on the complete separation of asymptotic structures in banach spaces abstract let e i i denote the unit vector basis of l p 1 p or c 0 we construct a reflexive banach space with an unconditional basis that admits e i i as a uniformly unique spreading model while it has no subspace with a unique asymptotic model and hence it has no asymptotic l p or c 0 subspace this solves a problem of e odell we also construct a space with a unique l 1 spreading model and no subspace with a uniformly unique l 1 spreading model these results are achieved with the utilization of a new version of the method of saturation under constraints that uses sequences of functionals with increasing weights

spectral analysis of time dependent market adjusted return correlation matrix abstract we present an adjusted method for calculating the eigenvalues of a time dependent return correlation matrix in a moving window first we compare the normalized maximum eigenvalue time series of the market adjusted return correlation matrix to that of the logarithmic return correlation matrix on an 18 year dataset of 310 s p 500 listed stocks for small and large window or memory sizes we observe that the resulting new eigenvalue time series is more stationary than the time series obtained without the adjustment second we perform this analysis while sweeping the window size 5 100 500 in order to examine the dependence on the choice of window size this approach demonstrates the multi modality of the eigenvalue distributions we find that the three dimensional distribution of the eigenvalue time series for our market adjusted return is significantly more stationary than that produced by classic method finally our model offers an approximate polarization domain characterized by a smooth l shaped strip the polarization with large amplitude is revealed while there is persistence in agreement of individual stock returns movement with market with small amplitude most of the time

3d extreme value analysis for stock return interest rate and speed of mean reversion it is important to analyze extreme cases of stock return interest rate and speed of mean reversion together while we explore strengths and limitations of heston stochastic volatility model based on behavior of its numerical solutions using milstein method simulations we suggest model improvements in the light of real data applications first we perform high peak and fat tail analysis for the impact of heston model parameters on the simulations of the extreme situations by using the first four standardized moments and extreme value tools such as quantile quantile qq mean excess me and hill plots to examine the fat tailness of the distributions later we illustrate high peak and fat tail analysis for bist 100 index between 02 01 2004 and 17 06 2013 moreover we investigate 3d dynamics of the average logarithmic stock return interest rate and speed of mean reversion variables together furthermore we believe that polarization and the transitions between polarizations and comovements are important part of extreme situation picture we investigate comovement and polarization of interest rates and daily returns of bist 100 index between 2010 and 2013 in order to understand the corresponding behavioral dynamics heston stochastic volatility model predicts that the average logarithmic stock return increases as interest rate rises actually we observe that there are also sufficiently large time intervals where interest rates were decreased and stock prices increased gradually in us stock markets and borsa istanbul unlike the heston stochastic volatility model suggests

a rate of convergence for monotone finite difference approximations to fully nonlinear uniformly elliptic pdes we obtain an algebraic rate of convergence for monotone and consistent finite difference approximations to lipschitz continuous viscosity solutions of uniformly elliptic partial differential equations 2007 wiley periodicals inc

a potential theory for the k curvature equation abstract in this paper we introduce a potential theory for the k curvature equation which can also be seen as a pde approach to curvature measures we assign a measure to a bounded upper semicontinuous function which is strictly subharmonic with respect to the k curvature operator and establish the weak continuity of the measure

dirichlet problems for general monge ampere equations here 172z denotes the hessian of z with respect to a given metric d s2 gu d u d u a in the local chart vzz z u f z where are the christoffel symbols there have been many papers to devoted to the boundary value problems of 1 1 for example cns1 2 t k tu h1 and most of them are only concerning the euclidean metric the purpose of the present paper is to deal with the dirichlet problem of 1 1 for any given metric as a moderate step in sect 2 and 3 we discuss the following problem

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

application of the heston stochastic volatility model for borsa istanbul using impression matrix norm we study the behavior of solutions for stochastic differential equations such as the heston stochastic volatility model we examine the numerical solutions using euler maruyama milstein and stochastic runge kutta methods to investigate whether there is a role of the methods for different volatility cases or not related to the impact of cumulative errors on this application we perform simulations for different stock market conditions by using the large data set from borsa istanbul 100 bist 100 between 04 01 2007 and 31 12 2012 we use volatilities in terms of extreme values at the overlapping case when we examine initial and long term volatilities for the application of the heston model we also apply unit volatility based on extreme values to approximate volatilities in our analysis we examine the advantages and limitations of the model moreover we introduce 3 dimensional matrix norms furthermore we define market impression matrix norm as an application to the 3 dimensional matrix norms we can benefit from it to quantify market impression approximately by means of the numerical solutions for the stochastic differential equations finally we analyze the simulation results for various parameters

monge ampere equations in big cohomology classes we define non pluripolar products of an arbitrary number of closed positive 1 1 currents on a compact kahler manifold x given a big 1 1 cohomology class on x i e a class that can be represented by a strictly positive current and a positive measure on x of total mass equal to the volume of and putting no mass on pluripolar sets we show that can be written in a unique way as the top degree self intersection in the non pluripolar sense of a closed positive current in we then extend kolodziedj s approach to sup norm estimates to show that the solution has minimal singularities in the sense of demailly if has l 1 e density with respect to lebesgue measure if is smooth and positive everywhere we prove that t is smooth on the ample locus of provided is nef using a fixed point theorem we finally explain how to construct singular kahler einstein volume forms with minimal singularities on varieties of general type

sharp regularity estimates for second order fully nonlinear parabolic equations we prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the form begin aligned u t f left d 2u du x t right f x t quad text in quad q 1 end aligned eq where f is elliptic with respect to the hessian argument and f in l p q q 1 the quantity xi n p q frac n p frac 2 q determines to which regularity regime a solution of eq belongs we prove that when 1 xi n p q 2 epsilon f solutions are parabolically alpha holder continuous for a sharp quantitative exponent 0 alpha n p q 1 precisely at the critical borderline case xi n p q 1 we obtain sharp parabolic log lipschitz regularity estimates when 0 xi n p q 1 solutions are locally of class c 1 sigma frac 1 sigma 2 and in the limiting case xi n p q 0 we show parabolic c 1 text log lip regularity estimates provided f has better a priori estimates

the neumann problem of special lagrangian equations with supercritical phase abstract in this paper we establish the global c 2 estimates of the neumann problem of special lagrangian equations with supercritical phase and the existence theorem by the method of continuity

the dirichlet problem for monge ampere equations in non convex domains and spacelike hypersurfaces of constant gauss curvature in this paper we extend the well known results on the existence and regularity of solutions of the dirichlet problem for monge ampere equations in a strictly convex domain to an arbitrary smooth bounded domain in r as well as in a general riemannian manifold we prove for the nondegenerate case that a sufficient and necessary condition for the classical solvability is the existence of a subsolution for the totally degenerate case we show that the solution is in c11 q if the given boundary data extends to a locally strictly convex c2 function ori q as an application we prove some existence results for spacelike hypersurfaces of constant gauss kronecker curvature in minkowski space spanning a prescribed boundary

liouville property and regularity of a hessian quotient equation we are concerned with properties of convex solutions to the hessian quotient equation sn c d2 p 1 k n as our first main result we prove some regularity of strong solutions while the second states that for p 1 an entire convex solution with a quadratic growth bound must be a quadratic polynomial

kahler einstein metrics with edge singularities this article considers the existence and regularity of kahler einstein metrics on a compact kahler manifold m with edge singularities with cone angle 2 along a smooth divisor d we prove existence of such metrics with negative zero and some positive cases for all cone angles 2 2 the results in the positive case parallel those in the smooth case we also establish that solutions of this problem are polyhomogeneous i e have a complete asymptotic expansion with smooth coecients along d for all 2 2

a banach space block finitely universal for monotone bases a reflexive banach space x with a basis ei is constructed having the property that every monotone basis is block finitely representable in each block basis of x

harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations abstract we present a new harnack inequality for non negative discrete supersolutions of fully nonlinear uniformly elliptic difference equations on rectangular lattices this estimate applies to all supersolutions and has the harnack constant depending on the graph distance on lattices for the proof we modify the proof of the weak harnack inequality applying the same idea to elliptic equations in a euclidean space we also derive a harnack type inequality for non negative viscosity supersolutions

meshfree finite difference approximations for functions of the eigenvalues of the hessian we introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the hessian approximations are defined on unstructured point clouds which allows for very complicated domains and a non uniform distribution of discretisation points the schemes are monotone which ensures that they converge to the viscosity solution of the underlying pde as long as the equation has a comparison principle numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds complex domains degenerate equations and singular solutions

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

the unconditional basic sequence problem we construct a banach space that does not contain any infinite un conditional basic sequence and investigate further properties of this space for example it has no subspace that can be written as a topological direct sum of two infinite dimensional spaces this property implies that every operator on the space is a strictly singular perturbation of a multiple of the identity in par ticular it is either strictly singular or fredholm with index zero this implies that the space is not isomorphic to any proper subspace trinity college cambridge cb2 1tq england current address department of mathematics university college london gower street lon don wc1e 6bt england e mail address wtg l o phx cam ac uk universit paris vii u f r de mathtmatiques 2 place jussieu 75251 paris cedex 05 france current address equipe d analyse et mathematiques appliquees universite de mamne la vallee 93160 noisy le grand france e mail address maurey logiquejussieu fr this content downloaded from 157 55 39 105 on fri 07 oct 2016 04 30 48 utc all use subject to http about jstor org terms

monge ampere equations relative to a riemannian metric we prove that in a bounded strictly convex open set q in ri the problem det v2u f x ulan 9 where f 0 f e c q p e c oq has a unique strictly convex solution u e c q this result extends to an arbitrary metric a theorem which has been proved by caffarelli nirenberg spruck in the case of the euclidean metric

a class of hessian quotient equations in euclidean space abstract in this paper we consider a class of hessian quotient equations in euclidean space under some sufficient conditions we obtain an existence result using standard degree theory based on a priori estimates for solutions to hessian quotient equations

the evans krylov theorem for nonlocal fully nonlinear equations we prove a regularity result for solutions of a purely integro differential bellman equation this regularity is enough for the solutions to be understood in the classical sense if we let the order of the equation approach two we recover the theorem of evans and krylov about the regularity of solutions to concave uniformly elliptic partial differential equations

gradient estimates for neumann boundary value problem of monge ampere type equations on riemannian manifolds abstract in this paper we consider the monge ampere type equations with the neumann boundary conditions on riemannian manifolds our main result is the gradient estimates for the degenerate elliptic solution which is the extension of the main theorem in jiang et al 2016 from r n to riemannian manifolds

a fully nonlinear flow for two convex hypersurfaces in riemannian manifolds we consider a one parameter family of closed embedded hypersurfaces moving with normal velocity g kappa big sum i j frac 1 lambda i lambda j 2 kappa big 1 where lambda 1 le cdots le lambda n denote the curvature eigenvalues and kappa is a nonnegative constant this defines a fully nonlinear parabolic equation provided that lambda 1 lambda 2 2 kappa in contrast to mean curvature flow this flow preserves the condition lambda 1 lambda 2 2 kappa in a general ambient manifold our main goal in this paper is to extend the surgery algorithm of huisken sinestrari to this fully nonlinear flow this is the first construction of this kind for a fully nonlinear flow as a corollary we show that a compact riemannian manifold satisfying overline r 1313 overline r 2323 ge 2 kappa 2 with non empty boundary satisfying lambda 1 lambda 2 2 kappa is diffeomorphic to a 1 handlebody the main technical advance is the pointwise curvature derivative estimate the proof of this estimate requires a new argument as the existing techniques for mean curvature flow due to huisken sinestrari haslhofer kleiner and brian white cannot be generalized to the fully nonlinear setting to establish this estimate we employ an induction on scales argument this relies on a combination of several ingredients including the almost convexity estimate the inscribed radius estimate as well as a regularity result for radial graphs we expect that this technique will be useful in other situations as well

estimates in the generalized campanato john nirenberg spaces for fully nonlinear elliptic equations we establish estimates in bmo and campanato john nirenberg spaces bmo for the second derivatives of solutions to the fully nonlinear elliptic equation f d2u x f x

hypersurfaces with mean curvature given by an ambient sobolev function we consider n hypersurfaces j with interior ej whose mean curvature are given by the trace of an ambient sobolev function uj w 1 p r h j uj ej on j 0 1 where ej denotes the inner normal of j we investigate 0 1 when j weakly as varifolds and prove that is an integral n varifold with bounded first variation which still satisfies 0 1 for uj u ej e p has to satisfy

conformal metrics with prescribed curvature functions on manifolds with boundary we study the dirichlet problem for a class of fully nonlinear elliptic equations related to conformal deformations of metrics on riemannian manifolds with boundary as a consequence we prove the existence of a conformal metric given its value on the boundary as a prescribed metric conformal to the induced background metric with a prescribed curvature function of the schouten tensor

differentiability properties of solutions of nondegenerate isaacs equations abstract in this expository paper we examine an open question concerning differentiability properties of viscosity solutions of uniformly elliptic isaacs equations in a bounded domain in euclidean space we look at specific examples of the simplest isaacs equations

two phase anisotropic free boundary problems and applications to the bellman equation in 2d we prove lipschitz continuity of solutions to a class of rather general two phase anisotropic free boundary problems in 2d and we classify global solutions as a consequence we obtain c 2 1 regularity of solutions to the bellman equation in 2d

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

twice differentiability of solutions to fully nonlinear parabolic equations near the boundary abstract in this paper we prove h 2 regularity for viscosity solutions to non convex fully nonlinear parabolic equations near the boundary this constitutes the parabolic counterpart of a similar c 2 regularity result due to silvestre and sirakov proved in 17 for solutions to non convex fully nonlinear elliptic equations

global regularity of solutions of nonlinear second order elliptic and parabolic differential equations on considere des solutions du probleme de dirichlet pour une equation elliptique totalement non lineaire dans un domaine borne lisse et du probleme parabolique analogue

the stabilized set of p s in krivine s theorem can be disconnected abstract for any closed subset f of 1 which is either finite or consists of the elements of an increasing sequence and its limit a reflexive banach space x with a 1 unconditional basis is constructed so that in each block subspace y of x l p is finitely block represented in y if and only if p f in particular this solves the question as to whether the stabilized krivine set for a banach space had to be connected we also prove that for every infinite dimensional subspace y of x there is a dense subset g of f such that the spreading models admitted by y are exactly the l p for p g

curvature estimates for immersed hypersurfaces in riemannian manifolds we establish mean curvature estimate for immersed hypersurface with nonnegative extrinsic scalar curvature in riemannian manifold n n 1 bar g through regularity study of a degenerate fully nonlinear curvature equation in general riemannian manifold the estimate has a direct consequence for the weyl isometric embedding problem of mathbb s 2 g in 3 dimensional warped product space n 3 bar g we also discuss isometric embedding problem in spaces with horizon in general relativity like the anti de sitter schwarzschild manifolds and the reissner nordstrom manifolds

the complex monge ampere equation on compact kahler manifolds we consider the complex monge ampere equation on a compact kahler manifold m g when the right hand side f has rather weak regularity in particular we prove that estimate of triangle phi and the gradient estimate hold when f is in w 1 p 0 for any p 0 2n as an application we show that there exists a classical solution in w 3 p 0 for the complex monge ampere equation when f is in w 1 p 0

the dirichlet problem for nonlinear second order elliptic equations ii complex monge ampere and uniformaly elliptic equations on considere le probleme de dirichlet pour des equations elliptiques non lineaires d ordre 2 pour une fonction reelle dans un domaine borne de r n a frontiere lisse

convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature we consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity we prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero we apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one for smooth strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point we prove that solutions remain smooth and strictly convex and become spherical in shape while contracting to points in finite time recently several papers have considered the flow of convex hypersurfaces by speeds which are homogeneous functions of the principal curvatures of degree 1 under suitable pinching conditions on the curvature of the initial hypersurface the aim is to prove that solutions become spherical as they contract to points this behaviour has been established for a wide range of flows where the speed is homogeneous of degree 1 in the principal curvatures including the mean curvature flow h1 the flow by nth root of gauss curvature ch1 square root of scalar curvature ch2 and a large family of other speeds a1 a3 a4 the first such result with degree of homogeneity higher than 1 was due to ben chow ch1 and concerned flow by powers of the gauss curvature he proved that flow by k with 1 n produces a spherical limiting shape provided the initial hypersurface is sufficiently pinched i n the sense that hij c hgij later such results were proved by schulze s3 for powers of the mean curvature by alessandroni and sinestrari for powers of the scalar curvature a as and by cabezas rivas and sinestrari crs for normalized flows by powers of elementary symmetric functions a feature of the results mentioned above is that the flows all have some divergence structure a point which is used crucially in deriving sufficient regularity of solutions to deduce the existence of a smooth limiting hypersurface in ch1 the divergence structure was used to deduce pinching of the principal curvatures using integral estimates in a manner similar to h1 in s3 a and as the curvature pinching was proved using maximum principle arguments but the divergence structure was still needed because higher regularity of solutions was established using results for divergence form porous medium equations to date we are not aware of any work which provides holder continuity for solutions of porous medium equations in non divergence form without assuming regularity of the coefficients in this paper we provide a geometric estimate which circumvents this difficulty we prove in section 3 that if the ratios of principal curvatures at each point are bounded in terms of the maximum principal curvature by a function which approaches one at infinity then the ratio of circumradius to inradius is as close to one as desired if the circumradius is sufficiently small using