A novel version of the Continuous-Time Random Walk (CTRW) model with memory is developed. This me... more A novel version of the Continuous-Time Random Walk (CTRW) model with memory is developed. This memory means the dependence between arbitrary number of successive jumps of the process, while waiting times between jumps are considered as i.i.d. random variables. The dependence was found by analysis of empirical histograms for the stochastic process of a single share price on a market within the high frequency time scale, and justified theoretically by considering bid-ask bounce mechanism containing some delay characteristic for any double-auction market. Our model turns out to be exactly analytically solvable, which enables a direct comparison of its predictions with their empirical counterparts, for instance, with empirical velocity autocorrelation function. Thus this paper significantly extends the capabilities of the CTRW formalism.
Social, technological, and economic time series are divided by events which are usually assumed t... more Social, technological, and economic time series are divided by events which are usually assumed to be random, albeit with some hierarchical structure. It is well known that the interevent statistics observed in these contexts differs from the Poissonian profile by being long-tailed distributed with resting and active periods interwoven. Understanding mechanisms generating consistent statistics has therefore become a central issue. The approach we present is taken from the continuous-time random-walk formalism and represents an analytical alternative to models of nontrivial priority that have been recently proposed. Our analysis also goes one step further by looking at the multifractal structure of the interevent times of human decisions. We here analyze the intertransaction time intervals of several financial markets. We observe that empirical data describe a subtle multifractal behavior. Our model explains this structure by taking the pausing-time density in the form of a superstatistics where the integral kernel quantifies the heterogeneous nature of the executed tasks. A stretched exponential kernel provides a multifractal profile valid for a certain limited range. A suggested heuristic analytical profile is capable of covering a broader region.
In this work we empirically verify the generic breaking of the Central Limit Theorem on the finan... more In this work we empirically verify the generic breaking of the Central Limit Theorem on the financial and commodity markets. We analysed the distributions of log-returns for typical indices and price of gold, for increasing time horizons. We considered Random Coarse Graining Transformation of the Continuous-Time Random Walk model, which can represent the non-Gaussian price dynamics of underlying assets and the corresponding derivatives, e.g., various options or future contracts. We confirmed that empirical data and predictions of the model quite well agree.
The three-state agent-based 2D model of nancial markets in the version proposed by Giulia Iori in... more The three-state agent-based 2D model of nancial markets in the version proposed by Giulia Iori in 2002 has been herein extended. We have introduced the increase of herding behaviour by modelling the altering trust of an agent in his nearest neighbours. The trust increases if the neighbour has foreseen the price change correctly and the trust decreases in the opposite case. Our version only slightly increases the number of parameters present in the Iori model. This version well reproduces the main stylized facts observed on nancial markets. That is, it reproduces log-returns clustering, fat-tail log-returns distribution and power-law decay in time of the volatility autocorrelation function.
Fractional Market Model and Its Verification on the Warsaw Stock Exchange
International Journal of Modern Physics C, Mar 1, 2008
We analyzed the rising and relaxation of the cusp-like local peaks superposed with oscillations w... more We analyzed the rising and relaxation of the cusp-like local peaks superposed with oscillations which were well defined by the Warsaw Stock Exchange index WIG in a daily time horizon. We found that the falling paths of all index peaks were described by a generalized exponential function or the Mittag-Leffler (ML) one superposed with various types of oscillations. However, the rising paths (except the first one of WIG which rises exponentially and the most important last one which rises again according to the ML function) can be better described by bullish anti-bubbles or inverted bubbles.2–4 The ML function superposed with oscillations is a solution of the nonhomogeneous fractional relaxation equation which defines here our Fractional Market Model (FMM) of index dynamics which can be also called the Rheological Model of Market. This solution is a generalized analog of an exactly solvable fractional version of the Standard or Zener Solid Model of viscoelastic materials commonly used in modern rheology.5 For example, we found that the falling paths of the index can be considered to be a system in the intermediate state lying between two complex ones, defined by short and long-time limits of the Mittag-Leffler function; these limits are given by the Kohlrausch-Williams-Watts (KWW) law for the initial times, and the power-law or the Nutting law for asymptotic time. Some rising paths (i.e., the bullish anti-bubbles) are a kind of log-periodic oscillations of the market in the bullish state initiated by a crash. The peaks of the index can be viewed as precritical or precrash ones since: (i) the financial market changes its state too early from the bullish to bearish one before it reaches a scaling region (defined by the diverging power-law of return per unit time), and (ii) they are affected by a finite size effect. These features could be a reminiscence of a significant risk aversion of the investors and their finite number, respectively. However, this means that the scaling region (where the relaxations of indexes are described by the KWW law or stretched exponential decay) was not observed. Hence, neither was the power-law of the instantaneous returns per unit time observed. Nevertheless, criticality or crash is in a natural way contained in our FMM and we found its "finger print".
Modern technology and innovations are becoming more crucial than ever for the survival of compani... more Modern technology and innovations are becoming more crucial than ever for the survival of companies in the market. Therefore, it is significant both from theoretical and practical points of view to understand how governments can influence technology growth and innovation diffusion (TGID) processes. We propose a simple but essential extension of Ausloos-Clippe-Pȩkalski and related Cichy numerical models of the TGID in the market. Both models are inspired by the nonlinear nonequilibrium statistical physics. Our extension involves a parameter describing the probability of government intervention in the TGID process in the company market. We show, using Monte Carlo simulations, the effects interventionism can have on the companies' market, depending on the segment of firms that are supported. The high intervention probability can result, paradoxically, in the destabilization of the market development. It lowers the market's technology level in the longtime limit compared to markets with a lower intervention parameter. We found that the intervention in the technologically weak and strong segments of the company market does not substantially influence the market dynamics, compared to the intervention helping the middle-level companies. However, this is still a simple model which can be extended further and made more realistic by including other factors. Namely, the cost and risk of innovation or limited government resources and capabilities to support companies.
This paper presents an exactly solvable (by applying the fractional calculus) the rheological mod... more This paper presents an exactly solvable (by applying the fractional calculus) the rheological model of fractional dynamics of financial market conformed to the principle of no arbitrage present on financial market. The rheological model of fractional dynamics of financial market describes some singular, empirical, speculative daily peaks of stock market indices, which define crashes as a kind of phase transition. In the frame of the model the plastic market hypothesis and financial uncertainty principle were formulated, which proposed possible scenarios of some market crashes. The brief presentation of the model was made in our earlier work (and references therein). The rheological model of fractional dynamics of financial market is a deterministic model and it is complementary to already existing other ones; together with them it offers possibility for thorough and widespread technical analysis of crashes. The constitutive, fractional integral equation of the model is an analogy of the corresponding one, which defines the fractional Zener model of plastic material. The fractional Zener model is the canonical one for modern rheology, polymer physics and biophysics concerning non-Debye relaxation of viscoelastic biopolymers. The useful approximate solution of the constitutive equation of the rheological model of fractional dynamics of financial market consists of two parts: (i) the first one connected with long-term memory present in the system, which is proportional to the generalized exponential function defined by the Mittag-Leffler function and (ii) the second one describing oscillations (e.g. beats or oscillations having two slightly shifted frequences). The shape exponent leading the Mittag-Leffler function, defines here the order of the phase transition between bullish and bearish states of the financial market, in particular, for recent hossa and bessa on some small, middle and large stock markets. It happened that this solution also successfully estimated some long-term price dynamics on the hypothetical market in United States.
In this work we compared the empirical data of annual income of Polish and European households as... more In this work we compared the empirical data of annual income of Polish and European households as well as annual income of individuals in United States (e.g. for years 2006 and 2008) with predictions of the most popular theoretical models. Particularly good agreements with Pareto distribution and prediction of the Yakovenko model were obtained. For the low society class well agreement with prediction of the cumulative exponential distribution was gained. However, it turned out that the cumulative distribution of annual income of Polish households can be described quite well by the Generalised Lotka-Volterra model.
'J Phys. Chem. Solids (to be published)' should be cited in references to material quoted from th... more 'J Phys. Chem. Solids (to be published)' should be cited in references to material quoted from this section prior to the publication of the relevant article.
Is Discretization of the Stochastic Continuous-Time Processes a Reason for the Non-Linear Long-Term Autocorrelations Observed in High-Frequency Financial Time-Series? ∗
Monte Carlo Simulations of Lattice Gases Exhibiting Quantum Statistical Distributions
International Journal of Modern Physics C, Mar 1, 1991
A non-interacting lattice gas with order preservation in a constant external field is studied by ... more A non-interacting lattice gas with order preservation in a constant external field is studied by numerical and analytical methods. The equilibrium distribution is of the Bose-Einstein type. If additional hard-core repulsion is imposed, it becomes a distribution of Fermi-Dirac type. When relaxing the order preservation condition the classical Boltzmann distribution is recovered.
Using the empirical data from the Norwegian tax office, we analyse the wealth and income of the r... more Using the empirical data from the Norwegian tax office, we analyse the wealth and income of the richest individuals in Norway during the period 2010-2013. We find that both annual income and wealth level of the richest individuals are describable using the Pareto law. We find that the robust mean Pareto exponent over the four-year period to be ≈ 2.3 for income and ≈ 1.5 for wealth.
We found a unified formula for description of the household incomes of all society classes, for i... more We found a unified formula for description of the household incomes of all society classes, for instance, for the European Union in years 2005-2010. The formula is more general than well known that of Yakovenko et al. because, it satisfactorily describes not only the household incomes of low-and medium-income society classes but also the household incomes of the high-income society class. As a stricking result, we found that the high-income society class almost disappeared in year 2009, in opposite to situation in remaining years, where this class played a significant role.
Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on ... more Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on the novel coronavirus COVID-19. The COVID-19 resource centre is hosted on Elsevier Connect, the company's public news and information website. Elsevier hereby grants permission to make all its COVID-19-related research that is available on the COVID-19 resource centre-including this research content-immediately available in PubMed Central and other publicly funded repositories, such as the WHO COVID database with rights for unrestricted research re-use and analyses in any form or by any means with acknowledgement of the original source. These permissions are granted for free by Elsevier for as long as the COVID-19 resource centre remains active.
Extreme events as foundation of Lévy walks with varying velocity
Chemical Physics, Nov 1, 2002
In this work we study the role of extreme events [E.W. Montroll, B.J. West, in: J.L. Lebowitz, E.... more In this work we study the role of extreme events [E.W. Montroll, B.J. West, in: J.L. Lebowitz, E.W. Montrell (Eds.), Fluctuation Phenomena, SSM, vol. VII, North-Holland, Amsterdam, 1979, p. 63; J.-P. Bouchaud, M. Potters, Theory of Financial Risks from Statistical Physics to Risk Management, Cambridge University Press, Cambridge, 2001; D. Sornette, Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and
The authors investigate the random walk of a particle on a one-dimensional chain which has been c... more The authors investigate the random walk of a particle on a one-dimensional chain which has been constructed by a random-walk procedure. Exact expressions are given for the mean-square displacement and the fourth moment after n steps. The probability density after n steps is derived in the saddle-point approximation, for large n. These quantities have also been studied by numerical simulation. The extension to continuous time has been made where the particle jumps according to a Poisson process. The exact solution for the self-correlation function has been obtained in the Fourier and Laplace domain. The resulting frequency-dependent diffusion coefficient and incoherent dynamical structure factor have been discussed. The model of random walk on a random walk is applied to self-diffusion in the concentrated one-dimensional lattice gas where the correct asymptotic behavior is found.
A novel version of the Continuous-Time Random Walk (CTRW) model with memory is developed. This me... more A novel version of the Continuous-Time Random Walk (CTRW) model with memory is developed. This memory means the dependence between arbitrary number of successive jumps of the process, while waiting times between jumps are considered as i.i.d. random variables. The dependence was found by analysis of empirical histograms for the stochastic process of a single share price on a market within the high frequency time scale, and justified theoretically by considering bid-ask bounce mechanism containing some delay characteristic for any double-auction market. Our model turns out to be exactly analytically solvable, which enables a direct comparison of its predictions with their empirical counterparts, for instance, with empirical velocity autocorrelation function. Thus this paper significantly extends the capabilities of the CTRW formalism.
Social, technological, and economic time series are divided by events which are usually assumed t... more Social, technological, and economic time series are divided by events which are usually assumed to be random, albeit with some hierarchical structure. It is well known that the interevent statistics observed in these contexts differs from the Poissonian profile by being long-tailed distributed with resting and active periods interwoven. Understanding mechanisms generating consistent statistics has therefore become a central issue. The approach we present is taken from the continuous-time random-walk formalism and represents an analytical alternative to models of nontrivial priority that have been recently proposed. Our analysis also goes one step further by looking at the multifractal structure of the interevent times of human decisions. We here analyze the intertransaction time intervals of several financial markets. We observe that empirical data describe a subtle multifractal behavior. Our model explains this structure by taking the pausing-time density in the form of a superstatistics where the integral kernel quantifies the heterogeneous nature of the executed tasks. A stretched exponential kernel provides a multifractal profile valid for a certain limited range. A suggested heuristic analytical profile is capable of covering a broader region.
In this work we empirically verify the generic breaking of the Central Limit Theorem on the finan... more In this work we empirically verify the generic breaking of the Central Limit Theorem on the financial and commodity markets. We analysed the distributions of log-returns for typical indices and price of gold, for increasing time horizons. We considered Random Coarse Graining Transformation of the Continuous-Time Random Walk model, which can represent the non-Gaussian price dynamics of underlying assets and the corresponding derivatives, e.g., various options or future contracts. We confirmed that empirical data and predictions of the model quite well agree.
The three-state agent-based 2D model of nancial markets in the version proposed by Giulia Iori in... more The three-state agent-based 2D model of nancial markets in the version proposed by Giulia Iori in 2002 has been herein extended. We have introduced the increase of herding behaviour by modelling the altering trust of an agent in his nearest neighbours. The trust increases if the neighbour has foreseen the price change correctly and the trust decreases in the opposite case. Our version only slightly increases the number of parameters present in the Iori model. This version well reproduces the main stylized facts observed on nancial markets. That is, it reproduces log-returns clustering, fat-tail log-returns distribution and power-law decay in time of the volatility autocorrelation function.
Fractional Market Model and Its Verification on the Warsaw Stock Exchange
International Journal of Modern Physics C, Mar 1, 2008
We analyzed the rising and relaxation of the cusp-like local peaks superposed with oscillations w... more We analyzed the rising and relaxation of the cusp-like local peaks superposed with oscillations which were well defined by the Warsaw Stock Exchange index WIG in a daily time horizon. We found that the falling paths of all index peaks were described by a generalized exponential function or the Mittag-Leffler (ML) one superposed with various types of oscillations. However, the rising paths (except the first one of WIG which rises exponentially and the most important last one which rises again according to the ML function) can be better described by bullish anti-bubbles or inverted bubbles.2–4 The ML function superposed with oscillations is a solution of the nonhomogeneous fractional relaxation equation which defines here our Fractional Market Model (FMM) of index dynamics which can be also called the Rheological Model of Market. This solution is a generalized analog of an exactly solvable fractional version of the Standard or Zener Solid Model of viscoelastic materials commonly used in modern rheology.5 For example, we found that the falling paths of the index can be considered to be a system in the intermediate state lying between two complex ones, defined by short and long-time limits of the Mittag-Leffler function; these limits are given by the Kohlrausch-Williams-Watts (KWW) law for the initial times, and the power-law or the Nutting law for asymptotic time. Some rising paths (i.e., the bullish anti-bubbles) are a kind of log-periodic oscillations of the market in the bullish state initiated by a crash. The peaks of the index can be viewed as precritical or precrash ones since: (i) the financial market changes its state too early from the bullish to bearish one before it reaches a scaling region (defined by the diverging power-law of return per unit time), and (ii) they are affected by a finite size effect. These features could be a reminiscence of a significant risk aversion of the investors and their finite number, respectively. However, this means that the scaling region (where the relaxations of indexes are described by the KWW law or stretched exponential decay) was not observed. Hence, neither was the power-law of the instantaneous returns per unit time observed. Nevertheless, criticality or crash is in a natural way contained in our FMM and we found its "finger print".
Modern technology and innovations are becoming more crucial than ever for the survival of compani... more Modern technology and innovations are becoming more crucial than ever for the survival of companies in the market. Therefore, it is significant both from theoretical and practical points of view to understand how governments can influence technology growth and innovation diffusion (TGID) processes. We propose a simple but essential extension of Ausloos-Clippe-Pȩkalski and related Cichy numerical models of the TGID in the market. Both models are inspired by the nonlinear nonequilibrium statistical physics. Our extension involves a parameter describing the probability of government intervention in the TGID process in the company market. We show, using Monte Carlo simulations, the effects interventionism can have on the companies' market, depending on the segment of firms that are supported. The high intervention probability can result, paradoxically, in the destabilization of the market development. It lowers the market's technology level in the longtime limit compared to markets with a lower intervention parameter. We found that the intervention in the technologically weak and strong segments of the company market does not substantially influence the market dynamics, compared to the intervention helping the middle-level companies. However, this is still a simple model which can be extended further and made more realistic by including other factors. Namely, the cost and risk of innovation or limited government resources and capabilities to support companies.
This paper presents an exactly solvable (by applying the fractional calculus) the rheological mod... more This paper presents an exactly solvable (by applying the fractional calculus) the rheological model of fractional dynamics of financial market conformed to the principle of no arbitrage present on financial market. The rheological model of fractional dynamics of financial market describes some singular, empirical, speculative daily peaks of stock market indices, which define crashes as a kind of phase transition. In the frame of the model the plastic market hypothesis and financial uncertainty principle were formulated, which proposed possible scenarios of some market crashes. The brief presentation of the model was made in our earlier work (and references therein). The rheological model of fractional dynamics of financial market is a deterministic model and it is complementary to already existing other ones; together with them it offers possibility for thorough and widespread technical analysis of crashes. The constitutive, fractional integral equation of the model is an analogy of the corresponding one, which defines the fractional Zener model of plastic material. The fractional Zener model is the canonical one for modern rheology, polymer physics and biophysics concerning non-Debye relaxation of viscoelastic biopolymers. The useful approximate solution of the constitutive equation of the rheological model of fractional dynamics of financial market consists of two parts: (i) the first one connected with long-term memory present in the system, which is proportional to the generalized exponential function defined by the Mittag-Leffler function and (ii) the second one describing oscillations (e.g. beats or oscillations having two slightly shifted frequences). The shape exponent leading the Mittag-Leffler function, defines here the order of the phase transition between bullish and bearish states of the financial market, in particular, for recent hossa and bessa on some small, middle and large stock markets. It happened that this solution also successfully estimated some long-term price dynamics on the hypothetical market in United States.
In this work we compared the empirical data of annual income of Polish and European households as... more In this work we compared the empirical data of annual income of Polish and European households as well as annual income of individuals in United States (e.g. for years 2006 and 2008) with predictions of the most popular theoretical models. Particularly good agreements with Pareto distribution and prediction of the Yakovenko model were obtained. For the low society class well agreement with prediction of the cumulative exponential distribution was gained. However, it turned out that the cumulative distribution of annual income of Polish households can be described quite well by the Generalised Lotka-Volterra model.
'J Phys. Chem. Solids (to be published)' should be cited in references to material quoted from th... more 'J Phys. Chem. Solids (to be published)' should be cited in references to material quoted from this section prior to the publication of the relevant article.
Is Discretization of the Stochastic Continuous-Time Processes a Reason for the Non-Linear Long-Term Autocorrelations Observed in High-Frequency Financial Time-Series? ∗
Monte Carlo Simulations of Lattice Gases Exhibiting Quantum Statistical Distributions
International Journal of Modern Physics C, Mar 1, 1991
A non-interacting lattice gas with order preservation in a constant external field is studied by ... more A non-interacting lattice gas with order preservation in a constant external field is studied by numerical and analytical methods. The equilibrium distribution is of the Bose-Einstein type. If additional hard-core repulsion is imposed, it becomes a distribution of Fermi-Dirac type. When relaxing the order preservation condition the classical Boltzmann distribution is recovered.
Using the empirical data from the Norwegian tax office, we analyse the wealth and income of the r... more Using the empirical data from the Norwegian tax office, we analyse the wealth and income of the richest individuals in Norway during the period 2010-2013. We find that both annual income and wealth level of the richest individuals are describable using the Pareto law. We find that the robust mean Pareto exponent over the four-year period to be ≈ 2.3 for income and ≈ 1.5 for wealth.
We found a unified formula for description of the household incomes of all society classes, for i... more We found a unified formula for description of the household incomes of all society classes, for instance, for the European Union in years 2005-2010. The formula is more general than well known that of Yakovenko et al. because, it satisfactorily describes not only the household incomes of low-and medium-income society classes but also the household incomes of the high-income society class. As a stricking result, we found that the high-income society class almost disappeared in year 2009, in opposite to situation in remaining years, where this class played a significant role.
Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on ... more Elsevier has created a COVID-19 resource centre with free information in English and Mandarin on the novel coronavirus COVID-19. The COVID-19 resource centre is hosted on Elsevier Connect, the company's public news and information website. Elsevier hereby grants permission to make all its COVID-19-related research that is available on the COVID-19 resource centre-including this research content-immediately available in PubMed Central and other publicly funded repositories, such as the WHO COVID database with rights for unrestricted research re-use and analyses in any form or by any means with acknowledgement of the original source. These permissions are granted for free by Elsevier for as long as the COVID-19 resource centre remains active.
Extreme events as foundation of Lévy walks with varying velocity
Chemical Physics, Nov 1, 2002
In this work we study the role of extreme events [E.W. Montroll, B.J. West, in: J.L. Lebowitz, E.... more In this work we study the role of extreme events [E.W. Montroll, B.J. West, in: J.L. Lebowitz, E.W. Montrell (Eds.), Fluctuation Phenomena, SSM, vol. VII, North-Holland, Amsterdam, 1979, p. 63; J.-P. Bouchaud, M. Potters, Theory of Financial Risks from Statistical Physics to Risk Management, Cambridge University Press, Cambridge, 2001; D. Sornette, Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and
The authors investigate the random walk of a particle on a one-dimensional chain which has been c... more The authors investigate the random walk of a particle on a one-dimensional chain which has been constructed by a random-walk procedure. Exact expressions are given for the mean-square displacement and the fourth moment after n steps. The probability density after n steps is derived in the saddle-point approximation, for large n. These quantities have also been studied by numerical simulation. The extension to continuous time has been made where the particle jumps according to a Poisson process. The exact solution for the self-correlation function has been obtained in the Fourier and Laplace domain. The resulting frequency-dependent diffusion coefficient and incoherent dynamical structure factor have been discussed. The model of random walk on a random walk is applied to self-diffusion in the concentrated one-dimensional lattice gas where the correct asymptotic behavior is found.
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