Analytical representation of stock and stock-indexes returns: Non-Gaussian random walks with various jump laws

Empirical data for stock and stock-indexes returns that is available for international markets as well as for the Russian stock market is introduced and discussed. Random walk process with a specific law of an elementary independent increment (jump) in some random walk space is proposed for a proper basic description. An analytical representation of a random walk distribution law is derived from a general conception of stochastic elementary jumps. This representation covers classic S. Chandrasekhar’s results for Gauss random walks with limited jumps and jumps having distributions with all moments as well as P. Levy flights with jump distributions without moments starting from the second one. Finally, it describes truncated Levy random walks for jumps having distributions with at least the second moment. The natural generalization of truncated Levy walks is done. The random walk process is compared with known empirical data for the above discussed return fluctuations of international assets and stock indexes. There is a very good agreement of the elaborated theory with empirical data.