@incollection{statphys23_1037,
	title = {A Stochastic Process with a Size-Dependent Standard Deviation 
 for Growth Rates},
	address = {Genova, Italy},
	author = {B.P. Boris Podobnik and H.E. Stanley and I.G. Grosse},
	booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics},
	editor = {Luciano Pietronero and Vittorio Loreto and Stefano Zapperi},
	month = {9-13 July},
	url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1037},
	year = {2007},
	biburl = {http://www.bibsonomy.org/bibtex/226d733662118f1784c159e846f6552d5/statphys23},
	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).},
	keywords = {analysis econophysics modeling random series statphys23 stochastic time topic-11 walks }
}