w2 p estimates for the parabolic monge ampere equation when u is a solution to the equation ut det dx2u f with f positive continuous and ft satisfying certain growth conditions we establish estimates in l for ut and show that dx2u satisfies uniform interior estimates in lp for 0 p

boundary regularity on the parabolic monge ampere equation abstract this paper establishes boundary estimates for solutions to the parabolic monge ampere equation u t det d 2 u 1 n f x t

geometric properties of boundary sections of solutions to the monge ampere equation and applications abstract in this paper we establish several geometric properties of boundary sections of convex solutions to the monge ampere equation the engulfing and separating properties and volume estimates as applications we prove a covering lemma of besicovitch type a covering theorem and a strong type p p estimate for the maximal function corresponding to boundary sections moreover we show that the monge ampere setting forms a space of homogeneous type

poincare and sobolev inequalities in the monge ampere quasi metric structure abstract we establish a number of poincare and sobolev inequalities in the monge ampere quasi metric structure that are either new or significant improvements upon known ones as applications we derive a host of new weighted poincare and sobolev inequalities

geometric properties of the sections of solutions to the monge ampere equation in this paper we establish several geometric properties of the cross sections of generalized solutions 4 to the monge ampere equation det d2 when the measure t satisfies a doubling property a main result is a characterization of the doubling measures t in terms of a geometric property of the cross sections of 4 this is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations 0 introduction let ri r be a convex function a supporting hyperplane to q at the point xo q xo is an affine function x q xo p x xo such that q x x for all x e r given t 0 a section of 0 at height t is the convex set so xo p t x e r x p ri defined by vq xo p q x xo p x xo vx e rn if e c rn then v e uxee vq x by a classical theorem of aleksandrov the class of sets e such that v e is lebesgue measurable is a o algebra that contains the borel sets and one can define the monge ampere measure associated with q as the borel measure u given by pt v e see ch y the purpose in this paper to analyze in detail several important geometric properties of the sections of the convex function q when its associated monge ampere measure u satisfies a doubling condition the interest in these properties comes from the study of the solutions of the monge ampere equation and its linearizations both elliptic and parabolic and from real harmonic analysis see cl c2 c g1 c g2 and h some of these properties have been used in those references to prove a lemma of besicovitch s type and a calderon zygmund decomposition ir terms of sections which allows us to establish estimates of the solutions to the linearized monge ampere equation also some of these properties imply that r received by the editors june 9 1997 1991 mathematics subject classification primary 35j60 35d10 secondary 26b25

characterization of efficient frontier for mean variance model with a drawdown constraint this paper examines an optimal portfolio selection problem with a drawndown constraint based on mean variance model when 1 there is no arbitrage 2 the market is complete 3 there are finitely many states first we obtain the necessary and sufficient condition for the existence of the optimal solution to the problem then we prove that the efficient frontier of the model is a continuous and convex function and the efficient frontier consists of a finite number of hyperbola segments and straight line segments with at most one subinterval corresponding to straight line segments moreover we give some investment strategies by the results we obtained finally we provide a numerical example to illustrate our results

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

properties of the solutions of the linearized monge ampere equation let r n r be a function strictly convex and smooth and det d 2 is the monge ampere generated by given x r n and t 0 let s x t y r n y x x y x t the purpose of this paper is to study the properties of the solutions of the linearized monge ampere equation given by a ij x d ij u 0 where the coefficients a ij x are the cofactors of the matrix d 2 x it is assumed that satisfies a doubling condition on the sets s x t and a uniform continuity condition at every scale with respect to lebesgue measure we establish that the distribution functions of nonnegative solutions u at altitude t decay like a negative power of t and prove an invariant harnack s inequality on the sections s x t all the estimates are independent of the regularity of and depend only on the constants in the hypotheses made on the measure

on the mean oscillation of the hessian of solutions to the monge ampere equation we give interior a priori estimates for the mean oscillation of second derivatives of solutions to the monge ampere equation detd2u f x with zero boundary values where f x is a non dini continuous function if the modulus of continuity of f x is f r such that limr 0f r log 1 r 0 then d2uvmo

properties of the solutions to the monge ampere equation abstract we consider solutions to the equation det d 2 when has a doubling property we prove new geometric characterizations for this doubling property by means of the so called engulfing property and deduce the quantitative behaviour of also a constructive approach to the celebrated c1 estimates proved by l caffarelli is presented settling one of the open questions posed by villani amer math soc 58 2003

the computation of the worst conditional expectation abstract recent advances in risk theory identify risk as a measure related to the tail of a probability distribution function since it represents the worst outcomes of the random variable measures like value at risk var conditional var expected shortfall and so on have become familiar operational tools for many financial applications in this paper one of these measures the worst conditional expectation with threshold of a discrete random variable z shortly wce z is considered it has been found that its computation can be formulated as a fractional integer programming problem with a single linear constraint but its complexity is np hard therefore it must be solved by implicit enumeration due to its similarity with the knapsack problem it has been found that a good upper bound and a sharp data structure allow the implementation of a branch bound that is able to solve realistic size problems in less than one hundredth of a second

the design of simulation studies in medical statistics by andrea burton douglas g altman patrick royston and roger l holder statistics in medicine 2006 25 4279 4292

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

multiple imputation under bayesianly smoothed pattern mixture models for non ignorable drop out conventional pattern mixture models can be highly sensitive to model misspecification in many longitudinal studies where the nature of the drop out and the form of the population model are unknown interval estimates from any single pattern mixture model may suffer from undercoverage because uncertainty about model misspecification is not taken into account in this article a new class of bayesian random coefficient pattern mixture models is developed to address potentially non ignorable drop out instead of imposing hard equality constraints to overcome inherent inestimability problems in pattern mixture models we propose to smooth the polynomial coefficient estimates across patterns using a hierarchical bayesian model that allows random variation across groups using real and simulated data we show that multiple imputation under a three level linear mixed effects model which accommodates a random level due to drop out groups can be an effective method to deal with non ignorable drop out by allowing model uncertainty to be incorporated into the imputation process

a generalization of the beta distribution with applications abstract this paper introduces a five parameter beta distribution gb which nests the generalized beta and gamma distributions and includes more than thirty distributions as limiting or special cases the generalized beta leads to an exponential generalized beta egb distribution which includes generalized forms of the logistics exponential gompertz and gumbell distributions and the normal as special cases the egb family of distributiosn provides the basis for partially adaptive estimation of econometric models with possibly skewed and leptokurtic error distributions applications of the models to investigating the distribution of income stock returns and in regression analysis are considered

harnackʼs inequality for solutions to the linearized monge ampere operator with lower order terms abstract under minimal geometric assumptions we prove a harnack inequality for non negative solutions to the non homogeneous linearized monge ampere equation with bounded lower order terms compatible with the maximum principle

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

on the regularity theory of fully nonlinear parabolic equations ii recently m crandall and p l lions 3 developed a very successful method for proving the existence of solutions of nonlinear second order partial differential equations their method called the theory of viscosity solutions also applies to fully nonlinear equations in which even the second order derivatives can enter in nonlinear fashion solutions produced by the viscosity method are guaranteed to be continuous but not necessarily smooth here we announce smoothness results for viscosity solutions our methods extend those of 1 we obtain krylov safonov i e c estimates 8 c 1 schauder c and w estimates for viscosity solutions of uniformly parabolic equations in general form the results can be viewed as a priori estimates on the classical c solutions our method produces in particular regularity results for a broad new array of nonlinear heat equations including the bellman equation 6

regularity results on the parabolic monge ampere equation with vmo type data abstract this paper establishes interior estimates for l p norms orlicz norms of solutions to the parabolic monge ampere equation u t det d 2 u f x t provided that f x t is positive bounded and satisfies a vmo type condition and some integrability conditions for f t our results improve the corresponding results in gutierrez and huang 2001 8 in some sense

tail variance of portfolio under generalized laplace distribution many popular downside risk measures such as tail conditional expectation and conditional value at risk only characterize the tail expectation but pay no attention to the tail variance beyond the value at risk this is a severe deficiency of risk management in finance and insurance industry especially in measuring extreme risk with large losses we derive the explicit formulae of the tail variance of portfolio under the assumption of generalized laplace distribution and mixture generalized laplace distribution as well some numerical results of parameters related to the tail variance of portfolio are also provided finally we present an example of application to optimization tail mean variance portfolio the empirical results show that the performance of optimal portfolio can be efficiently improved by controlling the tail variability of returns distribution

worst case conditional value at risk with application to robust portfolio management this paper considers the worst case conditional value at risk cvar in the situation where only partial information on the underlying probability distribution is available the minimization of the worst case cvar under mixture distribution uncertainty box uncertainty and ellipsoidal uncertainty are investigated the application of the worst case cvar to robust portfolio optimization is proposed and the corresponding problems are cast as linear programs and second order cone programs that can be solved efficiently market data simulation and monte carlo simulation examples are presented to illustrate the proposed approach

a generalized asymmetric student t distribution with application to financial econometrics this paper proposes a new class of asymmetric student t ast distributions and investigates its properties gives procedures for estimation and indicates applications in financial econometrics we derive analytical expressions for the cdf quantile function moments and quantities useful in financial econometric applications such as the expected shortfall a stochastic representation of the distribution is also given although the ast density does not satisfy the usual regularity conditions for maximum likelihood estimation we establish consistency asymptotic normality and efficiency of ml estimators and derive an explicit analytical expression for the asymptotic covariance matrix a monte carlo study indicates generally good finite sample conformity with these asymptotic properties

on a prioric 1 α andw 2 p estimates for a parabolic monge ampere equation in the gauss curvature flows this paper establishes holder estimates of du and l p estimates of d 2 u for solutions u to the parabolic monge ampere equation au t det d 2 u 1 n

w 2 p estimates for the linearized monge ampere equation let c r be a strictly convex domain and let c 2 be a convex function such that det d 2 0 and c 0 depending only on n a a and dist such that for all solutions u c 2 to the equation l u f

sharp regularity results on second derivatives of solutions to the monge ampere equation with vmo type data we establish interior estimates for lp norms orlicz norms and mean oscillation of second derivatives of solutions to the monge ampere equation det d2u f x with zero boundary value where f x is strictly positive bounded and satisfies a vmo type condition these estimates develop the regularity theory of the monge ampere equation in vmo type spaces our orlicz estimates also sharpen caffarelli s celebrated w2 p estimates 2008 wiley periodicals inc

the multivariate g and h distribution in this article we consider a generalization of the univariate g and h distribution to the multivariate situation with the aim of providing a flexible family of multivariate distributions that incorporate skewness and kurtosis the approach is to modify the underlying random variables and their quantiles directly giving rise to a family of distributions in which the quantiles rather than the densities are the foci of attention using the ideas of multivariate quantiles we show how to fit multivariate data to our multivariate g and h distribution this provides a more flexible family than the skew normal and skew elliptical distributions when quantiles are of principal interest unlike those families the distribution of quadratic forms from the multivariate g and h distribution depends on the underlying skewness we illustrate our methods on australian athletes data as well as on some wind speed data from the northwest pacific

a comparison of several approaches for choosing between working correlation structures in generalized estimating equation analysis of longitudinal binary data the method of generalized estimating equations gee models the association between the repeated observations on a subject with a patterned correlation matrix correct specification of the underlying structure is a potentially beneficial goal in terms of improving efficiency and enhancing scientific understanding we consider two sets of criteria that have previously been suggested respectively for selecting an appropriate working correlation structure and for ruling out a particular structure s in the gee analysis of longitudinal studies with binary outcomes the first selection criterion chooses the structure for which the model based and the sandwich based estimator of the covariance matrix of the regression parameter estimator are closest while the second selection criterion chooses the structure that minimizes the weighted error sum of squares the rule out criterion deselects structures for which the estimated correlation parameter violates standard constraints for binary data that depend on the marginal means in addition we remove structures from consideration if their estimated parameter values yield an estimated correlation structure that is not positive definite we investigate the performance of the two sets of criteria using both simulated and real data in the context of a longitudinal trial that compares two treatments for major depressive episode practical recommendations are also given on using these criteria to aid in the efficient selection of a working correlation structure in gee analysis of longitudinal binary data