We present examples of agent-based and stochastic models of competition and business processes in economics and finance. We start from as simple as possible models, which have microscopic, agent-based, versions and macroscopic treatment in behavior. Microscopic and macroscopic versions of herding model proposed by Kirman and Bass diffusion of new products are considered in this contribution as two basic ideas. Further we demonstrate that general herding behavior can be considered as a background of nonlinear stochastic model of financial fluctuations.
Another interesting problem tackled in this work is related to the dynamics of the intermittent behavior. This kind of behavior is observed in many different complex systems ranging from the geology (ex., earthquakes @cite_23 ) and astronomy (ex., sunspots @cite_40 ) to the biology (ex., neuron activity @cite_83 ) and finance @cite_68 . Great review of the universality of the bursty behavior is given by @cite_53 and Kleinberg @cite_62 . In @cite_53 the bursting behavior is considered as a point process with threshold mechanism. In this contribution we analyze the class of nonlinear SDE exhibiting power law statistics and bursting behavior, which was derived from the multiplicative point process @cite_65 @cite_33 @cite_82 with applications for the modeling of trading activity in financial markets @cite_74 @cite_66 . This provides a very general, via hitting time formalism @cite_42 @cite_37 @cite_77 , approach to the modeling of bursty behavior of trading activity and absolute return in the financial markets @cite_6 .
@cite_37 @cite_62 @cite_33 @cite_53 @cite_42 @cite_65 @cite_6 @cite_77 @cite_40 @cite_83 @cite_23 @cite_74 @cite_68 @cite_66 @cite_82
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The dynamics of a wide range of real systems, from email patterns to earthquakes, display a bursty, intermittent nature, characterized by short timeframes of intense activity followed by long times of no or reduced activity. The understanding of the origin of such bursty patterns is hindered by the lack of tools to compare different systems using a common framework. Here we propose to characterize the bursty nature of real signals using orthogonal measures quantifying two distinct mechanisms leading to burstiness: the interevent time distribution and the memory. We find that while the burstiness of natural phenomena is rooted in both the interevent time distribution and memory, for human dynamics memory is weak, and the bursty character is due to the changes in the interevent time distribution. Finally, we show that current models lack in their ability to reproduce the activity pattern observed in real systems, opening up avenues for future work. Copyright c �EPLA, 2008 We study the relaxation response of a social system after endogenous and exogenous bursts of activity using the time series of daily views for nearly 5 million videos on YouTube. We find that most activity can be described accurately as a Poisson process. However, we also find hundreds of thousands of examples in which a burst of activity is followed by an ubiquitous power-law relaxation governing the timing of views. We find that these relaxation exponents cluster into three distinct classes and allow for the classification of collective human dynamics. This is consistent with an epidemic model on a social network containing two ingredients: a power-law distribution of waiting times between cause and action and an epidemic cascade of actions becoming the cause of future actions. This model is a conceptual extension of the fluctuation-dissipation theorem to social systems [Ruelle, D (2004) Phys Today 57:48–53] and [Roehner BM, et al, (2004) Int J Mod Phys C 15:809–834], and provides a unique framework for the investigation of timing in complex systems. Volatility clustering, evaluated through slowly decaying auto-correlations, Hurst effect or 1 f noise for absolute returns, is a characteristic property ofmost financial assets return time series (1999). Statistical analysis alone is not able to provide a definite answer for the presence or absence of long-range dependence phenomenon in stock returns or volatility, unless economic mechanisms are proposed to understand the origin of such phenomenon Cont (2005); (1999). Whether results of statistical analysis correspond to longrange dependence is a difficult question and subject to an ongoing statistical debate Cont (2005). Extensive empirical analysis of the financial market data, supporting the idea that the longrange volatility correlations arise from trading activity, provides valuable background for further development of the long-ranged memory stochastic models (2003); (2001). The power-law behavior of the auto-regressive conditional duration process Sato (2004) based on the random multiplicative process and it’s special case the self-modulation process Takayasu (2003), exhibiting 1 f fluctuations, supported the idea of stochastic modeling with a power-law probability density function (PDF) and long-range memory. Thus the agent based economic models Kirman & Teyssiere (2002); Lux & Marchesi (2000) as well as the stochastic models Borland (2004); (2008; 2010); Queiros (2007) exhibiting long-range dependence phenomenon in volatility or trading volume are of great interest and remain an active topic of research. Properties of stochastic multiplicative point processes have been investigated analytically and numerically and the formula for the power spectrum has been derived Gontis & Kaulakys (2004). In themore recent papers (2006); Kaulakys A Ruseckas & Kaulakys (2010) the general form of the multiplicative stochastic differential equation (SDE) was derived in agreement with the model earlier proposed in Gontis & Kaulakys (2004). Since Gontis & Kaulakys (2004) a model of trading activity, based on a SDE driven Poisson-like process, was presented (2008) and in the most recent paper (2010) we proposed a double stochastic model, whose return time series yield two power-law statistics, i.e., the PDF and the power spectral density (PSD) of absolute return, mimicking the empirical data for the one-minute trading return in the NYSE. In this chapter we present theoretical arguments and empirical evidence for the non-linear double stochastic model of return in financial markets. With empirical data from NYSE and Vilnius Stock Exchange (VSE) demonstrating universal scaling of return statistical properties, 27 Inhomogeneous temporal processes, like those appearing in human communications, neuron spike trains, and seismic signals, consist of high-activity bursty intervals alternating with long low-activity periods. In recent studies such bursty behavior has been characterized by a fat-tailed inter-event time distribution, while temporal correlations were measured by the autocorrelation function. However, these characteristic functions are not capable to fully characterize temporally correlated heterogenous behavior. Here we show that the distribution of the number of events in a bursty period serves as a good indicator of the dependencies, leading to the universal observation of power-law distribution for a broad class of phenomena. We find that the correlations in these quite different systems can be commonly interpreted by memory effects and described by a simple phenomenological model, which displays temporal behavior qualitatively similar to that in real systems.