scaling and memory in the non poisson process of limit order cancelation abstract the order submission and cancelation processes are two crucial aspects in the price formation of stocks traded in order driven markets we investigate the dynamics of order cancelation by studying the statistical properties of inter cancelation durations defined as the waiting times between consecutive order cancelations of 22 liquid stocks traded on the shenzhen stock exchange of china in year 2003 three types of cancelations are considered including cancelation of any limit orders of buy limit orders and of sell limit orders we find that the distributions of the inter cancelation durations of individual stocks can be well modeled by weibulls for each type of cancelation and the distributions of rescaled durations of each type of cancelations exhibit a scaling behavior for different stocks complex intra day patterns are also unveiled in the inter cancelation durations the detrended fluctuation analysis dfa and the multifractal dfa show that the inter cancelation durations possess long term memory and multifractal nature which are not influenced by the intra day patterns no clear crossover phenomenon is observed in the detrended fluctuation functions with respect to the time scale these findings indicate that the cancelation of limit orders is a non poisson process which has potential worth in the construction of order driven market models

convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping we consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time asymptotically the hyperbolic model may be viewed as isentropic euler equations with friction term added to the momentum equation to model gas flow through a porous media in this case our result justifies darcy s law time asymptotically our model may also be viewed as an elastic model with damping

global existence of solutions for a weakly coupled system of semilinear damped wave equations abstract in this paper we consider the cauchy problem for a weakly coupled system of semilinear damped wave equations we prove the global existence of solutions for small data in the supercritical case for any space dimension we also give estimates of the weighted energy of solutions and in a special case we prove an almost optimal estimate moreover in the subcritical case we give an almost optimal estimate of the lifespan from both above and below

exponentially decayed double power law distribution of bitcoin trade sizes abstract bitcoin is the most important cryptocurrency that leads the cryptocurrency market the 776 629 356 bitcoin trades from 3 january 2009 to 31 december 2017 are retrieved it is found that the medium and large trade sizes comply with the exponentially decayed double power law distribution the tail exponent 1 of the medium trade sizes fluctuates in 2009 increases during the period from 2011 to 2014 and then converges to a stable level at 1 0 266 0 127 the tail exponent 2 of the large trade sizes fluctuates more and we have 2 1 752 1 688

leech roots and vinberg groups this chapter investigates the remarkable properties of the leech roots which are the fundamental roots for the even unimodular lattice in lorentzian space r 25 1 and correspond one for one with the points of the leech lattice we give an extensive table of the leech roots in both euclidean and hyperbolic coordinates we also provide the first of what promise to be many applications by showing that the leech roots simplify and explain the remarkable results of vinberg kaplinskaja and meyer on the reflection groups of unimodular lorentzian lattices in dimensions below 20 they also enable us to make some progress on the study of these groups in the next few dimensions

understanding the cubic and half cubic laws of financial fluctuations recent empirical research has uncovered regularities in nancial u ii the half cubic law of volumes volumes follow a power law distribution with exponent 3 iii approximate cubic law of number of trades the number of trades in a given time intervals follows a power law distribution with exponent around 3 we discuss a new theory that explains them as well as some related facts c 2003 published by elsevier science b v

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

an optimal estimate for the difference of solutions of two abstract evolution equations abstractikehata and nishihara have established that the difference between any solution u of alinearly damped abstract wave equation and a certain solution v of a related abstract heatequation decays at least like t 1 logt 12 e as time tends to in nity they conjectured that thedecayisinfactliket 1 weproveherethevalidityofthisconjecturebyrelyingonthespectraltheorem for unbounded self adjoint operators we also establish the optimality of thisestimate for the wave equation in an exterior domain r 2003 elsevier science usa all rights reserved msc primary34d05 secondary35l90keywords abstract heatequation abstract waveequation diffusionphenomenon 1 introductionthe main objective of this work is to establish an estimate of the differencebetween the solution u of the abstract dissipative wave equationu 00 u 0 au 0 u 0 u 0 u 0 0 u 1 1 1 and the solution v of the abstract heat equationv 0 av 0 v 0 u 0 u 1 1 2 article in press corresponding author e mail address haraux ann jussieu fr a haraux 0022 0396 03 seefrontmatterr 2003 elsevierscience usa allrights reserved doi 10 1016 s0022 0396 03 00057 3

exact hurst exponent and crossover behavior in a limit order market model abstract an exclusion particle model is considered as a highly simplified model of a limit order market its price behavior reproduces the well known crossover from over diffusion hurst exponent h 1 2 to diffusion h 1 2 when the time horizon is increased provided that orders are allowed to be canceled for early times a mapping to the totally asymmetric exclusion process yields the exact result h 2 3 which is in good agreement with empirical data the underlying universality class of the exclusion process suggests some robustness of the exponent with respect to changes in the trading rules in the crossover regime the hurst plot has a scaling property where the bulk deposition cancelation rate is the critical parameter analytical results are fully supported by numerical simulations

the automorphism group of the 26 dimensional even unimodular lorentzian lattice abstract the group mentioned in the title is shown to be a certain infinitely generated coxeter group extended by the negative of the identity operation and the group of all automorphisms of the notorious leech lattice

price fluctuations from the order book perspective empirical facts and a simple model abstract statistical properties of an order book and the effect they have on price dynamics were studied using the high frequency nasdaq level ii data it was observed that the size distribution of marketable orders transaction sizes has power law tails with an exponent 1 market 2 4 0 1 the distribution of limit order or quote sizes was found to be consistent with a power law with an exponent close to 2 a somewhat better fit to this distribution was obtained by using a log normal distribution which has an effective power law exponent equal to 2 in the middle of the observed range the depth of the order book measured as a price impact of a hypothetical large market order was observed to be a non linear function of its size a large imbalance in the number of limit orders placed at bid and ask sides of the book was shown to lead to a predictable short term price change which is in accord with the law of supply and demand

asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping we consider the asymptotic behavior of the solution of quasilinear hyperbolic equation with linear dampingvtt a vx x vt 0 x t r 0 subsequent to k nishihara j differential equations131 1996 171 188 in that article the system with dampingvt ux 0 ut p v x u p v 0 was treated and the convergence rates to the diffusion wave by l hsiao and t p liu comm math phys 143 1992 599 605 and l hsiao and t p liu chinese ann math ser b14 1993 465 480 were improved for initial data v0 u0 x satisfying v0 u0 v0 0 and v0 x v0 dx 0 this system is reduced to the quasilinear hyperbolic equation with linear dampingvtt p v0 vx p v0 x vt 0 and the decay rates v t l o t 1 2 vx t l o t 1 vt t l o t 3 2 were obtained provided that v vt t 0 h3 h2 is small and inl1 l1 from these decay rates vis expected to behave as a solution to the parabolic equation the aims of this paper are to seek the asymptotic profile x t satisfying t l o t 1 2 x t l o t 1 t t l o t 3 2 and to show that the solution of satisfies v t l o t 1 v x t l o t 3 2 v t t l o t 2 the proof is based on the elementary energy method and the green function method for the parabolic equation

existence and nonexistence of time global solutions to damped wave equation on half line on the half line r 0 infinity the initial boundary value problems with null dirichlet boundary for both the semilinear heat equation and damped wave equation are considered the critical exponent p c n k of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space d n d k r k rn k is given by p c n k 1 2 n k j appl math phys 39 1988 135 149 arch rational mech anal 109 1990 63 71 since the damped wave equation is expected to be close to the heat equation j differential equations 191 2003 445 469 math z 244 2003 631 649 the critical exponent for the semilinear damped wave equation is expected to be same as that of the semilinear heat equation however there is no blow up result on the halved space for the damped wave equation in this paper the exponent pc l 1 2 is shown to be critical for the existence and nonexistence of time global solution to both the semilinear heat equation and damped wave equation on the half line r together with the derivation of the blow up time for the proof the explicit formulas of solutions are used in a similar fashion to those in li and zhou discrete continuous dynamic systems 1 1995 503 520 c 2005 elsevier ltd all rights reserved

empirical shape function of limit order books in the chinese stock market abstract we have analyzed the statistical probabilities of limit order book lob shape through building the book using the ultra high frequency data from 23 liquid stocks traded on the shenzhen stock exchange in 2003 we find that the averaged lob shape has a maximum away from the same best price for both buy and sell sides of the lob the lob shape function has nice exponential form in the right tail the buy side of the lob is found to be abnormally thicker for the price levels close to the same best although there are much more sell orders on the book we also find that the lob shape functions for both buy and sell sides have periodic peaks with a period of five the 1 min averaged volumes at fixed tick level follow log normal distributions except for the left tails which display power law behaviors exhibit abnormal intraday patterns with increasing trend and possess long memory that cannot be explained by the intraday patterns academic implications of our empirical results are also briefly discussed

volatility driven market in a generalized lotka voltera formalism abstract the generalized lotka voltera glv formalism has been introduced in order to explain the power law distributions in the individual wealth w i t pareto law and financial markets returns fluctuations r as a result of the auto catalytic multiplicative random character of the individual capital dynamics as long as the multiplicative random factor is extracted from the same probability distribution for all the individuals the exponent of the power laws turns out to be independent on the time variations of the average this explains also the stability over the past century of experimentally measured pareto exponent in contrast to the scaling properties of the single time unconditional probability distributions the auto correlations between observables measured at different times are not correctly reproduced by the original glv if the variance 2 of is time independent in the glv formalism the volatility r 2 auto correlations decay exponentially while the measurements in real markets indicate a power law with a very small exponent we show in the present paper that by making the variance of the individual wealth changes 2 a function of the market volatility r 2 one correctly reproduces the market volatility long range correlations moreover we show that this non trivial feedback loop between the market price volatility and the variance of the investors wealth leads to non trivial patterns in the overall market trends if the feedback is too strong it may even endanger the market stability

diffusion phenomenon for linear dissipative wave equations in an exterior domain abstract under the general condition of the initial data we will derive the crucial estimates which imply the diffusion phenomenon for the dissipative linear wave equations in an exterior domain in order to derive the diffusion phenomenon for dissipative wave equations the time integral method which was developed by ikehata and matsuyama sci math japon 55 2002 33 plays an effective role

remarks on the asymptotic behavior of the solution to damped wave equations abstract we study the diffusion phenomena for damped wave equations we prove that the solution of an abstract damped wave equation becomes closer to the solution of a heat type equation as time tends to infinity under some assumption to the resolvent as an application of our approach we also study the asymptotic behavior of the damped wave equation in euclidean space under the geometric control condition

the diffusion phenomenon for dissipative wave equations in metric measure spaces abstract we study the long time behavior of the solution to a type of dissipative wave equation where the operator in the equation is time dependent and the solution is defined on a metric measure space x m satisfying appropriate conditions the operator is assumed to be self adjoint and is related to a time dependent dirichlet form we link hyperbolic pdes with the firmly established theories for parabolic pdes in metric measure spaces and dirichlet forms subsequently deriving the asymptotic behavior of the solution to the dissipative wave equation we present several nontrivial examples of dissipative wave equations in metric measure spaces where our theory works

more statistical properties of order books and price impact we investigate present some new statistical properties of order books we analyse data from the nasdaq and investigate a the statistics of incoming limit order prices b the shape of the average order book and c the typical life time of a limit order as a function of the distance from the best price we also determine the price impact function using french and british stocks and find a logarithmic rather than a power law dependence of the price response on the volume the weak time dependence of the response function shows that the impact is surprisingly quasi permanent and suggests that trading itself is interpreted by the market as new information

decay properties of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u p 1 u 0 t x r r n the behavior of u as t is expected to be the same as that for the corresponding heat equation t p 1 0 t x r r n which has the similarity solution w a t x with the form t 1 p 1 f x t depending on a lim x x 2 p 1 f x 0 provided that p is less than the fujita exponent p c n 1 2 n in this paper as a first step if 1 p p c n and the data u 0 u 1 x decays exponentially as x without smallness condition the solution is shown to decay with rates as t u t l 2 u t l p 1 u t l 2 o t 1 p 1 n 4 t 1 p 1 n 2 p 1 t 1 p 1 1 2 n 4 those of which seem to be reasonable because the similarity solution w a t x has the same decay rates as for the proof the weighted l 2 energy method will be employed with suitable weight similar to that in todorova and yordanov y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489

the interacting gaps model reconciling theoretical and numerical approaches to limit order models we consider the emergence of power law tails in the returns distribution of limit order driven markets we explain a previously observed clash between the theoretical and numerical studies of such models we introduce a solvable model that interpolates between the previous studies and agrees with each of them in the relevant limit c

statistical properties of share volume traded in financial markets we quantitatively investigate the ideas behind the often expressed adage it takes volume to move stock prices and study the statistical properties of the number of shares traded q deltat for a given stock in a fixed time interval deltat we analyze transaction data for the largest 1000 stocks for the two year period 1994 95 using a database that records every transaction for all securities in three major us stock markets we find that the distribution p q deltat displays a power law decay and that the time correlations in q deltat display long range persistence further we investigate the relation between q deltat and the number of transactions n deltat in a time interval deltat and find that the long range correlations in q deltat are largely due to those of n deltat our results are consistent with the interpretation that the large equal time correlation previously found between q deltat and the absolute value of price change g deltat related to volatility are largely due to n deltat