%0 Book Section %1 statphys23_1037 %A Podobnik, B.P. Boris %A Stanley, H.E. %A Grosse, I.G. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K analysis econophysics modeling random series statphys23 stochastic time topic-11 walks %T A Stochastic Process with a Size-Dependent Standard Deviation for Growth Rates %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1037 %X Stationary and nonstationary stochastic processes P. Jung, Rev. Mod. Phys. 234, 175 (1993) occur in a variety of phenomena as different as Brownian motion A. Einstein, Ann. Phys. 17, 549 (1905), Johnson noise J. Johnson, Phys. Rev. 32, 97 (1928), stellar dynamics S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943), and quantum optics H. Risken, in Progress in Optics. Besides in physics, stochastic processes have been successfully applied in economics for modeling and thus explaining diverse levels of economics systems, ranging from the ``micro'' level of company products to the ``macro'' level of company sizes and even national economies. Recently, Fu et al.\ D. Fu et al., Proc. Natl. Acad. Sci. USA\/ 102, 18801 (2005) show that for different economic variables from both the micro and the macro level, the distribution of logarithmic growth rates are approximately (i) exponential in the central part, (ii) power-law decaying in the tails, and that there is (iii) a monotonically decreasing power-law relation between the company sales and the standard deviation of logarithmic growth rates. Fu et al.\ propose a process recently cited in the Handbook of Industrial Organization Volume 3, edited by Robert Porter and Mark Armstrong for modeling the empirical observations (i) and (ii), but this model fails to reproduce observation (iii). For modeling observations (i)--(iii), we propose the multiplicative stochastic process of logarithmic growth rates equation R_t łnłeft(S_tS_t-1\right)=\mu_0\Delta t + (S_t-1)^\gamma\sigma_0\eta_t\Delta t, equation where $\sigma$, $\gamma$, and $\mu$ are three parameters, $\eta_t$ is an i.i.d.\ Gaussian noise, and $S_t $ is the random variable. When the parameter $\gamma$ introduced for modeling the dependence of the standard deviation $\sigma(R_t)$ on the size $S_t$ is set equal to zero, the stochastic process reduces to geometric Brownian motion, the most widely employed stochastic process in finance. The process can also be related to the Ornstein-Uhlenbeck process, a well-known stochastic process introduced in physics. For different time series of logarithmic growth rates $R_t$ with $\gamma=-0.15$, we calculate the average size $S \rangle$ and the standard deviation $\sigma(R_t)$. Fig.~1(a) shows that, due to $< 0$, $\sigma(R_t)$ versus $S\rangle$ scales as a power law $\sigma(R_t) S\rangle^\beta$, where $\beta=\gamma$. We find in Fig.~1(b) that for $\gamma=-0.15$ the central part of distribution $P(R_t|S_0)$ can be approximated by an exponential distribution, and Fig.~1(c) shows that the far tails of $P(R_t|S_0)$ can be approximated by power-laws, where the parameter $\sigma$ controls the power-law exponent. We also find in Fig.~1(d) that four important macroeconomic variables, (export, import, debt, and investments) exhibit the same properties (i)-(iii).

@incollection{statphys23_1037, abstract = {Stationary and nonstationary stochastic processes [P. Jung, {\it Rev. Mod. Phys.} {\bf 234}, 175 (1993)] occur in a variety of phenomena as different as Brownian motion [A. Einstein, {\it Ann. Phys.} {\bf 17}, 549 (1905)], Johnson noise [J. Johnson, {\it Phys. Rev.} {\bf 32}, 97 (1928)], stellar dynamics [S. Chandrasekhar, {\it Rev. Mod. Phys.} {\bf 15}, 1 (1943)], and quantum optics [H. Risken, in {\it Progress in Optics}. Besides in physics, stochastic processes have been successfully applied in economics for modeling and thus explaining diverse levels of economics systems, ranging from the ``micro'' level of company products to the ``macro'' level of company sizes and even national economies. Recently, Fu et al.\ [D. Fu et al., {\it Proc. Natl. Acad. Sci. USA\/} {\bf 102}, 18801 (2005)] show that for different economic variables from both the micro and the macro level, the distribution of logarithmic growth rates are approximately (i) exponential in the central part, (ii) power-law decaying in the tails, and that there is (iii) a monotonically decreasing power-law relation between the company sales and the standard deviation of logarithmic growth rates. Fu et al.\ propose a process recently cited in the Handbook of Industrial Organization [Volume 3, edited by Robert Porter and Mark Armstrong] for modeling the empirical observations (i) and (ii), but this model fails to reproduce observation (iii). For modeling observations (i)--(iii), we propose the multiplicative stochastic process of logarithmic growth rates \begin{equation} R_t \equiv \ln\left({S_t\over S_{t-1}}\right)=\mu_0\Delta t + (S_{t-1})^\gamma\sigma_0\eta_t\Delta t, \end{equation} where $\sigma$, $\gamma$, and $\mu$ are three parameters, $\eta_t$ is an i.i.d.\ Gaussian noise, and $S_t $ is the random variable. When the parameter $\gamma$ introduced for modeling the dependence of the standard deviation $\sigma(R_t)$ on the size $S_t$ is set equal to zero, the stochastic process reduces to geometric Brownian motion, the most widely employed stochastic process in finance. The process can also be related to the Ornstein-Uhlenbeck process, a well-known stochastic process introduced in physics. For different time series of logarithmic growth rates $R_{t}$ with $\gamma=-0.15$, we calculate the average size $\langle S \rangle$ and the standard deviation $\sigma(R_t)$. Fig.~1(a) shows that, due to $\gamma < 0$, $\sigma(R_t)$ versus $\langle S\rangle$ scales as a power law $\sigma(R_t) \propto \langle S\rangle^{\beta}$, where $\beta=\gamma$. We find in Fig.~1(b) that for $\gamma=-0.15$ the central part of distribution $P(R_t|S_0)$ can be approximated by an exponential distribution, and Fig.~1(c) shows that the far tails of $P(R_t|S_0)$ can be approximated by power-laws, where the parameter $\sigma$ controls the power-law exponent. We also find in Fig.~1(d) that four important macroeconomic variables, (export, import, debt, and investments) exhibit the same properties (i)-(iii).}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Podobnik, B.P. Boris and Stanley, H.E. and Grosse, I.G.}, biburl = {https://www.bibsonomy.org/bibtex/226d733662118f1784c159e846f6552d5/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {68b671e34df0ee731117b4eac0adf8d2}, intrahash = {26d733662118f1784c159e846f6552d5}, keywords = {analysis econophysics modeling random series statphys23 stochastic time topic-11 walks}, month = {9-13 July}, timestamp = {2007-06-20T10:16:37.000+0200}, title = {A Stochastic Process with a Size-Dependent Standard Deviation for Growth Rates}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1037}, year = 2007 }