson process representation that generalizes existing bivariate results we apply the new techniques to simulated and environmental data and demonstrate the marked advantage that the new approach offers for joint tail extrapolation

constrained rough paths we introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural on the way we develop a notion of rough integration and an ecient and intrinsic theory of rough dierential equations rdes on manifolds the theory of rdes is then used to construct parallel translation along manifold valued rough paths finally this framework is used to show there is a one to one correspondence between rough paths on a d dimensional manifold and rough paths on d dimensional euclidean space this last result is a rough path analogue of cartan s development map and its stochastic version which was developed by eells and elworthy and malliavin

a note on polynomial solvability of the cdt problem recent research has focused on the complexity of extensions of the classical trust region subproblem which addresses the minimization of a quadratic function over a unit ball in mathbb r n

smoothness of densities for area like processes of fractional brownian motion we consider a process given by a n dimensional fractional brownian motion with hurst parameter frac 1 4 h frac 1 2 along with an associated levy area like process and prove the smoothness of the density for this process with respect to lebesgue measure

solving the trust region subproblem by a generalized eigenvalue problem the state of the art algorithms for solving the trust region subproblem trs are based on an iterative process involving solutions of many linear systems eigenvalue problems subspace optimization or line search steps a relatively underappreciated fact due to gander golub and von matt linear algebra appl 114 1989 pp 815 839 is that trss can be solved by one generalized eigenvalue problem with no outer iterations in this paper we rediscover this fact and discover its great practicality which exhibits good performance both in accuracy and efficiency moreover we generalize the approach in various directions namely by allowing for an ellipsoidal constraint dealing with the so called hard case and obtaining approximate solutions efficiently when high accuracy is unnecessary we demonstrate that the resulting algorithm is a general purpose trs solver effective both for dense and large sparse problems including the so called hard case our algorithm is easy to implement its essence i

an m estimator for tail dependence in arbitrary dimensions consider a random sample in the max domain of attraction of a multivariate extreme value distribution such that the dependence structure of the attractor belongs to a parametric model a new estimator for the unknown parameter is defined as the value that minimizes the distance between a vector of weighted integrals of the tail dependence function and their empirical counterparts the minimization problem has with probability tending to one a unique global solution the estimator is consistent and asymptotically normal the spectral measures of the tail dependence models to which the method applies can be discrete or continuous examples demonstrate the applicability and the performance of the method

the trust region subproblem with non intersecting linear constraints this paper studies an extended trust region subproblem etrs in which the trust region intersects the unit ball with m m linear inequality constraints when m 0 m 1 m 0 m 1 or m 2 m 2 and the linear constraints are parallel it is known that the etrs optimal value equals the optimal value of a particular convex relaxation which is solvable in polynomial time however it is also known that when m ge 2 m 2 and at least two of the linear constraints intersect within the ball i e some feasible point of the etrs satisfies both linear constraints at equality then the same convex relaxation may admit a gap with etrs this paper shows that the convex relaxation has no gap for arbitrary m m as long as the linear constraints are non intersecting

euler estimates for rough differential equations abstract we consider controlled ordinary differential equations and give new estimates for higher order euler schemes our proofs are inspired by recent work of a m davie who considers first and second order schemes in order to implement the general case we make systematic use of geodesic approximations in the free nilpotent group such euler estimates have powerful applications by a simple limit argument they apply to rough path differential equations rdes in the sense of t lyons and hence also to stochastic differential equations driven by brownian motion or other random rough paths with sufficient integrability in the context of the latter we obtain strong remainder estimates in stochastic taylor expansions a la azencott ben arous castell and platen although our findings appear novel even in the case of driving brownian motion our main insight is the genuine rough path nature of quantitative remainder estimates in stochastic taylor expansions there are several other applications of which we discuss in detail l q convergence in lyons universal limit theorem and moment control of rde solutions

strong duality in nonconvex quadratic optimization with two quadratic constraints we consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints when the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions we then develop a connection between the image of the real and complex spaces under a quadratic mapping which together with the results in the complex case lead to a condition that ensures strong duality in the real setting preliminary numerical simulations suggest that for random instances of the extended trust region subproblem the sufficient condition is satisfied with a high probability furthermore we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems finally we discuss an application of our results to robust least squares problems

markov chain models for threshold exceedances in recent research on extreme value statistics there has been an extensive development of threshold methods first in the univariate case and subsequently in the multivariate case as well in this paper an alternative methodology for extreme values of univariate time series is developed by assuming that the time series is markovian and using bivariate extreme value theory to suggest appropriate models for the transition distributions a new likelihood representation for threshold methods is presented which we apply to a markovian time series an important motivation for developing this kind of theory is the possibility of calculating probability distributions for functionals of extreme events we address this issue by showing how a theory of compound poisson limits for additive functionals can be combined with simulation to obtain numerical solutions for problems of practical interest the methods are illustrated by application to temperature data

a mixture model for multivariate extremes this thesis is a contribution to multivariate extreme value statistics the tail of a multivariate distribution function is characterized by its spectral distribution for which we propose a new semi parametric model based on mixtures of dirichlet distributions to estimate the components of this model reversible jump monte carlo markov chain and em algorithms are developed their performances are illustrated on real and simulated data obtained using new representations of the extremal logistic and dirichlet models in parallel with the estimation of the spectral distribution extreme value statistic machinery requires the selection of a threshold in order to classify data as extreme or not this selection is achieved by a new method based on heuristic arguments it allows a selection independent of the dimension of the data its performance is illustrated on real and simulated data primal scientific interests behind a multivariate extreme value analysis reside in the estimation of quantiles of rare events and in the exploration of the dependence structure for which the estimation of the spectral measure is a means rather than an end these two issues are addressed for the first a monte carlo method is developed based on simulation of extremes it is compared with classical and new methods of the literature for the second one an original conditional dependence analysis is proposed which enlightens various aspects of the dependence structure of the data examples using real data sets are given in the last part the semi parametric model and the presented methods are extended to spatial extremes it is made possible by considering the spectral distribution as the distribution of a random probability an original viewpoint adopted throughout this thesis classical multivariate extremes are extended to extremes of random measures the application is illustrated on rainfall data in china

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

on the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues we derive some basic results on the geometry of semidefinite programming sdp and eigenvalue optimization i e the minimization of the sum of the k largest eigenvalues of a smooth matrix valued function we provide upper bounds on the rank of extreme matrices in sdps and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue optimization in the spectrum of an optimal matrix the kth and k 1 st largest eigenvalues tend to be equal and frequently have multiplicity greater than two this clustering is intuitively plausible and has been observed as early as 1975 when the matrix valued function is affine we prove that clustering must occur at extreme points of the set of optimal solutions if the number of variables is sufficiently large we also give a lower bound on the multiplicity of the critical eigenvalue these results generalize to the case of a general matrix valued function under appropriate conditions

reconstruction for the signature of a rough path recently it was proved that the group of rough paths modulo tree like equivalence is isomorphic to the corresponding signature group through the signature map s a generalized notion of taking iterated path integrals however the proof of this uniqueness result does not contain any information on how to see the trajectory of a tree reduced rough path from its signature and a constructive understanding on the uniqueness result in particular on the inverse of s has become an interesting and important question the aim of this paper is to reconstruct a rough path from its signature in an explicit and universal way

estimating the probability of a rare event let x 1 y 1 x 2 y 2 x n y n be a random sample from a bivariate distribution function f which is in the domain of attraction of a bivariate extreme value distribution function g a subset c of r 2 is given which contains none of the observations we shall give an asymptotic confidence interval for pr x i y i c under certain conditions

strong duality for the cdt subproblem a necessary and sufficient condition in this paper we consider the problem of minimizing a nonconvex quadratic function subject to two quadratic inequality constraints as an application such a quadratic program plays an important role in the trust region method for nonlinear optimization such a problem is known as the celis dennis and tapia cdt subproblem in the literature the lagrangian dual of the cdt subproblem is a semidefinite program sdp hence convex and solvable however a positive duality gap may exist between the cdt subproblem and its lagrangian dual because the cdt subproblem itself is nonconvex in this paper we present a necessary and sufficient condition to characterize when the cdt subproblem and its lagrangian dual admits no duality gap i e the strong duality holds this necessary and sufficient condition is easy verifiable and involves only one any optimal solution of the sdp relaxation for the cdt subproblem moreover the condition reveals that it is actually rare to render a positive duality gap for the cdt subproblems in general moreover if the strong duality holds then an optimal solution for the cdt problem can be retrieved from an optimal solution of the sdp relaxation by means of a matrix rank one decomposition procedure the same analysis is extended to the framework where the necessary and sufficient condition is presented in terms of the lagrangian multipliers at a kkt point furthermore we show that the condition is numerically easy to work with approximatively

smoothness of the density for solutions to gaussian rough differential equations we consider stochastic differential equations driven by a multi dimensional gaussian process under the assumption that the vector fields satisfy hormander s bracket condition we demonstrate that the solution admits a smooth density for any strictly positive time t provided the driving noise satisfies certain non degeneracy assumptions our analysis relies on an interplay of rough path theory malliavin calculus and the theory of gaussian processes our result applies to a broad range of examples including fractional brownian motion with hurst parameter greater than 1 4 the ornstein uhlenbeck process and the brownian bridge returning after time t

nonparametric tests for constant tail dependence with an application to energy and finance the present paper proposes new tests for detecting structural breaks in the tail dependence of multivariate time series using the concept of tail copulas to obtain asymptotic properties we derive a new limit result for the sequential empirical tail copula process moreover consistency of both the tests and a change point estimator are proven we analyze the finite sample behavior of the tests by monte carlo simulations finally and crucial from a risk management perspective we apply the new findings to datasets from energy and financial markets

a construction principle for multivariate extreme value distributions we present a construction principle for the spectral density of a multivariate extreme value distribution it generalizes the pairwise beta model introduced in the literature recently and may be used to obtain new parametric models from lower dimensional spectral densities we illustrate the flexibility of this new class of models and apply it to a wind speed dataset

bivariate extreme value theory models and estimation summary bivariate extreme value distributions arise as the limiting distributions of renormalized componentwise maxima no natural parametric family exists for the dependence between the marginal distributions but there are considerable restrictions on the dependence structure we consider modelling the dependence function with parametric models for which two new models are presented tests for independence and discriminating between models are also given the estimation procedure and the flexibility of the new models are illustrated with an application to sea level data extreme value theory has recently been an area of much theoretical and practical work univariate theory is a well documented area whereas bivariate multivariate extreme value theory has until recently received surprisingly little attention in the multivariate case no natural parametric family exists for the dependence structure so this must be modelled in some way in the analysis of environmental extreme value data there is a need for models of dependence between extremes from different sources for example at various sea ports or at various points of a river in this paper we consider bivariate extreme value distributions we assume without loss of generality that we have exponential marginal distributions with unit means the class of bivariate exponential distributions in which we are interested satisfy a strong stability relation exponential variables x y satisfy the stability relation if and only if w min ax by is also exponentially distributed for all a b 0 pickands 1981 therefore the models we will consider have particular application in reliability and survival analysis one approach to modelling the dependence structure is via parametric models this requires a flexible family of models which satisfy certain constraints models are of two kinds either differentiable or nondifferentiable all nondifferentiable models give distri butions which are singular with nonzero probability concentrated on a certain subspace the differentiable models have densities but the existing models are symmetric which leads to the variables being exchangeable here we present two new asymmetric differenti able models which have increased flexibility properties of the differentiable models are examined estimation of the parametric models has previously been by ad hoc methods because there is a nonregular estimation problem when the margins are independent for the

narrowing the difficulty gap for the celis dennis tapia problem we study the celis dennis tapia cdt problem minimize a non convex quadratic function over the intersection of two ellipsoids in contrast to the well studied trust region problem where the feasible set is just one ellipsoid the cdt problem is not yet fully understood our main objective in this paper is to narrow the difficulty gap that occurs when the hessian of the lagrangian is indefinite at all karush kuhn tucker points we prove new sufficient and necessary conditions both for local and global optimality based on copositivity giving a complete characterization in the degenerate case

indefinite trust region subproblems and nonsymmetric eigenvalue perturbations this paper extends the theory of trust region subproblems in two ways i it allows indefinite inner products in the quadratic constraint and ii it uses a two sided upper and lower bound quadratic constraint characterizations of optimality are presented that have no gap between necessity and sufficiency conditions for the existence of solutions are given in terms of the definiteness of a matrix pencil a simple dual program is introduced that involves the maximization of a strictly concave function on an interval this dual program simplifies the theory and algorithms for trust region subproblems it also illustrates that the trust region subproblems are implicit convex programming problems and thus explains why they are so tractable the duality theory also provides connections to eigenvalue perturbation theory trust region subproblems with zero linear term in the objective function correspond to eigenvalue problems and adding a linear term in the objective function is seen to correspond to a p

fisher information for a multivariate extreme value distribution summary explicit algebraic formulae for the fisher information matrix of the multivariate extreme value distribution with generalised extreme value margins and logistic dependence structure are given

flow of diffeomorphisms induced by a geometric multiplicative functional in this paper it is shown that the unique multiplicative functional solution to a differential equation driven by a geometric multiplicative functional consitutes a flow of local diffeomorphisms in the case where the driving geometric multiplicative functional is generated by a brownian motion the result in particular presents an answer to an open problem proposed in ikeda and watanabe 4

statistics for near independence in multivariate extreme values we propose a multivariate extreme value threshold model for joint tail estimation which overcomes the problems encountered with existing techniques when the variables are near independence we examine inference under the model and develop tests for independence of extremes of the marginal variables both when the thresholds are fixed and when they increase with the sample size motivated by results obtained from this model we give a new and widely applicable characterisation of dependence in the joint tail which includes existing models as special cases a new parameter which governs the form of dependence is of fundamental importance to this characterisation by estimating this parameter we develop a diagnostic test which assesses the applicability of bivariate extreme value joint tail models the methods are demonstrated through simulation and by analysing two previously published data sets

optimality conditions for the minimization of a quadratic with two quadratic constraints the trust region method has been proven to be very successful in both unconstrained and constrained optimization it requires the global minimum of a general quadratic function subject to ellipsoid constraints in this paper we generalize the trust region subproblem by allowing two general quadratic constraints conditions and properties of its solution are discussed

regional modelling of extreme storms via max stable processes asymptotic models for extremes of random processes often form the basis for estimating the extremal behaviour of environmental phenomena most such phenomena have a spatial dimension and the aim of this paper is to develop a procedure for modelling in continuous space the spatial dependence within extreme events a principal objective in the analysis as with other current research on extremes is to base inference on as much of the available data as possible the modelling procedures are justified on simulated data and subsequently applied to a series of rainfall data

densities for rough differential equations under hormander s condition we consider stochastic differential equations dy v y dx driven by a multidimensional gaussian process x in the rough path sense t lyons rev mat iberoamericana 14 1998 215 310 using malliavin calculus we show that y t admits a density for t 0 t provided i the vector fields v v 1 v d satisfy hormander s condition and ii the gaussian driving signal x satisfies certain conditions examples of driving signals include fractional brownian motion with hurst parameter h 1 4 the brownian bridge returning to zero after time t and the ornstein uhlenbeck process

weighted approximations of tail copula processes with application to testing the bivariate extreme value condition consider n i i d random vectors on r 2 with unknown common distribution function f under a sharpening of the extreme value condition on f we derive a weighted ap proximation of the corresponding tail copula process then we construct a test to check whether the extreme value condition holds by comparing two estimators of the limiting extreme value dis tribution one obtained from the tail copula process and the other obtained by first estimating the spectral measure which is then used as a building block for the limiting extreme value distribu tion we derive the limiting distribution of the test statistic from the aforementioned weighted approximation this limiting distribution contains unknown functional parameters therefore we show that a version with estimated parameters converges weakly to the true limiting distribution based on this result the finite sample properties of our testing procedure are investigated through a simulation study a real data application is also presented

stochastic analysis rough path analysis and fractional brownian motions in this paper we show by using dyadic approximations the existence of a geometric rough path associated with a fractional brownian motion with hurst parameter greater than 1 4 using the integral representation of fractional brownian motions we furthermore obtain a skohorod integral representation of the geometric rough path we constructed by the results in ly1 a stochastic integration theory may be established for fractional brownian motions and strong solutions and a wong zakai type limit theorem for stochastic differential equations driven by fractional brownian motions can be deduced accordingly the method can actually be applied to a larger class of gaussian processes with covariance functions satisfying a simple decay condition

an m estimator of spatial tail dependence tail dependence models for distributions attracted to a max stable law are tted using observations above a high threshold to cope with spatial high dimensional data a rank based m estimator is proposed relying on bivariate margins only a data driven weight matrix is used to minimize the asymptotic variance empirical process arguments show that the estimator is consistent and asymptotically normal its nite sample performance is assessed in simulation experiments involving popular max stable processes perturbed with additive noise an analysis of wind speed data from the netherlands illustrates the method

trust region problems with linear inequality constraints exact sdp relaxation global optimality and robust optimization the trust region problem which minimizes a nonconvex quadratic function over a ball is a key subproblem in trust region methods for solving nonlinear optimization problems it enjoys many attractive properties such as an exact semi definite linear programming relaxation sdp relaxation and strong duality unfortunately such properties do not in general hold for an extended trust region problem having extra linear constraints this paper shows that two useful and powerful features of the classical trust region problem continue to hold for an extended trust region problem with linear inequality constraints under a new dimension condition first we establish that the class of extended trust region problems has an exact sdp relaxation which holds without the slater constraint qualification this is achieved by proving that a system of quadratic and affine functions involved in the model satisfies a range convexity whenever the dimension condition is fulfilled second we show that the dimension condition together with the slater condition ensures that a set of combined first and second order lagrange multiplier conditions is necessary and sufficient for global optimality of the extended trust region problem and consequently for strong duality through simple examples we also provide an insightful account of our development from sdp relaxation to strong duality finally we show that the dimension condition is easily satisfied for the extended trust region model that arises from the reformulation of a robust least squares problem lsp as well as a robust second order cone programming model problem socp as an equivalent semi definite linear programming problem this leads us to conclude that under mild assumptions solving a robust lsp or socp under matrix norm uncertainty or polyhedral uncertainty is equivalent to solving a semi definite linear programming problem and so their solutions can be validated in polynomial time

quadratic programs with hollows let mathcal f be a quadratically constrained possibly nonconvex bounded set and let mathcal e 1 ldots mathcal e l denote ellipsoids contained in mathcal f with non intersecting interiors we prove that minimizing an arbitrary quadratic q cdot over mathcal g mathcal f setminus cup k 1 ell mathrm int mathcal e k is no more difficult than minimizing q cdot over mathcal f in the following sense if a given semidefinite programming sdp relaxation for min q x x in mathcal f is tight then the addition of l linear constraints derived from mathcal e 1 ldots mathcal e l yields a tight sdp relaxation for min q x x in mathcal g we also prove that the convex hull of x xx t x in mathcal g equals the intersection of the convex hull of x xx t x in mathcal f with the same l linear constraints inspired by these results we resolve a related question in a seemingly unrelated area mixed integer nonconvex quadratic programming

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

a semidefinite framework for trust region subproblems with applications to large scale minimization primal dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem trs this latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and therefore it is a generalization of the minimum eigenvalue problem the importance of trs is due to the fact that it provides the step in trust region minimization algorithms the semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for trs in particular a dual simplex type method is studied that solves trs as a parametric eigenvalue problem this method uses the lanczos algorithm for the smallest eigenvalue as a black box therefore the essential cost of the algorithm is the matrix vector multiplication and thus sparsity can be exploited a primal simplex type method provides steps for the so called hard case extensive numerical tests for large sparse problems are discussed these tests show that the cost of the algorithm is 1 n times the cost of finding a minimum eigenvalue using the lanczos algorithm where 0 n 1 is a fraction which decreases as the dimension increases

second order cone constraints for extended trust region subproblems the classical trust region subproblem trs minimizes a nonconvex quadratic objective over the unit ball in this paper we consider extensions of trs having extra constraints when two parallel cuts are added to trs we show that the resulting nonconvex problem has an exact representation as a semidenite program with additional linear and second order cone constraints for the case where an additional ellipsoidal constraint is added to trs resulting in the two trust region subproblem ttrs we provide a new relaxation including second order cone constraints that strengthens the usual sdp relaxation

rough differential equations driven by signals in besov spaces rough differential equations are solved for signals in general besov spaces unifying in particular the known results in holder and p variation topology to this end the paracontrolled distribution approach which has been introduced by gubinelli imkeller and perkowski 24 to analyze singular stochastic pdes is extended from holder to besov spaces as an application we solve stochastic differential equations driven by random functions in besov spaces and gaussian processes in a pathwise sense

local minimizers of quadratic functions on euclidean balls and spheres in this paper a characterization of the local nonglobal minimizes of a quadratic function defined on a euclidean ball or a sphere is given it is proven that there exists at most one local nonglobal minimizes and that the lagrange multiplier that corresponds to this minimizes is the largest solution of a nonlinear scalar equation an algorithm is proposed for computing the local nonglobal minimizes

ergodicity of stochastic differential equations driven by fractional brownian motion we study the ergodic properties of finite dimensional syste ms of sdes driven by non degenerate additive fractional brownian motion with arbitrary hurst parameter h 2 0 1 a general framework is constructed to make precise the notions of invariant measure and stationary state for such a system we then prove under rather weak dissipativity conditions that such an sde possesses a unique stationary solution and that the convergence rate of an arbitrary solut ion towards the stationary one is at least algebraic a lower bound on the exponent is also given

evolving communities with individual preferences the goal of this paper is to provide mathematically rigorous tools for modelling the evolution of a community of interacting individuals we model the population by a measure space f where determines the abundance of individual preferences the preferences of an individual are described by a measurable choice x of a rough path we aim to identify for each individual a choice for the forward evolution yt for an individual in the community these choices yt m ust be consistent so thatyt correctly accounts for the individual s preference and correctly models their interaction with the aggregate behaviour of the community in general solutions are continuum of interacting threads analogous to the huge number of individual atomic trajectories that together make up the motion of a fluid the evolution of the population need not be governed by any over arching partial differential equation pde although one can match the standard non linear parabolic pdes of mckean vlasov type with specific examples of communities in this case the bulk behaviour of the evolving population provides a solution to the pde we focus on the case of weakly interacting systems where we are able to exhibit the existence and uniqueness of consistent solutions an important technical result is continuity of the behaviour of the system with respect to changes in the measure assigning weight to individuals replacing the deterministic with the empirical distribution of an independent and identically distributed sample from leads to many standard models and applying the continuity result allows easy proofs for propagation of chaos the rigorous underpinning presented here leads to uncomplicated models which have wide applicability in both the physical and social sciences we make no presumption that the macroscopic dynamics are modelled by a pde this work builds on the fine probability literature considering the limit behaviour for systems where a large number of particles are interacting with independent preferences there is also work on continuum models with preferences described by a semi martingale measure we mention some of the key papers

solving generalized cdt problems via two parameter eigenvalues we consider solving a nonconvex quadratic minimization problem with two quadratic constraints one of which being convex this problem is a generalization of the celis denis tapia cdt problem and thus we refer to it as gcdt generalized cdt the cdt problem has been widely studied but no polynomial time algorithm was known until bienstock s recent work his algorithm solves the cdt problem in polynomial time with respect to the number of bits in data and log epsilon 1 by admitting an epsilon error in the constraints the algorithm however appears to be difficult to implement in this paper we present another algorithm for gcdt which is guaranteed to find a global solution for almost all gcdt instances and slightly perturbed ones in some exceptionally rare cases in exact arithmetic including eigenvalue computation our algorithm is based on the approach proposed by iwata nakatsukasa and takeda 2015 for computing the signed distance between overlapping ellipsoids our algorithm computes all the lagrange multipliers of gcdt by solving a two parameter linear eigenvalue problem obtains the corresponding kkt points and finds a global solution as the kkt point with the smallest objective value in practice in finite precision arithmetic our algorithm requires o n 6 log log u 1 computational time where n is the number of variables and u is the unit roundoff although we derive our algorithm under the unrealistic assumption that exact eigenvalues can be computed numerical experiments illustrate that our algorithm performs well in finite precision arithmetic

tail estimates for markovian rough paths we work in the context of markovian rough paths associated to a class of uniformly subelliptic dirichlet forms 26 and prove a better than exponential tail estimate for the accumulated local p variation functional which has been introduced and studied in 17 we comment on the significance of these estimates to a range of currently studied problems including the recent results of ni hao 32 and chevyrev and lyons 18

estimation of quantiles of the maximum of n observations summary given independent and identically distributed random variables from an unknown distribution f we want to estimate quantiles of fn using asymptotic extreme value theory two parametric approximations to fn are considered ahd approximate maximum likelihood estimates are obtained conditions for consistency and asymptotic normality of these estimates are given and properties of the estimates are obtained through a small simulation study

how to convexify the intersection of a second order cone and a nonconvex quadratic a recent series of papers has examined the extension of disjunctive programming techniques to mixed integer second order cone programming for example it has been shown by several authors using different techniques that the convex hull of the intersection of an ellipsoid mathcal e e and a split disjunction l x j x j u le 0 l xj xj u 0 with l u l

ozone concentrations a robust analysis of multivariate extremes data on ozone concentrations at four monitoring sites in central canada are examined univariate extreme value methods do not allow for the required inference and multivariate methods exploiting the joint dependence of the data are necessary families of multivariate extreme value copulas with a flexible dependence structure and closed form cumulative distribution functions have recently been derived this article shows that maximum likelihood estimates of these parametric models may be obtained numerically and that their use can lead to increased efficiency in the estimation of margins we do not however have any assurance that all data are well modeled by these distributions because the space of multivariate extreme value copulas is infinite dimensional and data on ozone levels are prone to outliers this article addresses the robust methods required for proper analysis

exact semidefinite formulations for a class of random and non random nonconvex quadratic programs we study a class of quadratically constrained quadratic programs qcqps called diagonal qcqps which contain no off diagonal terms x j x k for j ne k and we provide a sufficient condition on the problem data guaranteeing that the basic shor semidefinite relaxation is exact our condition complements and refines those already present in the literature and can be checked in polynomial time we then extend our analysis from diagonal qcqps to general qcqps i e ones with no particular structure by reformulating a general qcqp into diagonal form we establish new polynomial time checkable sufficient conditions for the semidefinite relaxations of general qcqps to be exact finally these ideas are extended to show that a class of random general qcqps has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables to the best of our knowledge this is the first result establishing the exactness of the semidefinite relaxation for random general qcqps

a version of hormander s theorem for the fractional brownian motion it is shown that the law of an sde driven by fractional brownian motion with hurst parameter greater than 1 2 has a smooth density with respect to lebesgue measure provided that the driving vector fields satisfy hormander s condition the main new ingredient of the proof is an extension of norris lemma to this situation

solving the trust region subproblem using the lanczos method the approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods when the number of variables is large the most widely used strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trust region boundary in this paper we investigate ways of continuing the process once the boundary has been encountered the key is to observe that the trust region problem within the currently generated krylov subspace has a very special structure which enables it to be solved very efficiently we compare the new strategy with existing methods the resulting software package is available as hsl vf05 within the harwell subroutine library

bivariate threshold methods for extremes summary an extension of the threshold method for extreme values is developed to consider the joint distribution of extremes of two variables the methodology is based on the point process representation of bivariate extremes both parametric and nonparametric models are considered the simplest case to handle is that in which both marginal distributions are known for the more realistic case in which the marginal distributions are unknown a mixed parametric nonparametric method is proposed the techniques are illustrated with data on sulphate and nitrate levels taken from a major study of acid rain

enhanced proximal dc algorithms with extrapolation for a class of structured nonsmooth dc minimization in this paper we consider a class of structured nonsmooth difference of convex dc minimization in which the first convex component is the sum of a smooth and nonsmooth functions while the second convex component is the supremum of possibly infinitely many convex smooth functions we first propose an inexact enhanced dc algorithm for solving this problem in which the second convex component is the supremum of finitely many convex smooth functions and show that every accumulation point of the generated sequence is an alpha eta d stationary point of the problem which is generally stronger than an ordinary d stationary point in addition inspired by the recent work pang et al in math oper res 42 1 95 118 2017 wen et al in comput optim appl 69 2 297 324 2018 we propose two proximal dc algorithms with extrapolation for solving this problem we show that every accumulation point of the solution sequence generated by them is an alpha eta d stationary point of the problem and establish the convergence of the entire sequence under some suitable assumption we also introduce a concept of approximate alpha eta d stationary point and derive iteration complexity of the proposed algorithms for finding an approximate alpha eta d stationary point in contrast with the dc algorithm pang et al 2017 our proximal dc algorithms have much simpler subproblems and also incorporate the extrapolation for possible acceleration moreover one of our proximal dc algorithms is potentially applicable to the dc problem in which the second convex component is the supremum of infinitely many convex smooth functions in addition our algorithms have stronger convergence results than the proximal dc algorithm in wen et al 2018

a trust region algorithm for equality constrained optimization a trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function under certain conditions global convergence and local superlinear convergence results are proved

a second order cone based approach for solving the trust region subproblem and its variants we study the trust region subproblem trs of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints despite having a nonconvex objective it is known that the classical trs and a number of its variants are polynomial time solvable in this paper we follow a second order cone soc based approach to derive an exact convex reformulation of the trs under a structural condition on the conic constraint our structural condition is immediately satisfied when there are no additional conic constraints and it generalizes several such conditions studied in the literature as a result our study highlights an explicit connection between the classical nonconvex trs and smooth convex quadratic minimization which allows for the application of cheap iterative methods such as nesterov s accelerated gradient descent to the trs furthermore under slightly stronger conditions we give a low complexity characterization of the convex hull of the epigraph of the nonconvex quadrati

new results on quadratic minimization in this paper we present several new results on minimizing an indefinite quadratic function under quadratic linear constraints the emphasis is placed on the case in which the constraints are two quadratic inequalities this formulation is termed the extended trust region subproblem in this paper to distinguish it from the ordinary trust region subproblem in which the constraint is a single ellipsoid the computational complexity of the extended trust region subproblem in general is still unknown in this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization for the extended trust region subproblem itself we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution combining this with a result obtained in the first part of the paper we propose a polynomial time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint

when uniform weak convergence fails empirical processes for dependence functions via epi and hypographs for copulas whose partial derivatives are not continuous everywhere on the interior of the unit cube the empirical copula process does not converge weakly with respect to the supremum distance this makes it hard to verify asymptotic properties of inference procedures for such copulas to resolve the issue a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed convergence with respect to the new metric is related to epi and hypoconvergence and is weaker than uniform convergence still for continuous limits it is equivalent to locally uniform convergence whereas under mild side conditions it implies lp convergence even in cases where uniform convergence fails weak convergence with respect to the new metric is established for empirical copula and tail dependence processes no additional assumptions are needed for tail dependence functions and for copulas the assumptions reduce to existence and continuity of the partial derivatives almost everywhere on the unit cube the results are applied to obtain asymptotic properties of minimum distance estimators goodness of fit tests and resampling procedures

a gentle geometric introduction to copositive optimization this paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way we focus on examples having just a few variables or a few constraints for which the quadratic problem can be formulated as a copositive style problem which itself can be recast in terms of linear second order cone and semidefinite optimization a particular highlight is the role played by the geometry of the feasible set

a sum characterization of hidden regular variation with likelihood inference via expectation maximization a fundamental deficiency of classical multivariate extreme value theory is the inability to distinguish between asymptotic independence and exact independence in this work we examine multivariate threshold modelling in the framework of regular variation on cones tail dependence is described by a limiting measure which in some cases is degenerate on joint tail regions despite strong subasymptotic dependence in such regions hidden regular variation a higher order tail decay on these regions offers a refinement of the classical theory we develop a representation of random vectors possessing hidden regular variation as the sum of independent regular varying components the representation is shown to be asymptotically valid via a multivariate tail equivalence result we develop a likelihood based estimation procedure from this representation via a monte carlo expectation maximization algorithm which has been modified for tail estimation the method is demonstrated on simulated data and applied to air pollution measurements

hypoelliptic non homogeneous diffusions let be a time dependent second order operator written in usual or hormander form we study the regularity of the law of the associated non homogeneous time dependent diffusion process under hormander s like conditions coefficients are only holder continuous in time the main tool is malliavin calculus our results extend and correct previous ones 17 and related works 15 related topics like filtering theory killed or reflected processes parabolic hypoellipticity are also discussed

malliavin calculus for regularity structures the case of gpam malliavin calculus is implemented in the context of hairer 2014 16 this involves some constructions of independent interest notably an extension of the structure which accommodates a robust and purely deterministic translation operator in l2l2 directions between models in the concrete context of the generalized parabolic anderson model in 2d one of the singular spdes discussed in the afore mentioned article we establish existence of a density at positive times

malliavin differentiability of solutions of rough differential equations abstract in this paper we study rough differential equations driven by gaussian rough paths from the viewpoint of malliavin calculus under mild assumptions on coefficient vector fields and underlying gaussian processes we prove that solutions at a fixed time are smooth in the sense of malliavin calculus examples of gaussian processes include fractional brownian motion with hurst parameter larger than 1 4

a two variable approach to the two trust region subproblem the trust region subproblem minimizes a general quadratic function over an ellipsoid and can be solved in polynomial time using a semidefinite programming sdp relaxation intersecting the feasible set with a second ellipsoid results in the two trust region subproblem ttrs even though ttrs can also be solved in polynomial time existing algorithms do not use sdp in this paper we investigate the use of sdp for ttrs starting from the basic sdp relaxation of ttrs which admits a gap recent research has tightened the basic relaxation using valid second order cone inequalities even still closing the gap requires more for the special case of ttrs in dimension n 2 we fully characterize the remaining valid inequalities which can be viewed as strengthened versions of the second order cone inequalities just mentioned we also demonstrate that these valid inequalities can be used computationally even when n 2 to solve ttrs instances that were previously unsolved using sdp based techniques

on the complexity of quadratic programming with two quadratic constraints the complexity of quadratic programming problems with two quadratic constraints is an open problem in this paper we show that when one constraint is a ball constraint and the hessian of the quadratic function defining the other constraint is positive definite then under quite general conditions the problem can be solved in polynomial time in the real number model of computation through an approach based on the analysis of the dual space of the lagrange multipliers however the degree of the polynomial is rather large thus making the result mostly of theoretical interest

from rough path estimates to multilevel monte carlo new classes of stochastic differential equations can now be studied using rough path theory see e g t j lyons m caruana and t levy differential equations driven by rough paths springer berlin 2007 or p k friz and m hairer a course on rough paths universitext springer cham 2014 in this paper we investigate from a numerical analysis point of view stochastic differential equations driven by gaussian noise in the aforementioned sense our focus lies on numerical implementations and more specifically on the savings possible via multilevel methods our analysis relies on a subtle combination of pathwise estimates gaussian concentration and multilevel ideas numerical examples are given which both illustrate and confirm our findings

the jain monrad criterion for rough paths and applications to random fourier series and non markovian hormander theory we discuss stochastic calculus for large classes of gaussian processes based on rough path analysis our key condition is a covariance measure structure combined with a classical criterion due to jain and monrad ann probab 11 1983 46 57 this condition is verified in many examples even in absence of explicit expressions for the covariance or volterra kernels of special interest are random fourier series with covariance given as fourier series itself and we formulate conditions directly in terms of the fourier coefficients we also establish convergence and rates of convergence in rough path metrics of approximations to such random fourier series an application to spde is given our criterion also leads to an embedding result for cameron martin paths and complementary young regularity cyr of the cameron martin space and gaussian sample paths cyr is known to imply malliavin regularity and also ito like probabilistic estimates for stochastic integrals resp stochastic differential equations despite their rough pathwise construction at last we give an application in the context of non markovian hormander theory

sensitivity of rough differential equations an approach through the omega lemma the ito map assigns the solution of a rough differential equation a generalization of an ordinary differential equation driven by an irregular path when existence and uniqueness hold by studying how a path is transformed through the vector field which is integrated we prove that the ito map is holder or lipschitz continuous with respect to all its parameters this result unifies and weaken the hypotheses of the regularity results already established in the literature

a note on the notion of geometric rough paths we use simple sub riemannian techniques to prove that every weak geometric p rough path a geometric p rough path in the sense of 20 is the limit in sup norm of a sequence of canonically lifted smooth paths uniformly bounded in p variation thus clarifying the two different definitions of a geometric p rough path our proofs are sufficiently general to include the case of holder and modulus type regularity this allows us to extend a few classical results on holder spaces and p variation spaces to the non commutative setting necessary for the theory of rough paths as an application we give a precise description of the support of enhanced fractional brownian motion and prove a conjecture by ledoux et al

non parametric bayesian inference on bivariate extremes the tail of a bivariate distribution function in the domain of attraction of a bivariate extreme value distribution may be approximated by the one of its extreme value attractor the extreme value attractor has margins that belong to a three parameter family and a dependence structure which is characterised by a probability measure on the unit interval with mean equal to one half called spectral measure inference is done in a bayesian framework using a censored likelihood approach a prior distribution is constructed on an in nite dimensional model for this measure the model being at the same time dense and computationally manageable a trans dimensional markov chain monte carlo algorithm is developed and convergence to the posterior distribution is established in simulations the bayes estimator for the spectral measure is shown to compare favorably with frequentist nonparametric estimators an application to a data set of danish re insurance claims is provided

models for the extremes of markov chains the modelling of extremes of a time series has progressed from the assumption of independent observations to more realistic forms of temporal dependence in this paper we focus on markov chains deriving a class of models for their joint tail which allows the degree of clustering of extremes to decrease at high levels overcoming a key limitation in current methodologies theoretical aspects of the model are examined and a simulation algorithm is developed through which the stochastic properties of summaries of the extremal txhaviour of the chain are evaluated the approach is illustrated through a simulation study of extremal events of gaussian autoregressive processes and an application to temperature data