Langevin-equation approach
The dynamics of conventional economic models is specifiedby an n-dimensional deterministic dynamical equation
where x is an n-dimensional state vector.
Its stochastic version, especially in econometric models, is often proposed by tacking noises, usually additive, onto deterministic equations. Instead, we consider
as its stochastic version, with g(∙, ∙) some smooth function of the two arguments, where X(t) is a stochastic state vector, and eξ(t) a vector-valued white noise process. Here, ∈ is a small positive constant, and ξ (t) is standardized to have variance 1. We drop the time argument for simplicity.
Its linearization is called the Langevin equation:
where dWi(t) = Wi(t + dt) - Wi(t) is the Wiener-process increment. See Todorovic (1992) for a brief account of Langevin’s approach. Cox and Miller (1965, p. 298) make a brief comment on it as well. Soize (1994) has more details.
Consider a scalar stochastic process
for a twice continously differentiable f on a compact set in the real line, with
Substituting dx out, we have
Suppose that a probability density function p(x, t) exists. Taking the expectation of the above with this density function, and interchanging the order of differentiation and integration, we evaluate the partial derivative with respect to t of the expectation of f (x) by integration by parts.
Noting that f and its partial derivatives have compact support, the resulting expression evaluated at the limits of integrations is zero, e.g.,
and so on. The resulting expression is 
Setting the left-hand side of the Kolmogorov equation equal to zero, the stationary probability density is obtained. See Soize (1994, Sec. VI.5) for example.
The density must have probability mass 1. In the case of scalar equations defined on the real line and where the positive and negative regions are distinct, and there is no probability flow from one side to the other at the origin, x = 0; then the probability current is zero at x = 0.
In the steady-state case, the current is constant; hence it is zero throughout if it is zero at any point, such as the origin. Given that J = 0 at all x,it follows
where ps denotes the stationary probability density. Integrating this equation, we obtain
8.7.2 The exponential distribution of the growth rates offirms
Let S(t) be a stochastic process of the size of a firm. Here the word “size” is to be interpreted broadly as meaning some quantity related to the scale of firm's activities, such as the number of employees, sales in dollars, or plant and equipment or capitalization in dollar terms. The parameter values of the distributions vary somewhat depending on the S being used, but the functional form of the distribution remains the same. See Amaral et al. (1997) for detail.
Here, we postulate that s(t) = ln S(t) grows by the rule
We next change the variables to put it into a standard form. Define
The solution of this equation gives the time-dependent probability density for s or its normalized version x.
The steady-state probability density in the original variable, denoted by 
is 
8.8