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Prices of financial assets (stock-market prices or currency exchange rates) obviously resemble trajectories of stochastic processes.

What is known about their mathematical properties ?

I know there is huge (too huge) literature (see e.g. MO-Financial Mathematics Books) around it, I am familiar with some ideas, like below, but would be grateful for any comments/suggestions.

1) The individual distributions are better modelled by heavy-tailed distributions, rather than by normal distribution, reflecting that sometimes prices change heavily in short time periods. (See e.g. MO: Is there any straightforward way to substitute for Gaussian/Brownian assumptions in financial mathematics?).

2) To some extent they are similar to Brownian (log-Brownian) motion, more precisely the increments are independent at least at some time scales (which means that you cannot win money), (however there are other claims that short time increments are correlated).

3) There are claims that fractal (Hausdorff ) dimension is near to 1.5 (the same as for Brownian motion).

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    $\begingroup$ The sister site quant.stackexchange.com might be a better place for this question. $\endgroup$
    – Andrey Rekalo
    Commented Apr 28, 2013 at 13:10
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    $\begingroup$ @Stefan, the same objection could be used against any physical science. Yet, as a community, we do not give up hope of modeling in the face of other very complicated systems like particle physics or the global climate. $\endgroup$
    – BSteinhurst
    Commented Apr 28, 2013 at 13:55

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The question is too open-ended. There is indeed a plethora of literature available..

For stock market, it started with Bachelier who assumed normally distributed stock prices (not even returns, but a stock price itself was assumed normal). Then Black-Scholes-Merton moved one step up assuming that stock prices are lognormal or, more precisely, that returns are normal and independent. The actual stocks returns distribution is of course different and seems to be of a (truncated) Levy distribution.

Then there are topics of autocorrelation, volatility clustering etc. and some results on the speed of autocorrelation function decay etc.

http://en.wikipedia.org/wiki/Financial_models_with_long-tailed_distributions_and_volatility_clustering

Also, as expected, the mathematical properties of price changes, depend on the time scale (frequency) e.g. some results show that for some (not too short) time scales it is more likely to have a positive price change after a sequence of two positive changes than after a sequence of mixed changes (e.g. up, then down. or down then up).

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    $\begingroup$ I only put it as an answer because the comment does not allow long posts :) ... this question can not be really "answered" :) $\endgroup$
    – Alexey Kvashchuk
    Commented Apr 28, 2013 at 14:03

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