Random walk of annihilating particles on the ring

We obtain the formula for the transition probability of an arbitrary number of annihilating particles wandering along a ring-shaped one-dimensional lattice from the initial coordinates to the final coordinates during an arbitrary number of discrete time steps. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 488–498, March, 2006.     

Random allocation for a class of polya plans

We consider N + 1-dimensional Polya random walks with stopping limit described by a symmetric function of N coordinates. For such plans we obtain the asymptotic distribution of the number of exits realized r times.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 130–143, 1988.     

Local Regularity of Nonsmooth Wavelet Expansions and Application to the Polya Function

We give criteria of pointwise regularity for expansions on Haar or Schauder basis (or spline-type wavelets) corresponding to large Hölder exponents. As an application, we determine the exact Hölder regularity of the Polya function at every point and show that it is multifractal.     

The equivalence between the Schur order and the stochastic orders generated by the scheme of allocation of particles into cells

In 1982, it was proved that the Schur partial-order relation on the set of distributions on {1, 2, ...} is equivalent to the order relation generated by the number of nonempty cell distributions in the scheme of independent allocation of particles into cells. Here it is shown that the same partialorder relation is generated by distributions of numbers of cells occupied by at least r particles for any r ≥ 2.     

Distributional analysis of a generalization of the Polya process

A nonhomogeneous birth process generalizing the Polya process is analyzed, and the distribution of the transition probabilities is shown to be the convolution of a negative binomial distribution and a compound Poisson distribution, whose secondary distribution is a mixture of zero-truncated geometric distributions. A simplified form of the secondary distribution is obtained when the transition intensities have a particular structure, and may sometimes be expressed in terms of Stirling numbers and special functions such as the incomplete gamma function, the incomplete beta function, and the exponential integral. Conditions under which the compound Poisson form of the marginal distributions may be improved to a geometric mixture are also given.     

The distribution of points on the sphere and corresponding cubature formulae

In applications, for instance in optics and astrophysics, thereis a need for high-accuracy integration formulae for functionson the sphere. To construct better formulae than previouslyused, almost equidistantly spaced nodes on the sphere and weightsbelonging to these nodes are required. This problem is closelyrelated to an optimal dispersion problem on the sphere and tothe theories of spherical designs and multivariate Gauss quadratureformulae. We propose a two-stage algorithm to compute optimal point locationson the unit sphere and an appropriate algorithm to calculatethe corresponding weights of the cubature formulae. Points aswell as weights are computed to high accuracy. These algorithmscan be extended to other integration problems. Numerical examplesshow that the constructed formulae yield impressively smallintegration errors of up to 10^-12.     

Product allocation to different types of distribution center in retail logistics networks

We consider the problem of assigning stockkeeping units to distribution centers (DCs) belonging to different DC types of a retail network, e.g., central, regional, and local DCs. The problem is motivated by the real situation of a retail company and solved by an MIP solution approach. The MIP model reflects the interdependencies between inbound transportation, outbound transportation and instore logistics as well as capital tied up in inventories and differences in picking costs between the warehouses. A novel solution approach is developed and applied to a real-life case of a leading European grocery retail chain. The application of the new approach results in cost savings of 6% of total operational costs compared to the present assignment. These savings amount to several million euros per year. In-depth analyses of the results and sensitivity analyses provide insights into the solution structure and the major related issues.     

Time required to unify all particles in the scheme of equiprobable allocation into a sequence of cell layers

We study the scheme of equiprobable allocations of particles into a sequence of cell layers, where the particles put into the same cell are considered as a single particle. We present conditions under which there exist, with positive probability, nonunified particles at each of the layers. For the case in which the number of cells at each of the layers is equal to the number of original particles, we prove the limit theorem for the time instant at which all the particles are unified into a single particle.     

Tracking models and the optimal regret distribution in asset allocation

We present a tracking model for asset allocation that tracks desired investment goals. The model is shown to be optimal with respect to an investor's ‘regret distribution’, the cumulative distribution of the difference between the revenue under perfect foresight and that possible without foresight. Relationships with Markowitz mean/variance models are also explored.     

Estimate for the accuracy of the Poisson approximation for the number of empty cells in an equiprobable scheme for group allocation of particles, and applications

The properties of the distribution of the number of empty cells are analyzed for a natural generalization of an equiprobable scheme for group allocation of particles. An error estimate is obtained for the Chen-Stein method of Poisson approximation for the distribution of the number of empty cells in this scheme. This estimate is used to derive sufficient conditions for the distribution of the number of empty cells to converge to the convolutions of the Poisson distribution and two-point distributions. On the basis of these results, asymptotic properties of the solution set of a perturbed system of linear Boolean equations are studied (in the case of consistent increase in the number of unknowns and the number of equations).     

Minimum of the modulus of an entire function on the sequence of Polya peaks

Let f be an entire function of genus zero, let (r_k) be a sequence of Pólya peaks for N(r, f) of order <1. Then there exists a sequence r _k ~r_k such that If for (r_k) one takes a sequence of Pólya peaks for ln M(r, f) or for T(r, f), 1/2<<1, then the result ceases to be true.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 45, pp. 26–40, 1986.     

On quasiconformal mappings corresponding to the beltrami equation

By using methods of integral equations, we investigate problems of conformal and quasiconformal mappings of close domains. Institute of Computational Mathematics, Georgian Academy of Sciences, Tbilisi. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1391–1397, October, 1999.     

Sharp distribution free lower bounds for spread options and the corresponding optimal subreplicating portfolios

We derive in closed form distribution free lower bounds and optimal subreplicating strategies for spread options in a one-period static arbitrage setting. In the case of a continuum of strikes, we complement the optimal lower bound for spread options obtained in [Rapuch, G., Roncalli, T., 2002. Pricing multiasset options and credit derivatives with copula, Credit Lyonnais, Working Papers] by describing its corresponding subreplicating strategy. This result is explored numerically in a Black-Scholes and in a CEV setting. In the case of discrete strikes, we solve in closed form the optimization problem in which, for each asset S₁ and S₂, forward prices and the price of one option are used as constraints on the marginal distributions of each asset. We provide a partial solution in the case where the marginal distributions are constrained by two strikes per asset. Numerical results on real NYMEX (New York Mercantile Exchange) crack spread option data show that the one discrete lower bound can be far and also very close to the traded price. In addition, the one strike closed form solution is very close to the two strike.     

Random distribution kernels and three types of defaultable contingent payoffs

We introduce the random distribution kernel on a product probability space and obtain the representation results connecting the product and base probability spaces. Using the random variable with the random distribution kernel to model default/death time, we then consider three types of defaultable contingent payoffs. By allowing the survival conditioning time to be anytime before the start time of the payoffs, between the start time and end time, or after the end time of the payoffs, we provide the complete treatment of three types of defaultable contingent payoffs. As the application of the general results developed in this paper, we also provide the more general results for three types of defaultable contingent payoffs than the ones in the literature under the stochastic intensity framework.     

The convergence to a Gaussian process of the number of empty cells in the classical problem of distributing particles among cells

We consider a case in which n particles are distributed independently of one another in N cells. We examine the behavior of the number of empty cells, ₀ (n), as a random function of the parameter n when n, N. We prove that for suitable variation of the time parameter, ₀ (n) will converge to a Gaussian process in the following cases: a) n/N, n/N^– l _nN; b) n/N0, n²/N.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp, 97–104, July, 1968.