Multiplicative Stochastic Process Research Articles

Analysis and Application of Multiplicative Stochastic Process with a Sample-Dependent Lower Bound

A multiplicative stochastic process with the lower bound lognormally distributed is investigated. For the process, the model is constructed, and its distribution function (involving four parameters) and the related statistical properties are derived. By adjusting the parameters, it is confirmed that the theoretical distribution is consistent with empirical distributions of some real data.

Valuation of Decision Flexibility and Strategic Value in Coal Gasification Projects with the Option-To-Switch between Different Outputs

Coal gasification has been promoted as a sophisticated clean energy technology alternative to coal burning these days. Aside from the usual technical difficulties, economic issues of such projects—especially valuation challenges—are important problems that practitioners usually struggle with. This is because of the major extent of managerial flexibility linked with specific characteristics of coal gasification projects, in particular, possibilities to mothball/restart manufacturing lines, or change between different outputs. The value of such flexibilities may be well assessed by real options valuations. The aim of this paper is to show that for the coal gasification technologies the real options valuation is more suitable than traditional discounted cash flow technique. This approach was applied to calculate an integrated plant that can produce either electricity or methanol. As the valuation approach the multiplicative stochastic process was used. As a consequence, binomial lattices of end-product (electricity and methanol) were developed. Then, in regard to them (reference instruments), two corresponding lattices of net cash flows (consecutive instruments) were created. In the end, two trees of switching option value were developed—one for electricity production as an initial mode, and the second for methanol production, delivering expanded net present (strategic) value.

Stochastic explosion and non-uniqueness for α-Riccati equation

We consider the problem of global in time existence and uniqueness for the initial value problems u′(t)=−u(t)+u2(αt),u(0)=u0≥0 for scaling parameters α≥0 and a large class of initial data u0. Much of the focus is on the “super-critical” case α>1 where by using the multiplicative stochastic cascade techniques we prove global existence for small initial data and finite-time blow-up for large initial data. However, while uniqueness holds for solutions satisfying a growth condition, in general it fails even for arbitrary small initial data. We demonstrate that this lack of uniqueness is directly connected to the stochastic explosion of an associated multiplicative stochastic cascade process. The key tool in establishing the above-mentioned results is an iterative algorithm that allows one to exploit the stochastic explosion of the underlying multiplicative cascade to establish both existence and lack of uniqueness of solutions.

Hard coal project valuation based on real options approach: multiplicative vs. arithmetic stochastic process

Abstract Precise valuation of the economic efficiency of risky investment projects in the mineral sector has a direct impact on the range of future investments. Since the mid-90s, a number of enterprises have also been giving increased attention to the valuation of managerial flexibility that cannot normally be estimated with classical discounted cash flow (DCF) analysis. This has been the result of a development in the real options analysis (ROA) and the simplification of its algorithms, most of which have been achieved through: ♦ incorporating lattice models, ♦ introducing a single uncertain project parameter (gross present value, PV) as an underlying instrument, ♦ assuming that the underlying asset follows the multiplicative stochastic process, ♦ introducing the ‘marketed asset disclaimer’ (MAD) assumption. Unfortunately, in most cases, models constructed on the abovementioned assumptions and modifications are not consistent with real projects. Some analysts recognize that project PVs might not follow the multiplicative process, which could have a direct impact on the project’s value. In order to improve the MAD approach, the paper proposes a modified model where the multiplicative tree is replaced with an additive one. In addition, methods of ‘additive volatility’ calculation and ‘dividend’ adjustments were suggested.

Dynamics of polymer ejection from capsid.

Polymer ejection from a capsid through a nanoscale pore is an important biological process with relevance to modern biotechnology. Here, we study generic capsid ejection using Langevin dynamics. We show that even when the ejection takes place within the drift-dominated region there is a very high probability for the ejection process not to be completed. Introducing a small aligning force at the pore entrance enhances ejection dramatically. Such a pore asymmetry is a candidate for a mechanism by which viral ejection is completed. By detailed high-resolution simulations we show that such capsid ejection is an out-of-equilibrium process that shares many common features with the much studied driven polymer translocation through a pore in a wall or a membrane. We find that the ejection times scale with polymer length, τ ∼ N(α). We show that for the pore without the asymmetry the previous predictions corroborated by Monte Carlo simulations do not hold. For the pore with the asymmetry the scaling exponent varies with the initial monomer density (monomers per capsid volume) ρ inside the capsid. For very low densities ρ ≤ 0.002 the polymer is only weakly confined by the capsid, and we measure α = 1.33, which is close to α=1.4 obtained for polymer translocation. At intermediate densities the scaling exponents α = 1.25 and 1.21 for ρ = 0.01 and 0.02, respectively. These scalings are in accord with a crude derivation for the lower limit α = 1.2. For the asymmetrical pore precise scaling breaks down, when the density exceeds the value for complete confinement by the capsid, ρ ⪆ 0.25. The high-resolution data show that the capsid ejection for both pores, analogously to polymer translocation, can be characterized as a multiplicative stochastic process that is dominated by small-scale transitions.

Stochastic model of Zipf's law and the universality of the power-law exponent.

We propose a stochastic model of Zipf's law, namely a power-law relation between rank and size, and clarify as to why a specific value of its power-law exponent is quite universal. We focus on the successive total of a multiplicative stochastic process. By employing properties of a well-known stochastic process, we concisely show that the successive total follows a stationary power-law distribution, which is directly related to Zipf's law. The formula of the power-law exponent is also derived. Finally, we conclude that the universality of the rank-size exponent is brought about by symmetry between an increase and a decrease in the random growth rate.

Emergence of Heavy-Tailed Distributions in a Random Multiplicative Model Driven by a Gaussian Stochastic Process

We consider a random multiplicative stochastic process with multipliers given by the exponential of a Brownian motion. The positive integer moments of the distribution function can be computed exactly, and can be represented as the grand partition function of an equivalent lattice gas with attractive 2-body interactions. The numerical results for the positive integer moments display a sharp transition at a critical value of the model parameters, which corresponds to a phase transition in the equivalent lattice gas model. The shape of the terminal distribution changes suddenly at the critical point to a heavy-tailed distribution. The transition can be related to the position of the complex zeros of the grand partition function of the lattice gas, in analogy with the Lee, Yang picture of phase transitions in statistical mechanics. We study the properties of the equivalent lattice gas in the thermodynamical limit, which corresponds to the continuous time limit of the random multiplicative model, and derive the asymptotics of the approach to the continuous time limit. The results can be generalized to a wider class of random multiplicative processes, driven by the exponential of a Gaussian stochastic process.

Application of Non-Gaussian Statistics to Heart Rate Variability

Non-Gaussian fluctuations are observed in a wide variety of biomedical time series. To characterize such nonGaussian time series, we introduce a multiplicative stochastic process in which an observed time series is assumed to be described by the multiplication of Gaussian and amplitude random variables. In this framework, we propose an analysis method using log-amplitude cumulants, which is capable of characterizing a wide class of symmetric unimodal distributions with fat tails. As an application of this method, we will discuss non-Gaussian property of heart rate variability.

Event distributions of polymer translocation

We present event distributions for the polymer translocation obtained by extensive Langevin dynamics simulations. Such distributions have not been reported previously and they provide new understanding of the stochastic characteristics of the process. We extract at a high length scale resolution distributions of polymer segments that continuously traverse through a nanoscale pore. The obtained log-normal distributions together with the characteristics of polymer translocation suggest that it is describable as a multiplicative stochastic process. In spite of its clear out-of-equilibrium nature the forced translocation is surprisingly similar to the unforced case. We find forms for the distributions almost unaltered with a common cut-off length. We show that the individual short-segment and short-time movements inside the pore give the scaling relations τ∼Nα and τ∼f-β for the polymer translocation.

Redistribution Spurs Growth by Using a Portfolio Effect on Risky Human Capital

We demonstrate by mathematical analysis and systematic computer simulations that redistribution can lead to sustainable growth in a society. In accordance with economic models of risky human capital, we assume that dynamics of human capital is modeled as a multiplicative stochastic process which, in the long run, leads to the destruction of individual human capital. When agents are linked by fully-redistributive taxation the situation might turn to individual growth in the long run. We consider that a government collects a proportion of income and reduces it by a fraction as costs for administration (efficiency losses). The remaining public good is equally redistributed to all agents. Sustainable growth is induced by redistribution despite the losses from the random growth process and despite administrative costs. Growth results from a portfolio effect. The findings are verified for three different tax schemes: proportional tax, taking proportional more from the rich, and proportionally more from the poor. We discuss which of these tax schemes performs better with respect to maximize growth under a fixed rate of administrative costs, and with respect to maximize the governmental income. This leads us to some general conclusions about governmental decisions, the relation to public good games with free-riding, and the function of taxation in a risk taking society.

The statistical laws of popularity: universal properties of the box-office dynamics of motion pictures

Are there general principles governing the process by which certain products or ideas become popular relative to other (often qualitatively similar) competitors? To investigate this question in detail, we have focused on the popularity of movies as measured by their box-office income. We observe that the log-normal distribution describes well the tail (corresponding to the most successful movies) of the empirical distributions for the total income, the income on the opening week, as well as the weekly income per theater. This observation suggests that popularity may be the outcome of a linear multiplicative stochastic process. In addition, the distributions of the total income and the opening income show a bimodal form, with the majority of movies either performing very well or very poorly in theaters. We also observe that the gross income per theater for a movie at any point during its lifetime is, on average, inversely proportional to the period that has elapsed after its release. We argue that (i) the log-normal nature of the tail, (ii) the bimodal form of the overall gross income distribution and (iii) the decay of gross income per theater with time as a power law, constitute the fundamental set of stylized facts (i.e. empirical ‘laws’) that can be used to explain other observations about movie popularity. We show that, in conjunction with an assumption of a fixed lower cut-off for income per theater below which a movie is withdrawn from a cinema, these laws can be used to derive a Weibull distribution for the survival probability of movies that agrees with empirical data. The connection to extreme-value distributions suggests that popularity can be viewed as a process where a product becomes popular by avoiding ‘failure’ (i.e. being pulled out from circulation) for many successive time periods. We suggest that these results may apply beyond the particular case of movies to popularity in general.

Functional integral approach for multiplicative stochastic processes

We present a functional formalism to derive a generating functional for correlation functions of a multiplicative stochastic process represented by a Langevin equation. We deduce a path integral over a set of fermionic and bosonic variables without performing any time discretization. The usual prescriptions to define the Wiener integral appear in our formalism in the definition of Green's functions in the Grassman sector of the theory. We also study nonperturbative constraints imposed by Becchi, Rouet and Stora symmetry (BRS) and supersymmetry on correlation functions. We show that the specific prescription to define the stochastic process is wholly contained in tadpole diagrams. Therefore, in a supersymmetric theory, the stochastic process is uniquely defined since tadpole contributions cancels at all order of perturbation theory.

A multiplicative stochastic process deriving the probability distribution in exact form

A simple multiplicative stochastic process is proposed in order to clarify directly the origin of the power–law. In this model, the probability distribution of the stochastic variables is calculated in an exact form. The necessary conditions that the multiplicative stochastic process generates the positive–negative asymmetric power–law distribution are investigated. Finally, comparisons between the distribution of stochastic variables simulated numerically and the stable distribution are discussed.

Understanding and Modeling the Internet Topology: Economics and Evolution Perspective

In this paper, we seek to understand the intrinsic reasons for the well-known phenomenon of heavy-tailed degree in the Internet AS graph and argue that in contrast to traditional models based on preferential attachment and centralized optimization, the Pareto degree of the Internet can be explained by the evolution of wealth associated with each ISP. The proposed topology model utilizes a simple multiplicative stochastic process that determines each ISP's wealth at different points in time and several maintenance rules that keep the degree of each node proportional to its wealth. Actual link formation is determined in a decentralized fashion based on random walks, where each ISP individually decides when and how to increase its degree. Simulations show that the proposed model, which we call Wealth-based Internet Topology (WIT), produces scale-free random graphs with tunable exponent α and high clustering coefficients (between 0.35 and 0.5) that stay invariant as the size of the graph increases. This evolution closely mimics that of the Internet observed since 1997.

Large-scale lognormal fluctuations in turbulence velocity fields

For several flows of laboratory turbulence, we obtain long records of velocity data. These records are divided into numerous segments. In each segment, we calculate the mean rate of energy dissipation, the mean energy at each scale, and the mean total energy. Their values fluctuate significantly among the segments. The fluctuations are lognormal, if the segment length lies within the range of large scales where the velocity correlations are weak but not yet absent. Since the lognormality is observed regardless of the Reynolds number and the configuration for turbulence production, it is expected to be universal. The likely origin is some multiplicative stochastic process related to interactions among scales through the energy transfer.

Investments in random environments

We present analytical investigations of a multiplicative stochastic process that models a simple investor dynamics in a random environment. The dynamics of the investor’s budget, x(t), depends on the stochasticity of the return on investment, r(t), for which different model assumptions are discussed. The fat-tail distribution of the budget is investigated and compared with theoretical predictions. We are mainly interested in the most probable value xmp of the budget that reaches a constant value over time. Based on an analytical investigation of the dynamics, we are able to predict xmpstat. We find a scaling law that relates the most probable value to the characteristic parameters describing the stochastic process. Our analytical results are confirmed by stochastic computer simulations that show a very good agreement with the predictions.

Evidence for power-law tail of the wealth distribution in India

The higher-end tail of the wealth distribution in India is studied using recently published lists of the wealth of richest Indians between the years 2002–2004. The resulting rank distribution seems to imply a power-law tail for the wealth distribution, with a Pareto exponent between 0.81 and 0.92 (depending on the year under analysis). This provides a comparison with previous studies of wealth distribution, which have all been confined to Western advanced capitalist economies. We conclude with a discussion on the appropriateness of multiplicative stochastic process as a model for asset accumulation, the relation between the wealth and income distributions (we estimate the Pareto exponent for the latter to be around 1.5 for India), as well as possible sources of error in measuring the Pareto exponent for wealth.

Interpreting wind-driven Southern Ocean variability in a stochastic framework

A stochastic model is derived from wind stress and bottom pressure gauge data to examine the response of the Antarctic Circumpolar Current (ACC) transport to wind stress forcing. A general method is used to estimate the drift and diffusion coefe cients of a continuous stationary Markovian system. As a e rst approximation, the response of the ACC to wind stress forcing can be described by a multivariate Ornstein-Uhlenbeck process: Gaussian red noise wind stress drives the evolution of the ACC transport, which is damped by a linear drag term. The model indicates that about 30( 610)% of ACC variability is directly driven by the wind stress. This stochastic model can serve as a null hypothesis for studies of wind driven ACC variability. A more accurate stochastic description of the wind stress over the Southern Ocean requires a multiplicative noise component. The variability of the wind stress increases approximately linearly with increasing wind stress values. A multiplicative stochastic process generates a power-law distribution rather than a Gaussian distribution. A simple stochastic model shows that non-Gaussian forcing could have a signie cant impact on the velocity (or transport) probability density functions (PDFs) of the wind-driven circulation. The net oceanic damping determines whether the distribution of the oceanic e ow is Gaussian (small damping) or resembles the distribution of the atmospheric forcing (large damping).

Leverage effect in financial markets: the retarded volatility model.

We investigate quantitatively the so-called "leverage effect," which corresponds to a negative correlation between past returns and future volatility. For individual stocks this correlation is moderate and decays over 50 days, while for stock indices it is much stronger but decays faster. For individual stocks the magnitude of this correlation has a universal value that can be rationalized in terms of a new "retarded" model which interpolates between a purely additive and a purely multiplicative stochastic process. For stock indices a specific amplification phenomenon seems to be necessary to account for the observed amplitude of the effect.

More stylized facts of financial markets: leverage effect and downside correlations

We discuss two more universal features of stock markets: the so-called leverage effect (a negative correlation between past returns and future volatility), and the increased downside correlations. For individual stocks, the leverage correlation can be rationalized in terms of a new ‘retarded’ model which interpolates between a purely additive and a purely multiplicative stochastic process. For stock indices a specific market panic phenomenon seems to be necessary to account for the observed amplitude of the effect. As for the increase of correlations in highly volatile periods, we investigate how much of this effect can be explained within a simple non-Gaussian one-factor description with time independent correlations. In particular, this one-factor model can explain the level and asymmetry of empirical exceedance correlations, which reflects the fat-tailed and negatively skewed distribution of market returns.