SEWELL, Martin, 2011. Characterization of financial time series. Research Note RN/11/01, University College London, London.
| Stochastic process | Description | Applicability to real markets | Notes |
|---|---|---|---|
| diffusion process | satisfies the diffusion equation | poor | Regnault (1863) and Osborne (1959) discovered that price deviation is proportional to the square root of time, but the nonstationarity found by Kendall (1953), Houthakker (1961) and Osborne (1962) compromises the significance of the process. |
| Gaussian process | increments normally distributed | poor | Financial markets exhibit leptokurtosis (Mitchell (1915, 1921), Olivier (1926), Mills (1927), Osborne (1959), Larson (1960), Alexander (1961)). |
| Lévy process | stationary independent increments | poor | Kendall (1953), Houthakker (1961) and Osborne (1962) found nonstationarities in markets in the form of positive autocorrelation in the variance of returns. |
| Markov process | memoryless | poor | Kendall (1953), Houthakker (1961) and Osborne (1962) found positive autocorrelation in the variance of returns. |
| martingale | zero expected return | submartingale: good for stock market | Bachelier (1900) and Samuelson (1965) recognised the importance of the martingale in relation to an efficient market. Whilst Cox and Ross (1976), Lucas (1978) and Harrison and Kreps (1979) pointed out that in practice investors are risk averse, so (presumably as compensation for the time value of money and systematic risk) they demand a positive expected return. In a long-only market like a stock market this implies that the price of a stock follows a submartingale (a martingale being a special case when investors are risk-neutral). |
| random walk | discrete version of Brownian motion | poor | LeRoy (1973) and (especially) Lucas (1978) pointed out that a random walk is neither necessary nor sufficient for an efficient market. |
| Wiener process/Brownian motion | continuous-time, Gaussian independent increments | poor | Bachelier (1900) developed the mathematics of Brownian motion and used it to model financial markets. Note that Brownian motion is a diffusion process, a Gaussian process, a Lévy process, a Markov process and a martingale. On the one hand this makes it a very strong condition (and therefore the least realistic), on the other hand it makes it a very important ‘generic’ stochastic process and is therefore used extensively for modelling financial markets (for example, see Black and Scholes (1973)). |
Note that above we are interested in the logarithm of the price of an asset (Osborne (1959)).