levy processes capacity and hausdorff dimension we use the recently developed multiparameter theory of additive levy processes to establish novel connections between an arbitrary levy process x in r d and a new class of energy forms and their corresponding capacities we then apply these connections to solve two long standing problems in the folklore of the theory of levy processes first we compute the hausdorff dimension of the image x g of a nonrandom linear borel set g c r where x is an arbitrary levy process in r d our work completes the various earlier efforts of taylor proc cambridge phil soc 49 1953 31 39 mckean duke math j 22 1955 229 234 blumenthal and getoor illinois j math 4 1960 370 375 j math mech 10 1961 493 516 millar z wahrsch verw gebiete 17 1971 53 73 pruitt j math mech 19 1969 371 378 pruitt and taylor z wahrsch verw gebiete 12 1969 267 289 hawkes z wahrsch verw gebiete 19 1971 90 102 j london math soc 2 17 1978 567 576 probah theory related fields 112 1998 1 11 hendricks ann math stat 43 1972 690 694 ann probab 1 1973 849 853 kahane publ math orsay 83 02 1983 74 105 recent progress in fourier analysis 1985b 65 121 becker kern meerschaert and scheffler monatsh math 14 2003 91 101 and khoshnevisan xiao and zhong ann probab 31 2003a 1097 1141 where dimx g is computed under various conditions on g x or both we next solve the following problem kahane 1983 publ math orsay 83 02 74 105 when x is an isotropic stable process what is a necessary and sufficient analytic condition on any two disjoint borel sets f g c r such that with positive probability x f n x g is nonempty prior to this article this was understood only in the case that x is a brownian motion khoshnevisan 1999 trans amer math soc 351 2607 2622 here we present a solution to kahane s problem for an arbitrary levy process x provided the distribution of x t is mutually absolutely continuous with respect to the lebesgue measure on r d for all t 0 as a third application of these methods we compute the hausdorff dimension and capacity of the preimage x 1 f of a nonrandom borel set f r d under very mild conditions on the process x this completes the work of hawkes probab theory related fields 112 1998 1 11 that covers the special case where x is a subordinator

on the irreducibility lyapunov rank and automorphisms of special bishop phelps cones abstract motivated by optimization considerations we consider cones in r n to be called special bishop phelps cones of the form t x t x where is a norm on r n 1 we show that when n 3 such cones are always irreducible defining the lyapunov rank of a proper cone k as the dimension of the lie algebra of the automorphism group of k we show that the lyapunov rank of any special bishop phelps polyhedral cone is one extending an earlier known result for the l 1 cone which is a special bishop phelps cone with 1 norm we show that any l p cone for 1 p p 2 has lyapunov rank one we also study automorphisms of special bishop phelps cones in particular giving a complete description of the automorphisms of the l 1 cone

on multifractality and time subordination for continuous functions let be a continuous function this paper relates to the existence of a decomposition of z as z g f where is a monofractal function with exponent 0 h 1 and f 0 1 0 1 is a time subordinator i e the integral of a positive borel measure supported by 0 1 an equivalent question consists of searching for a multifractal parametrization of z which transforms z into a monofractal function we establish that such a decomposition can be found for a large class of functions which includes the usual examples of multifractal functions we find an interesting relationship between self similar functions and self similar measures as an application of our results our theorems yield new insights in the understanding of the multifractal behaviour of functions giving a significant role to the regularity analysis of borel measures

scaling in currency exchange we study the scaling behavior in currency exchange rates our results suggest that they satisfy scaling with an exponent close to 0 5 but that it differs qualitatively from that of a simple random walk indeed price variations cannot be considered as independent variables and subtle correlations are present furthermore we introduce a novel statistical analysis for economic data which makes the physical properties of a signal more evident and eliminates the systematic effects of time periodicity

dimension results for gaussian vector fields and index alpha stable fields the hausdorff dimension and packing dimension of the image and graph of gaussian vector fields and index a stable fields are obtained under general conditions which allow for different local behavior of the components and for dependence among them these results correct some errors in print the condition for an index a stable field to have k multiple points are considered and in the case of nk k 1 e 1ap the hausdorff dimension of the set of k multiple times is given

on the hausdorff dimension of some graphs consider the functions wb x b cn 1 bnx on 1 0n n oo where b 1 0 0 such that if b is large enough then the hausdorff dimension of the graph of wb is bounded below by 2a c ln b we also show that if a function f is convex lipschitz of order a then the graph of f has a finite measure with respect to hausdorff s measure in dimension 2 a the convex lipschitz functions of order a include zygmund s class a our analysis shows that the graph of the classical van der waerdentagaki nowhere differentiable function has a finite measure with respect to h t t in 1 t we consider the hausdorff dimension of the graphs of various continuous functions we introduce a new geometric property of a function convex lipschitz of some order and obtain an upper bound on the dimension of a graph with this property in particular our analysis includes functions of the form

the multifractal nature of levy processes we show that the sample paths of most levy processes are multifractal functions and we determine their spectrum of singularities

the singularity spectrum of levy processes in multifractal time abstract let x x t t 0 be a levy process and a positive borel measure on r suppose that the integral of defines a continuous increasing multifractal time f t 0 0 t under suitable assumptions on we compute the singularity spectrum of the sample paths of the process x in time defined as the process x f t t 0 a fundamental example consists in taking a measure equal to an independent random cascade and independently of a suitable stable levy process x then the associated process x in time is naturally related to the so called fixed points of the smoothing transformation in interacting particles systems our results rely on recent heterogeneous ubiquity theorems

the p cones in dimension n 3 are not homogeneous when p 2 abstract using the t algebra machinery we show that up to linear isomorphism the only strictly convex homogeneous cones in r n n 3 are the 2 cones also known as lorentz cones or second order cones in particular this shows that the p cones are not homogeneous when p 2 1 p and n 3 thus answering a problem proposed by gowda and trott

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

scaling exponents and multifractal dimensions for independent random cascades this paper is concerned with mandelbrot s stochastic cascade measures the problems of i scaling exponents of structure functions of the measure q and ii multifractal dimensions are considered for cascades with a generator vector w1 wc of the general type these problems were previously studied for independent strongly bounded variableswi 0 a wi c consequently a broad class of models used in applications including kolmogorov s long normal model in turbulence long stable universal cascades in atmospheric dynamics has not been covered roughly speaking problems i ii are here solved under the condition that the scaling exists the function is calculated for all arguments previously this was done for positiveq and a new effect emerges the function can generally involve discontinuities in the first derivative as well as in the second

z transformations on proper and symmetric cones z transformations motivated by the similarities between the properties of z matrices on r n and lyapunov and stein transformations on the semidefinite cone mathcal s n we introduce and study z transformations on proper cones we show that many properties of z matrices extend to z transformations we describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations finally we study the equivalence of q and p properties of z transformations on symmetric cones in particular we prove such an equivalence on the lorentz cone

a dynamical model describing stock market price distributions abstract high frequency data in finance have led to a deeper understanding on probability distributions of market prices several facts seem to be well established by empirical evidence specifically probability distributions have the following properties i they are not gaussian and their center is well adjusted by levy distributions ii they are long tailed but have finite moments of any order iii they are self similar on many time scales finally iv at small time scales price volatility follows a non diffusive behavior we extend merton s ideas on speculative price formation and present a dynamical model resulting in a characteristic function that explains in a natural way all of the above features the knowledge of such a distribution opens a new and useful way of quantifying financial risk the results of the model agree with high degree of accuracy with empirical data taken from historical records of the standard poor s 500 cash index

fat tails and colored noise in financial derivatives abstract we study the effect of heavy tails and correlations on the price of the one of the simplest financial derivative the european call option we see that both effects have opposite and nontrivial consequences on the price of the derivatives

a contribution to the systematics of stochastic volatility models abstract we systematically compare several classes of stochastic volatility models of stock market fluctuations we show that the long time return distribution is either gaussian or develops a power law tail while the short time return distribution has generically a stretched exponential form but can also assume an algebraic decay in the family of models which we call garch type the intermediate regime is found in the exponential ornstein uhlenbeck process we also calculate the decay of the autocorrelation function of volatility

the metric entropy of diffeomorphisms soit m une variete de riemann compacte c sans bord soit f un diffeomorphisme c 2 de m et soit m une mesure de probabilite de borel f invariante sur m soit h m f l entropie de f on donne des relations entre h m f et les proprietes geometriques de f et de m

fractal dimensions and singularities of the weierstrass type functions a new type of fractal measures k s 1 s 2 defined on the subsets of the graph of a continuous function is introduced the k dimension defined by this measure is closer to the hausdorff dimension than the other fractal dimensions in recent literatures for the weierstrass type functions defined by w x 0 i g i x where 1 0 1 and g is an almost periodic lipschitz function of order greater than it is shown that the k dimension of the graph of w equals to 2 this conclusion is also equivalent to certain rate of the local oscillation of the function some problems on the knot points and the nondifferentiability of w are also discussed

the automorphism group and the non self duality of p cones abstract in this paper we determine the automorphism group of the p cones p 2 in dimension greater than two in particular we show that the automorphism group of those p cones are the positive scalar multiples of the generalized permutation matrices that fix the main axis of the cone next we take a look at a problem related to the duality theory of the p cones under the euclidean inner product it is well known that a p cone is self dual only when p 2 however it was not known whether it is possible to construct an inner product depending on p which makes the p cone self dual our results show that no matter which inner product is considered a p cone will never become self dual unless p 2 or the dimension is less than three

the graph and range singularity spectra of b adic independent cascade functions abstract with the iso holder sets of a function we naturally associate subsets of the graph and the range of the function we compute the hausdorff dimension of these subsets for a class of statistically self similar multifractal functions namely the b adic independent cascade functions

stochastic model for market stocks with floors we present a model to describe the stochastic evolution of stocks that show a strong resistance at some level and generalize to this situation the evolution based upon geometric brownian motion if volatility and drift are related in a certain way we show that our model can be integrated in an exact way the related problem of how to prize general securities that pay dividends at a continuous rate and earn a terminal payoff at maturity t is solved via the martingale probability approach r 2007 published by elsevier b v

the limited rademacher functions and bernoulli convolutions associated with pisot numbers abstract in this paper we give a systematical study of the local structures and fractal indices of the limited rademacher functions and bernoulli convolutions associated with pisot numbers for a given pisot number in the interval 1 2 we construct a finite family of non negative matrices maybe non square such that the corresponding fractal indices can be re expressed as some limits in terms of products of these non negative matrices we are especially interested in the case that the associated pisot number is a simple pisot number i e the unique positive root of the polynomial x k x k 1 x 1 k 2 3 in this case the corresponding products of matrices can be decomposed into the products of scalars based on which the precise formulas of fractal indices as well as the multifractal formalism are obtained

hausdorff dimension of the limit sets of some planar geometric constructions abstract we determine the hausdorff and box dimension of the limit sets for some class of planar non moran like geometric constructions generalizing the bedford mcmullen general sierpinski carpets the class includes affine constructions generated by an arbitrary partition of the unit square by a finite number of horizontal and vertical lines as well as some non affine examples e g the flexed sierpinski gasket

integrated random processes exhibiting long tails finite moments and power law spectra a dynamical model based on a continuous addition of colored shot noises is presented the resulting process is colored and non gaussian a general expression for the characteristic function of the process is obtained which after a scaling assumption takes on a form that is the basis of the results derived in the rest of the paper one of these is an expansion for the cumulants which are all finite subject to mild conditions on the functions

multifractal analysis of complex random cascades we achieve the multifractal analysis of a class of complex valued statistically self similar continuous functions for we use multifractal formalisms associated with pointwise oscillation exponents of all orders our study exhibits new phenomena in multifractal analysis of continuous functions in particular we find examples of statistically self similar such functions obeying the multifractal formalism and for which the support of the singularity spectrum is the whole interval 0

on a theorem of arrow barankin and blackwell a well known theorem of arrow barankin and blackwell states that if bf r n is equipped with the natural ordering then for every compact convex subset s of bf r n the set of properly minimal elements of s is dense in the set of minimal elements of s in this note a result of jahn is used to show a generalization of the density theorem of arrow barankin and blackwell it will be shown that this theorem holds in a real normed space that is partially ordered by a convex cone with a closed bounded base