Clive Granger writes (Financial Analysts Journal): Most of the old literature on prices, returns, and volatility had, basically, a linear foundation. From studying the models suggested by these approaches, researchers have accumulated a number of "stylized facts"; they are empiricial "facts" that have been observed to occur for many (possibly all) assets in most (possibly all) markes in most time periods and for most data frequencies. A list of these stylized facts would include the following:

• Returns are nearly white noise; that is, they have no serial or autocorrelation.
• The autocorrelations of r²_t and |r_t|^d decline slowly with increasing lag (the long-memory effect), with the slowest decline for d = 1 (the Taylor effect).
• Autocorrelations of sign r_t are all small, insignificant.
• If one fits a GARCH(1,1) [that is a generalized autoregressive conditional heteroscedasticity (1,1)] model to the series, then α + β ≈ 1, with the usual notation.

In a remarkable paper, Yoon (2003) showed, largely by simulation, that the simple stochastic unit root model

P_t = (1 + a_t)P_t-1 + ε_t,

where P_t is log stock price and a_t and ε_t are independent white-noise series, produces return series that have all of the stylized facts observed in actual data. This outcome does not imply that actual log stock prices are genereated by the model, but it does suggest that a simple model can capture many realistic properties. So, the model deserves further study. Yoon's model is an example of a "stochastic unit root process" as discussed by Granger and Swanson (1997) and by Leybourne, McCabe, and Mills (1996). <>

Into the Future: The immediate future in any active academic field always involves endeavors that have already started. Conditional distributions will thus continue to be a major subject as finance learns how to convert more of ts fundamental theories into distributional forms: arbitrage and portfolio theory, efficient market theory and its consequences, the Black-Scholes formula, and so forth. <> Also, structural breaks are likely in the present framework. Such breaks are, by nature, difficult to forecast, but two possibilities may already be visible. The first is a new approach to volatility, and the second is a reformulation of basic functional theory.