Additive Stochastic Processes Research Articles

Modelling the chernobyl radioactive fallout (II): A multifractal approach in some European countries

This paper deals with the 137Cs cumulative soil deposition measured in some European Countries after the Chernobyl accident. We devised a multifractal model to aid in describing the spatial distribution of radioactivity. The model is based the Fractal Sum of Pulses theory, involving additive stochastic processes. We use, as input source of information, the available data of radioactive deposition measured in some European Countries. The results look promising, since realistic scenarios of environmental pollution are produced.

Modelling the Chernobyl radioactive fallout (I): A fractal approach in Northern Italy

This paper deals with the 137Cs airborne radioactive concentration measured in Northern Italy after the Chernobyl accident. We devised a fractal model to aid in describing the space-time distribution of radioactivity. The model is based on the Fractal Sum of Pulses theory, involving additive stochastic processes. We use, as input source of information, the parametrization of the time trend of radioactive concentration in a few Italian Provinces we calculated in previous work. The results look promising, since realistic scenarios of environmental pollution are produced.

Computing durations for bond portfolios

A lthough most applications of duration we for bond portfolios, duration measures generally we derived for individual securities, not for portfo'ios. Portfolio durations usually are computed by av?raging the durations of the composite individual jecurities obtained by one of two ways: 1) from fornulas based on projected future cash flows and an assumed stochastic process, or 2) empirically as a ,rice elasticity from past data. The literature with one ?xception has considered only portfolios in which all ,ends of equal maturity trade at the same interest .ate, that is, all bonds trade on the same term strucure. This article expands the analysis to portfolios if bonds that trade at different interest rates even hough they have the same maturity, that is, trade )n different term structures, because of, say, differmces in credit quality. For such portfolios, two conlusions emerge: 1) portfolio duration is dependent an the future value of portfolio cash flows at the duation date, and 2) the weights used to obtain the tortfolio duration from the individual durations of .le composite securities are also dependent on the Jture values of the securities at the duration date. l e s e results suggest that portfolio durations ob2ined by applying present value weights to average idividual bond durations, the usual practice, are inorrect and introduce additional stochastic process :sk. For the sake of simplicity, the analysis in this aper is restricted to Macaulay durations and sto.iastic processes that are consistent with Macaulay durations. At the same time, the results can be generalized to include stochastic processes that are consistent with other measures of duration.'

Additive functions and stochastic processes. II

Additive functions and stochastic processes. II

Additive functions and stochastic processes

Additive functions and stochastic processes

Stochastic Optimal Control for Monetary Planning

A method for solving a stochastic optimal control problem, by revising the system random parameters and their covariance matrix, is suggested The method mainly comprises the Bayesian way of updating the reduced form coefficients of an econometric model and their covariance matrix, when implementing optimal control. Such a control problem is then formulated, based on a model for the monetary sector of the Indian economy. The derived plan provides the optimal set of policies and compares the differences in the results under two separate optimisation methods, i.e. deterministic (with additive vector stochastic process) and stochastic (passive adaptive) controls

Statistical-dynamical theory of nonlinear stochastic processes: II. Time-convolutionless projector method in nonequilibrium open systems

A new statistical-dynamical theory of nonlinear stochastic processes in nonequilibrium open systems is presented. It is shown by means of the time-convolutionless projector method that multiplicative type stochastic equations of motion for relevant variables A i ( t ) can always be transformed exactly into additive type (Langevin type) stochastic equations of motion for A i ( t ) and the corresponding master equation for the probability distribution. The Langevin type equation consists of a drift term and a fluctuating force. The fluctuating force is shown to give a new kind of additive stochastic process which has a quite different feature from the ordinary additive processes. The importance of Langevin equations of this type in the multiplicative stochastic process is pointed out from the statistical-dynamical viewpoint. A new cumulant expansion of the master equation in powers of stochastic forces is also found. The application of the dynamic renormalization group method to steady states far from thermal equilibrium is discussed from the statistical-dynamical standpoint developed in the present paper. The present theory is applied to renormalize nonlinear stochastic equations by eliminating the short-wavelength modes and thus to derive nonlinear Langevin equations with renormalized kinetic coefficients and the corresponding master equation. The renormalized coefficients are then used not only to find recursion relations but also to calculate explicitly a cutoff dependence of the kinetic coefficients. As an actual example, fluctuations in the generalized time-dependent Ginzburg-Landau model are investigated near its nonequilibrium tricritical point.

Multiplicative stochastic processes in statistical physics

For a large class of nonlinear stochastic processes with pure multiplicative fluctuations the corresponding time-dependent Fokker-Planck-equation is solved exactly by analytic methods. A universal eigenvalue spectrum and the corresponding set of eigenfunctions are obtained in closed form. The eigenvalue spectrum consists of a discrete as well as a continuous part. To emphasize the significance of the model proposed for the description of more-general stochastic processes the authors investigate its stability with respect to the inclusion of weak additive fluctuations. A discussion of the differences in the static as well as the dynamic behavior of multiplicative and additive stochastic processes is given in detail. It is shown explicitly how internal as well as externally imposed fluctuations can lead to multiplicative stochastic processes. The applications of the results to various fields such as nonlinear optics---subharmonic generation, parametric three-wave mixing, Raman scattering---electronic devices, autocatalytic chemical reactions, and population dynamics are given. In particular, a comparison with recent experiments by S. Kabashima et al., who investigated the statistical properties of electronic parametric oscillators driven by external noise, is carried out.

Laser scattering from droplets: A theory of multiplicative and additive stochastic processes

The scattering function S (Q,ω) for droplets of superfluid helium near absolute zero is calculated. Assuming that the surface tension τ=τ0+τ̃ (t) is fluctuating with correlation strength α, a wave diffusion decay mechanism for a nondissipative system is found. For each lth multipole of droplet shape oscillation, S (Q,ω) contains two Lorentzian peaks at frequency shifts ± (Clτ0)1/2, with a width αCl/2τ0, where Cl is the Rayleigh form factor. Applying Bose statistics, it is found that the magnitude of τ̃ is not greatly different from that of τ0. Estimates of the scattering intensity show that with present day techniques measurements are possible.

A characterization of the families of finite-dimensional distributions associated with countably additive stochastic processes whose sample paths are inD

Associated with any countably additive probability measureP on the well-known Skorohod spaceD is the family {Pα} of P's finite-dimensional distributions. This paper characterizes all such {Pα}.

Which families of finite-dimensional joint distributions are associated with continuous-path, countably additive, stochastic processes?

Necessary and sufficient conditions on a family of finite-dimensional distributions for it to be the family of finite-dimensional joint distributions of a countably additive stochastic process with continuous paths have been given by Bartoszynski in 1962. The present paper gives such conditions when the parameter space is any compact metric space and, in particular, any compact manifold.

Niveaudurchagangszeiten zur charakterisierung sequentieller schätzverfahren

Let θ be a parameter of a homogenous additive stochastic process. In order to get an unbiased and efficient estimator for a function h(v) one has often to use sequential procedures. In this paper we consider processes of the socalled exponential class. We study level crossing times, which characterize certain sequential estimations. It is shown that the family of level crossing times for an increasing sequence of levels is also a process of the exponential class. The density function of the one-dimensional probability distributions of this new process is given Examples and applications conclude the paper.

On almost sure convergence in a finitely additive setting

In their book “How To Gable If You Must”, Dubins and Savage introduced finitely additive stochastic processes in discrete time and they obtained some results about finitely additive probability measures on infinite product spaces. In the paper, “Some Finitely Additive Probability”, Purves and Sudderth showed how to extend these finitely additive probability measures and it thus became possible to consider many of the classical strong convergence theorems. In this paper, we extend many of the classical strong convergence theorems to a finitely additive setting. Since all proofs in this paper are valid for a countably additive setting if we consider the problem on a coordinate representation process, the results in this paper are generalizations of the classical results on such a process. Some examples are also provided for contrasting a finitely additive setting with a countably additive setting.

Signal processing for precise ocean mapping

In this work a bottom return signal model and accompanying signal processor are described for a wide swath bottom mapping system. An incoherent scattering model is employed under the assumptions that the bottom is a random rough surface composed of a large number of independent scatters with spatial correlation distance negligible relative to the ensonified area. The envelope of the signal received from the various spatial directions is modeled as a smooth, nearly Gaussian-shaped function representing the effects of the twoway spatial beam pattern, angle of incidence, and depth corrupted by multiplicative and additive noise stochastic processes. A signal processor is derived which makes use of the a priori information vested in this smooth function to provide a matched filter for the received signal envelope for each spatial direction. Computer simulation results are presented and the performance of the signal processor examined in a qualitative fashion.

Wald's identity and random walk models for neuron firing

A class of probability models for the firing of neurons is introduced and treated analytically. The cell membrane potential is assumed to be a one-dimensional random walk on the continuum; the first passage of the moving boundary triggers a nerve spike, an all-or-none event. The interspike interval distribution of first-passage time is shown to satisfy a Volterra integral equation suitable for numerical evaluation. Explicit solutions as well as their Laplace (Mellin) transforms are obtained for some special cases. The main technique used in this paper is the extension of Wald's fundamental identity of sequential analysis to a wide range of additive stochastic processes with an (eventually) linear boundary function. This identity is also useful in evaluating model parameters in terms of observed firing times, as well as providing a unified exposition of many earlier results.

Wald's identity and random walk models for neuron firing

A class of probability models for the firing of neurons is introduced and treated analytically. The cell membrane potential is assumed to be a one-dimensional random walk on the continuum; the first passage of the moving boundary triggers a nerve spike, an all-or-none event. The interspike interval distribution of first-passage time is shown to satisfy a Volterra integral equation suitable for numerical evaluation. Explicit solutions as well as their Laplace (Mellin) transforms are obtained for some special cases. The main technique used in this paper is the extension of Wald's fundamental identity of sequential analysis to a wide range of additive stochastic processes with an (eventually) linear boundary function. This identity is also useful in evaluating model parameters in terms of observed firing times, as well as providing a unified exposition of many earlier results.

Multiplicative stochastic processes, Fokker-Planck equations, and a possible dynamical mechanism for critical behavior

A derivation of the Fokker-Planck equations for additive stochastic processes is given which involves treating the continuity equation in the configuration space representation of the additive stochastic process as a multiplicative stochastic process. The average of the continuity equation becomes the Fokker-Planck equation. A presentation of the ``multiplicative stochastic, Markov approximation'' follows. This approximation is applied to the analysis of the dynamics of a heavy particle in a molecular fluid as described by Hamilton's equations. The nonperturbative approximation technique leads to the Fokker-Planck equation for simple Brownian motion. As part of the analysis, ``intrinsic diffusion'' is discovered and used to show ergodicity for the autocorrelation formula which appears during the Brownian motion calculation. An account of how these methods might be used to study the dynamical origins of critical behavior is given.

Signal detection and extraction by cepstrum techniques

A technique for decomposing a composite signal of unknown multiple wavelets overlapping in time is described. The computation algorithm incorporates the power cepstrum and complex cepstrum techniques. It has been found that the power cepstrum is most efficient in recognizing wavelet arrival times and amplitudes while the complex cepstrum is invaluable in estimating the form of the basic wavelet and its echoes, even if the latter are distorted. Digital data-processing problems such as the detection of multiple echoes, various methods of linear filtering the complex cepstrum, the picket-fence phenomenon, minimum-maximum phase situations, and amplitude- versus phase-smoothing for the additive-noise case are examined empirically and where possible theoretically, and are discussed. A similar investigation is performed for some of the preceding problems when the echo or echoes are distorted versions of the wavelet, thereby giving some insight into the complex problem of separating a composite signal composed of several additive stochastic processes. The threshold results are still empirical and the results should be extended to multi-dimensional data. Applications are the decomposition or resolution of signals (e.g., echoes) in radar and sonar, seismology, speech, brain waves, and neuroelectric spike data. Examples of results are presented for decomposition in the absence and presence of noise for specified signals. Results are tendered for the decomposition of pulse-type data appropriate to many systems and for the decomposition of brain waves evoked by visual stimulation.

Contributions to the Theory of Multiplicative Stochastic Processes

The theory of multiplicative stochastic processes is contrasted with the theory of additive stochastic processes. The case of multiplicative factors which are purely random, Gaussian, stochastic processes is treated in detail. In a spirit originally introduced by theoretical work in nuclear magnetic resonance and greatly extended by Kubo, dissipative behavior is demonstrated, on the average, for dynamical equations which do not show dissipative behavior without averaging. It is suggested that multiplicative stochastic processes lead to a conceptual foundation for nonequilibrium thermodynamics and nonequilibrium statistical mechanics, of marked generality.

Finitely Additive Set Functions and Stochastic Processes

Finitely Additive Set Functions and Stochastic Processes