european option pricing with transaction costs the authors consider the problem of pricing european options in a market model similar to the black scholes one except that proportional transaction charges are levied on all sales and purchases o

on the valuation of asian options by variational methods we consider the valuation of asian options by variational methods in particular the value function is seen to satisfy an ultraparabolic variational inequality a theoretical framework is developed as is a numerical analysis of a finite element implementation computations are presented that validate the applicability and efficiency of the method

a sufficient condition for global regularity of the neumann operator abstract a theory of global regularity of the neumann operator is developed which unifies the two principal approaches to date namely the one via compactness due to kohn nirenberg j j kohn l nirenberg non coercive boundary value problems comm pure appl math 18 1965 443 492 and catlin david catlin global regularity of the neumann problem in y t siu ed complex analysis of several variables in proc sympos pure math vol 41 amer math soc providence ri 1984 pp 39 49 and the one via plurisubharmonic defining functions and or vector fields that commute approximately with due to boas and the author harold p boas emil j straube sobolev estimates for the neumann operator on domains in c n admitting a defining function that is plurisubharmonic on the boundary math z 206 1 1991 81 88 harold p boas emil j straube de rham cohomology of manifolds containing the points of infinite type and sobolev estimates for the neumann problem j geom anal 3 3 1993 225 235

stochastic optimal control of ultradiffusion processes with application to dynamic portfolio management we consider theoretical and approximation aspects of the stochastic optimal control of ultradiffusion processes in the context of a prototype model for the selling price of a european call option within a continuous time framework the dynamic management of a portfolio of assets is effected through continuous or point control activation costs and phase delay the performance index is derived from the unique weak variational solution to the ultraparabolic hamilton jacobi equation the value function is the optimal realization of the performance index relative to all feasible portfolios an approximation procedure based upon a temporal box scheme finite element method is analyzed numerical examples are presented in order to demonstrate the viability of the approach

universal confidence sets for solutions of optimization problems we consider random approximations to deterministic optimization problems the objective function and the constraint set can be approximated simultaneously relying on concentration of measure results we derive universal confidence sets for the constraint set the optimal value and the solution set special attention is paid to solution sets which are not single valued with many statistical estimators being solutions to random optimization problems the approach can also be employed to derive confidence sets for constrained estimation problems

a multiplier condition for hypoellipticity of complex vector fields with optimal loss of derivatives abstract in c z r t we consider the function g g z set g 1 z g g 1 1 z z g and define the operator l g z i g 1 t we discuss estimates with loss of derivatives in the sense of kohn for the system l g f k l g where l g l g is 1 2 m subelliptic at 0 and f 0 0 d f 0 0 we prove estimates with a loss l k 1 2 m if the multiplier condition f g 1 1 1 2 m 1 is fulfilled for estimates without cut off subellipticity can be weakened to compactness and this results in a loss of l k 2 m 1 for the choice g f k z 2 m z k this result was obtained by kohn and bove derridj kohn tartakoff for m 1 and m 1 respectively also the loss l k 1 2 m was proven to be optimal we show that it remains optimal for the model g f k x 2 m x k instead for the model g f k z 2 m x k in which the multiplier condition is violated the loss is not lowered by the type and must be k 1 2

lp and weighted estimates for barred derivatives abstract we generalize the basic l 2 inequality for barred derivatives that holds on bounded pseudoconvex domains the analogue in l p sobolev norms is established for smooth bounded pseudoconvex domains of finite type with comparable levi eigenvalues a weighted l 2 version and l p estimates with loss are obtained on any smooth bounded weakly q convex domain where the weights are fractional powers of the distance to the boundary

characterization of lil behavior in banach space in a recent paper by the authors a general result characterizing two sided lil behavior for real valued random variables has been established in this paper we look at the corresponding problem in the banach space setting we show that there are analogous results in this more general setting in particularly we provide a necessary and sufficient condition for lil behavior with respect to sequences of the form p nh n where h is from a suitable subclass of the positive nondecreasing slowly varying functions to prove these results we have to use a different method one of our main tools is an improved fuk nagaev type inequality in banach space which should be of independent interest

the diederich fornaess index and the global regularity of the bar partial neumann problem we describe along the guidelines of kohn quantitative estimates for global regularity analysis and geometry in several complex variables pp 97 128 trend math birkhauser boston 1999 the constant mathcal e s which is needed to control the commutator of a totally real vector field t mathcal e with bar partial in order to have h s a priori estimates for the bergman projection b k k ge q 1 on a smooth q convex domain d subset subset mathbb c n this statement not explicit in kohn quantitative estimates for global regularity analysis and geometry in several complex variables pp 97 128 trend math birkhauser boston 1999 yields regularity of b k in specific sobolev degree s next we refine the pseudodifferential calculus at the boundary in order to relate for a defining function r of d the operators t frac delta 2 and r frac delta 2 we are thus able to extend to general degree k ge 0 of b k the conclusion of quantitative estimates for global regularity analysis and geometry in several complex variables pp 97 128 trend math birkhauser boston 1999 which only holds for q 1 and k 0 if for the diederich fornaess index delta of d we have 1 delta frac 1 2 le mathcal e s then b k is h s regular

a note on plurisubharmonic defining functions in mathbb c n let omega subset subset mathbb c n n 3 be a smoothly bounded domain suppose that admits a smooth defining function which is plurisubharmonic on the boundary of then a diederich fornaess exponent can be chosen arbitrarily close to 1 and the closure of admits a stein neighborhood basis

on stability in multiobjective programming a stochastic approach in this paper we assume that a deterministic multiobjective programming problem is approximated by surrogate problems based on estimations for the objective functions and the constraints making use of a large deviations approach we investigate the behaviour of the constraint sets the sets of efficient points and the solution sets if the size of the underlying sample tends to infinity the results are illustrated by applying them to stochastic programming with chance constraints where i the distribution function of the random variable is estimated by the empirical distribution function ii certain parameters have to be estimated

a note on plurisubharmonic defining functions in mathbb c 2 let mathbb c 2 be a smoothly bounded domain suppose that admits a smooth defining function which is plurisubharmonic on the boundary of then the diederich fornaess exponent can be chosen arbitrarily close to 1 and the closure of admits a stein neighborhood basis

on the rate of convergence of optimal solutions of monte carlo approximations of stochastic programs in this paper we discuss monte carlo simulation based approximations of a stochastic programming problem we show that if the corresponding random functions are convex piecewise linear and the distribution is discrete then an optimal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size moreover by using the theory of large deviations we show that the probability of such an event approaches one exponentially fast with increase of the sample size in particular this happens in the case of linear two or multi stage stochastic programming with recourse if the corresponding distributions are discrete the obtained results suggest that in such cases monte carlo simulation based methods could be very efficient we present some numerical examples to illustrate the ideas involved

propagation of holomorphic extendibility and non hypoellipticity of the neumann problem in an exponentially degenerate boundary abstract we prove that cr lines in an exponentially degenerate boundary are propagators of holomorphic extendibility this explains in the context of the cr geometry why in this situation the induced kohn laplacian b is not hypoelliptic christ 2000 2

asset liquidity and the valuation of derivative securities we consider the valuation of european style derivative securities under limited asset liquidity through the dynamic management of a portfolio of assets effected through continuous transaction the valuation arises from the optimal realization of a performance index relative to the set of all feasible portfolio trajectories an approximation procedure based upon the method of lines finite element method is developed and analyzed numerical examples are presented in order to demonstrate the viability of the approach

analysis of sample path optimization sample path optimization is a method for optimizing limit functions occurring in stochastic modeling problems such as steady state functions in discrete event dynamic systems it is closely related to retrospective optimization techniques and to m estimation the method has been computationally tested elsewhere on problems arising in production and in project planning with apparent success in this paper we provide a mathematical justification for sample path optimization by showing that under certain assumptions which hold for the problems just mentioned the method will almost surely find a point that is in a specified sense sufficiently close to the set of optimizers of the limit function

necessary geometric and analytic conditions for general estimates in the bar partial neumann problem we show the geometric and analytic consequences of a general estimate in the bar partial neumann problem a gain in the estimate yields a bound in the type of the boundary that is in its order of contact with an analytic curve as well as in the rate of the bergman metric we also discuss the potential theoretical consequence a gain implies a lower bound for the levi form of a bounded weight

maximal hypoellipticity for the neumann problem abstract we establish maximal hypoellipticity for the neumann problem on smooth bounded pseudoconvex domains in c n under the weakest possible condition on the levi form in particular maximal hypoellipticity holds on the level of n 1 forms for pseudoconvex domains of finite commutator type

regularity of the neumann problem at point of infinite type abstract we introduce general estimates for gain of regularity of solutions of the neumann problem and relate it to the existence of weights with large levi form at the boundary this enables us to discuss in a unified framework the classical results on fractional ellipticity subellipticity superlogarithmic ellipticity and compactness for each case we exhibit a corresponding class of domains

regularity of a class of sublaplacians on the 3 dimensional torus abstract in this paper we present a necessary and sufficient condition for a family of sums of squares operators to be globally hypoelliptic on a torus this condition says that either a diophantine condition is satisfied or there exists a point of finite type also we describe the analytic and gevrey versions of this result the proof is based on l 2 estimates and microlocal analysis

regularity of canonical operators and nebenhulle hartogs domains abstract we relate the regularity of the bergman projection operator and the canonical solution operator to the nebenhulle of complete hartogs domains

a sufficient condition for compactness of the neumann operator we show that the existence of a certain family of plurisubharmonic functions on a bounded domain in cn implies that the neumann operator associated to the domain is compact our condition generalizes previous work of catlin on compactness of the neumann operator

characterization of the law of the iterated logarithm in banach spaces on utilise une technique de randomisation gaussienne pour demontrer qu une variable aleatoire a valeurs dans un espace de banach satisfait la loi du logarithme itere sous une certaine condition necessaire et suffisante

the complex green operator on cr submanifolds of ℂⁿ of hypersurface type compactness we establish compactness estimates for b on a compact pseudoconvex cr submanifold of c of hypersurface type that satisfies property p when the submanifold is orientable these estimates were proved by a raich in 32 using microlocal methods our proof deduces the estimates from a slight extension when q 1 of those known on hypersurfaces via the fact that locally cr submanifolds of hypersurface type are cr equivalent to a hypersurface the relationship between two potential theoretic conditions is also clarified

hypoellipticity of the kohn laplacian box _b and of the bar partial neumann problem by means of subelliptic multipliers we prove local hypoellipticity of the complex laplacian box and of the kohn laplacian box b in a pseudoconvex boundary when for a system of cut off eta the gradient partial b eta and the levi form frac 1 2 partial b bar partial b bar partial b partial b eta are subelliptic multipliers in the sense of kohn acta math 142 79 122 1979

some results on the lil in banach space with applications to weighted empirical processes we examine the cluster set of s n a n for banach space valued random variables and investigate the relationship between the central limit theorem and the law of the iterated logarithm in this setting in the case of hilbert space valued random variables necessary and sufficient conditions are given for the law of the iterated logarithm some interesting examples are also included we then apply our results to weighted empiricals both in the supremum norm and the l 2 lbrack 0 1 rbrack norm

asymptotic theory for solutions in statistical estimation and stochastic programming new techniques of local sensitivity analysis for nonsmooth generalized equations are applied to the study of sequences of statistical estimates and empirical approximations to solutions of stochastic programs consistency is shown to follow from a certain local invertibility property and asymptotic distributions are derived from a generalized implicit function theorem that characterizes asymptotic behavior in situations where estimates are subjected to constraints and estimation functionals are nonsmooth

regularity of the bergman projection on forms and plurisubharmonicity conditions let omega subset subset mathbb c n be a smoothly bounded domain suppose has a defining function such that the sum of any q eigenvalues of its complex hessian is non negative we show that this implies global regularity of the bergman projection b j 1 and the bar partial neumann operator n j acting on 0 j forms for j in q ldots n

the hardy littlewood lemma and the estimate of the neumann problem in a general norm abstract we prove a generalized hardy littlewood lemma on a non smooth domain in the f norm and give an application to a corresponding estimate for the neumann problem by means of suitable weights

asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems we study the asymptotic behavior of the statistical estimators that maximize a not necessarily dieren tiable criterion function possibly subject to side constraints equalities and inequalities the consistency results generalize those of wald and huber conditions are also given under which one is still able to obtain asymptotic normality the analysis brings to the fore the relationship between the problem of nding statistical estimators and that of nding the optimal solutions of stochastic optimization problems with partial information the last section is devoted to the properties of the saddle points of the associated lagrangians

sobolev estimates for the cauchy riemann complex on c1 pseudoconvex domains let omega subset mathbb c n be a bounded pseudoconvex domain with c k boundary k 1 in this paper we will prove that the cauchy riemann operator overline partial has a bounded solution operator in the sobolev space w s p q omega for all 0 leq s k frac 1 2

a sobolev mapping property of the bergman kernel we prove that if d is a pseudoconvex domain with lipschitz boundary having an exhaustion function rho such that rho eta is plurisubharmonic then the bergman projection maps the sobolev space w s boundedly to itself for any s eta 2

on differential stability in stochastic programming in this paper optimal solutions of a stochastic programming problem are considered as functions of the underlying probability distribution their directional derivatives in the sense of g teaux are calculated by applying some recent results from the sensitivity analysis of nonlinear programs these derivatives are employed as a heuristic device in order to derive the asymptotic distribution of statistical estimators of the optimal solutions

compactness of the complex green operator on cr manifolds of hypersurface type the purpose of this article is to study compactness of the complex green operator on cr manifolds of hypersurface type we introduce cr pq a potential theoretic condition on 0 q forms that generalizes catlin s property pq to cr manifolds of arbitrary codimension we prove that if an embedded cr manifold of hypersurface type of real dimension at least five satisfies cr pq and cr pn 1 q then the complex green operator is a compact operator on the sobolev spaces h s 0 q m and h s 0 n 1 q m if 1 q n 2 and s 0 we use cr plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle 1 introduction and results in this article we introduce property cr pq a potential theoretic condition on 0 q forms we show that if an embedded cr manifold of hypersurface type satisfies cr pq and cr pn 1 q then the complex green operator is a compact operator on the sobolev spaces h s 0 q m and h s 0 n 1 q m if 1 q n 2 we use cr plurisubharmonic functions to build a microlocal norm that controls the bad direction of the tangent bundle we first prove the closed range and compactness results on l 2 q m and use an elliptic regularization

the bergman projection on hartogs domains in estimates in l 2 sobolev norms are proved for the bergman projection in certain smooth bounded hartogs domains in c 2 in particular 1 if the domain is pseudoconvex and nonwormlike the normal vector does not wind on a critical set in the boundary then the bergman projection is regular and 2 barrett s counter example domains with irregular bergman projection nevertheless admit a priori estimates

on rates of convergence for stochastic optimization problems under non independent and identically distributed sampling in this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size this is a well studied problem in case the samples are independent and identically distributed i e when standard monte carlo simulation is used here we study the case where that assumption is dropped broadly speaking our results show that under appropriate assumptions the rates of convergence for pointwise estimators under a sampling scheme carry over to the optimization case in the sense that convergence of approximating optimal solutions and optimal values to their true counterparts has the same rates as in pointwise estimation we apply our results to two well established sampling schemes namely latin hypercube sampling and randomized quasi monte carlo qmc the novelty of our work arises from the fact that while there has been some work on the use of variance reduction techniques and qmc methods in stochastic optimization none of the existing work to the best of our knowledge has provided a theoretical study on the effect of these techniques on rates of convergence for the optimization problem we present numerical results for some two stage stochastic programs from the literature to illustrate the discussed ideas

supnorm and f holder estimates for on convex domains of general type in c2 abstract in this paper we study supnorm and modified holder estimates for the integral solution of the equation on a class of convex domains of general type in c 2 that includes many infinite type examples

sobolev estimates for the bar partial neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary

weighted and boundary l p estimates for solutions of the equation on lineally convex domains of finite type and applications we obtain sharp weighted estimates for solutions of the equation u f in a lineally convex domain of finite type precisely we obtain estimates in the spaces l p being the distance to the boundary with two different types of hypothesis on the form f first if the data f belongs to l p 1 we have a mixed gain on the index p and the exponent secondly we obtain a similar estimate when the data f satisfies an apropriate anisotropic l p estimate with weight 1 moreover we extend those results to 1 and obtain l p and bmo estimates these results allow us to extend the l p regularity results for weighted bergman projection obtained in cdm14b for convex domains to more general weights

global regularity for the neumann operator and bounded plurisubharmonic exhaustion functions abstract for a smooth bounded pseudoconvex domain c n we derive a new sufficient condition for global regularity of the neumann operator that generalizes mcneal s property p the approximately holomorphic vector fields of boas and straube and a condition involving bounded plurisubharmonic exhaustion functions due to kohn

on rates of convergence for sample average approximations in the almost sure sense and in mean we study the rates at which optimal estimators in the sample average approximation approach converge to their deterministic counterparts in the almost sure sense and in mean to be able to quantify these rates we consider the law of the iterated logarithm in a banach space setting and first establish under relatively mild assumptions almost sure convergence rates for the approximating objective functions which can then be transferred to the estimators for optimal values and solutions of the approximated problem by exploiting a characterisation of the law of the iterated logarithm in banach spaces we are further able to derive under the same assumptions that the estimators also converge in mean at a rate which essentially coincides with the one in the almost sure sense this in turn allows to quantify the asymptotic bias of optimal estimators as well as to draw conclusive insights on their mean squared error and on the estimators for the optimality gap finally we address the notion of convergence in probability to derive rates in probability for the deviation of optimal estimators and weak rates of error probabilities without imposing strong conditions on exponential moments we discuss the possibility to construct confidence sets for the optimal values and solutions from our obtained results and provide a numerical illustration of the most relevant findings

moderate deviations and associated laplace approximations for sums of independent random vectors let xj be an i i d sequence of banach space valued r v s and let sn z 1 xj for certain positive sequences bn oo we determine the exact asymptotic behavior of e exp b2 n d sn bn where d is a smooth function we also prove a large deviation principle for 2y sn bn

a smoothing property of the bergman projection let b be the bergman projection associated to a domain on which the bar partial neumann operator is compact we show that arbitrary l 2 derivatives of bf are controlled by derivatives of f taken in a single distinguished direction as a consequence functions not contained in c infty overline omega that are mapped by b to c infty overline omega are explicitly described

comparison of the bergman and szego kernels abstract the quotient of the szego and bergman kernels for a smooth bounded pseudoconvex domains in c n is bounded from above by a constant multiple of log p for any p n where is the distance to the boundary for a class of domains that includes those of d angelo finite type and those with plurisubharmonic defining functions the quotient is also bounded from below by a constant multiple of log p for any p 1 moreover for convex domains the quotient is bounded from above and below by constant multiples of

uniform approximation of holomorphic functions on bounded hartogs domains in ℂ2 on etudie l algebre a des fonctions continues sur qui sont holomorphes dans approximation uniforme des fonctions de a par des fonctions holomorphes dans un voisinage de question de la densite uniforme dans a de l algebre a des fonctions regulieres sur qui sont holomorphes dans

real hypersurfaces orders of contact and applications a fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the cauchy riemann equations suppose that d is an open domain in cn and that its boundary m is a smooth real submanifold of cn how does the geometry of m influence the function theory on d one approach to this question is the a neumann problem 6 there one studies the inhomogeneous cauchy riemann equations au a where a is a 0 q form with distribution coefficients on the closed domain d u m satisfying the necessary compatibility condition a a 0 the a neumann problem constructs a particular solution u which is orthogonal to the null space of the complex laplacian kohn 10a has solved this problem and has shown that good local regularity properties for u in terms of a follow from so called subelliptic estimates it is natural to seek geometric conditions on m that are necessary and sufficient for these estimates in this paper we will be concerned with a certain geometric condition on m that we feel is the right one for subellipticity although the motivation for the questions discussed here comes from partial differential equations the techniques come from algebraic geometry to see why we state theorems of kohn greiner and catlin concerning subellipticity kohn 1oa has discovered a sufficient condition for a subelliptic estimate near a fixed point p in m in case d is pseudoconvex and is contained in c2 this condition is equivalent to the finiteness of order of tangency of every complex analytic one dimensional manifold with m at p greiner 13 established the necessity of this condition in

superlogarithmic estimates on pseudoconvex domains and cr manifolds this paper is concerned with proving superlogarithmic estimates for the operator r b on pseudoconvex cr manifolds and using them to establish hypoellipticity of lb and of the neumann problem these estimates are established under the assumption that subellipticity degenerates in certain specified ways

random approximations in multiobjective optimization often decision makers have to cope with a programming problem with unknown quantitities then they will estimate these quantities and solve the problem as it then appears the approximate problem thus there is a need to establish conditions which will ensure that the solutions to the approximate problem will come close to the solutions to the true problem in a suitable manner confidence sets i e sets that cover the true sets with a given prescribed probability provide useful quantitative information in this paper we consider multiobjective problems and derive confidence sets for the sets of efficient points weakly efficient points and the corresponding solution sets besides the crucial convergence conditions for the objective and or constraint functions one approach for the derivation of confidence sets requires some knowledge about the true problem which may be not available therefore also another method called relaxation is suggested this approach works without any knowledge about the true problem the results are applied to the markowitz model of portfolio optimization

compact and subelliptic estimates for the overline partial neumann operator on c2 pseudoconvex domains in this paper we will show that catlin s property p implies regularity for the overline partial neumann operator in w 1 on c 2 domains we will also look at the subelliptic case and derive the natural generalization to c k domains when k 2

behavior of the bergman projection on the diederich fornaess worm w 1 introduction in this paper we show that the bergman projection operator for certain smooth bounded pseudoconvex domains does not preserve smoothness as measured by sobolev norms let q be a smooth bounded pseudoconvex domain in c and let p t denote the orthogonal projection from l2 f onto the bergman subspace b ff l2 f3 r with respect to the weighted norm i fll j olf z l 2 e t lzll2dv l 2 let wk t denote the sobolev space consisting of functions whose derivatives of order to k g2 on the other hand there is a large collection of results implying that for certain types of domains the unweighted bergman projection p pr preserves w k for all k 0 see fk for the strictly pseudoconvex case for results on weakly pseudoconvex domains the reader may consult ko2 ca si and the recent bstl ch as well as the references cited therein most of these results are focused on the a neumann operator rather than p see bst2 for the connection also except for kol positive results in this area are typically valid for any choice of smooth positive weight function on 0 the question of whether or not p is similarly well behaved for all weakly pseudoconvex domains has remained open for many years in this paper we show that this is not the case in fact when q is the so called worm domain of diederich and forna ss then p does not map w to w k when k zr total amount of winding this latter quantity is explained in section 4 below where the construction of the worm is reviewed and the main result is proved the proof depends on computations for a piecewise levi flat model domain depending in turn on certain one dimensional computations these are treated in sections 3 and 2 respectively section 5 contains additional remarks and questions

extrapolation discontinuous galerkin method for ultraparabolic equations ultraparabolic equations arise from the characterization of the performance index of stochastic optimal control relative to ultradiffusion processes they evidence multiple temporal variables and may be regarded as parabolic along characteristic directions we consider theoretical and approximation aspects of a temporally order and step size adaptive extrapolation discontinuous galerkin method coupled with a spatial lagrange second order finite element approximation for a prototype ultraparabolic problem as an application we value a so called asian option from mathematical finance

portfolio selection with transaction costs in this paper optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log normal diffusion this problem was solved by merton and others when transactions between bank and stock are costless here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted it is shown that the optimal buying and selling policies are the local times of the two dimensional process of bank and stock holdings at the boundaries of a wedge shaped region which is determined by the solution of a nonlinear free boundary problem an algorithm for solving the free boundary problem is given