Often, we need to analyze nonstationary probability distributions to investigate, for example, how the distributions behave as time progresses. For instance, we may be interested in knowing how the distributions of market shares of firms behave in some sector as the sector or industry matures.

If we can't solve master equations directly in the time domain, we may try to solve them by the method of probability generating functions. In cases where that approach does not work, we may try solving ordinary differential equations for the first few moments of the distributions by the method of cumulant generating functions; see Cox and Miller (1965, p. 159). Alternatively, we may be content with deriving probabilities such as P0(t), this being the probability for extinction of certain types (of their sizes being reduced to zero).

This section describes the probability- and cumulant-generating-function methods for solving the master equations. In those cases where the transition rates are more general nonlinear functions of state variables than polynomials, we can try Taylor series expansions of transition rates to solve the master equations approximately.

In this chapter, we illustrate some procedures to obtain nonstationary probability distributions on some elementary models. (See also the method of Langevin equations, which is discussed in Section 8.7.) This leads to (approximate) solutions of Fokker–Planck equations.

Example: Open models with two types of agents

The next example is an open market or sector model with two types of agents.