Critical comparison of several order-book models for stock-market fluctuations

Far-from-equilibrium models of interacting particles in one dimension are used as a basis for modelling the stock-market fluctuations. Particle types and their positions are interpreted as buy and sel orders placed on a price axis in the order book. We revisit some modifications of well-known models, starting with the Bak-Paczuski-Shubik model. We look at the four decades old Stigler model and investigate its variants. One of them is the simplified version of the Genoa artificial market. The list of studied models is completed by the models of Maslov and Daniels et al. Generically, in all cases we compare the return distribution, absolute return autocorrelation and the value of the Hurst exponent. It turns out that none of the models reproduces satisfactorily all the empirical data, but the most promising candidates for further development are the Genoa artificial market and the Maslov model with moderate order evaporation.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.     

Enveloppes convexes des processus gaussiens

Let X={X(t), t[0,1]} be a process on [0,1] and V_X=Conv{(t,x)t[0,1], x=X(t)} be the convex hull of its path.The structure of the set ext(V_X) of extreme points of V_X is studied. For a Gaussian process X with stationary increments it is proved that:

• The set ext(V_X) is negligible if X is non-differentiable.

• If X is absolutely continuous process and its derivative X′ is continuous but non-differentiable, then ext(V_X) is also negligible and moreover it is a Cantor set.

It is proved also that these properties are stable under the transformations of the type Y(t)=f(X(t)), if f is a sufficiently smooth function.     

Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula

We prove a uniform bound for the density, p _t (x), of the solution at time t(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p _t (x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al. ⁽⁵⁾     

Stability and Attraction to Normality for Lévy Processes at Zero and at Infinity

We prove some limiting results for a Lévy process X _t as t0 or t, with a view to their ultimate application in boundary crossing problems for continuous time processes. In the present paper we are mostly concerned with ideas related to relative stability and attraction to the normal distribution on the one hand and divergence to large values of the Lévy process on the other. The aim is to find analytical conditions for these kinds of behaviour which are in terms of the characteristics of the process, rather than its distribution. Some surprising results occur, especially for the case t0; for example, we may have X _t /t ^P + (t0) (weak divergence to +), whereas X _t /t a.s. (t0) is impossible (both are possible when t), and the former can occur when the negative Lévy spectral component dominates the positive, in a certain sense. Almost sure stability of X _t , i.e., X _t tending to a nonzero constant a.s. as t or as t0, after normalisation by a non-stochastic measurable function, reduces to the same type of convergence but with normalisation by t, thus is equivalent to strong law behaviour. Boundary crossing problems which are amenable to the methods we develop arise in areas such as sequential analysis and option pricing problems in finance.     

Blow-up of semilinear pde's at the critical dimension. A probabilistic approach

We present a probabilistic approach which proves blow-up of solutions of the Fujita equation in the critical dimension . By using the Feynman-Kac representation twice, we construct a subsolution which locally grows to infinity as . In this way, we cover results proved earlier by analytic methods. Our method also applies to extend a blow-up result for systems proved for the Laplacian case by Escobedo and Levine (1995) to the case of -Laplacians with possibly different parameters .

     

Fuzzy Markov Chains and Decision-Making

Decision-making in an environment of uncertainty and imprecision for real-world problems is a complex task. In this paper it is introduced general finite state fuzzy Markov chains that have a finite convergence to a stationary (may be periodic) solution. The Cesaro average and the -potential for fuzzy Markov chains are defined, then it is shown that the relationship between them corresponds to the Blackwell formula in the classical theory of Markov decision processes. Furthermore, it is pointed out that recurrency does not necessarily imply ergodicity. However, if a fuzzy Markov chain is ergodic, then the rows of its ergodic projection equal the greatest eigen fuzzy set of the transition matrix. Then, the fuzzy Markov chain is shown to be a robust system with respect to small perturbations of the transition matrix, which is not the case for the classical probabilistic Markov chains. Fuzzy Markov decision processes are finally introduced and discussed.     

Mechanistic aspects of the dehydration and dehydrohalogenation of halo-hydroxyformaldoxime conformers. A quantum chemical model study

A detailed exploration of the configurational and conformational space of chloro- and bromo-hydroxyformaldoximes, Xhfaox (X = Cl, Br) has been carried out with the aid of the B3LYP level of density functional theory, using the 6-31G(d,p) basis set. The most stable configuration in each series of the Clhfaox and Brhfaox conformers corresponds to the Z-s-cis, s-trans configuration, while the highest energy Z-(s-trans, s-cis) conformers were found at 7.0(7.6) and 6.0(6.6) kcal mol(-1), respectively, at the B3LYP(QCISD(T)) levels of theory. Saddle points were also located on the PES of the Clhfaox and Brhfaox compounds corresponding to Z-(s-cis, s-cis) conformers at 13.8(14.9) and 13.6(14.6) kcal mol(-1), respectively, at the B3LYP(QCISD(T)) levels. Upon dehydration Xhfaox could afford a number of isomeric CXNO species. The dehydration processes of Xhfaox are predicted to be endothermic, the computed heats of reactions found in the range of 20.5 to 86.2 kcal mol(-1) and 15.9 to 100.4 kcal mol(-1) at the B3LYP and QCISD(T) levels, respectively. The reaction pathways for the addition of water to halo-fulminates yielding the most stable Xhfaox conformers was predicted to be concerted with a single transition structure, but are asynchronous with activation barriers of 32.8 and 43.0 kcal mol(-1) for the chloro- and bromo-derivatives, respectively. The PES governing the isomerization reactions of the CXNO isomers have also been calculated, and possible isomerization pathways have been delineated. Upon dehydrohalogenation the Xhfaox conformers yield hydroxy-isocyanate or hydroxy-fulminate, the former being more stable by 31.8(18.8) kcal mol(-1) at the B3LYP(QCISD(T)) levels of theory. The reaction pathways for the addition of HX to hydroxy-isocyanate were predicted to be slightly exothermic, the heats of reactions being -3.2 and -5.5 kcal mol(-1), respectively, and have to surmount high activation barriers of 39.7 and 35.0 kcal mol(-1), respectively. Similarly, the addition of HX to hydroxy-fulminate was predicted to be much more exothermic, the heats of reactions being -34.7 and -37.3 kcal mol(-1), respectively, and have to surmount much lower activation barriers of only 10.5 and 7.5 kcal mol(-1) respectively, at the B3LYP level. Finally, calculated structures, relative stability, and bonding properties of all stationary points located on the PES of the systems and reactions studied are thoroughly discussed with respect to computed electronic properties.     

Parametric amplification at low-frequency pumping and generation of four-mode entangled states

We show that parametric amplification at low-frequency pumping can be implemented in aperiodic nonlinear photonic crystals. In this process, the intensities of the waves with frequencies higher than the pumping frequency increase with increase in the interaction length as in the case of the standard process of parametric amplification at high-frequency pumping. The process under consideration includes a nondegenerate parametric down-conversion followed by two parametric up-conversions in the same pumping wave. As a result, generation at four new frequencies arises. Quantum analysis of the process demonstrates the presence of entanglement in the states of four modes. Talk presented at the oral issue of J. Russ. Laser Res. dedicated to the memory of Professor Vladimir A. Isakov, Professor Alexander S. Shumovsky, and Professor Andrei V. Vinogradov held in Moscow February 21–22, 2008.