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.
Our work in the modeling of complex social and economic systems has begun from the applications of nonlinear stochastic differential equations (abbr. SDE) seeking reproduce statistics of financial market data. The proposed class of equations has power law statistics evidently very similar to the ones observed in the empirical data. As all of this work (for broad review see @cite_12 ) was done by relying on the macroscopic phenomenological reasoning, we are now motivated to find the microscopic reasoning for the proposed equations. The development of the macroscopic treatments for the well established agent-based models appears to be the most consistent approach, as the movement in the opposite direction seems to be very complex and ambiguitious task. Thus we decided that we should select the simple agent-based models, which would have an expected macroscopic description. In this contribution we present a few examples of the agent-based modeling, based on the Kirman's model, in the business and finance while showing that the examples have useful and informative macroscopic treatments.
@cite_12
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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