  An empirical algorithm is used here to study the stochastic and multifractal
nature of nonlinear time series. A parameter can be defined to quantitatively
measure the deviation of the time series from a Wiener process so that the
stochasticity of different time series can be compared. The local volatility of
the time series under study can be constructed using this algorithm and the
multifractal structure of the time series can be analyzed by using this local
volatility. As an example, we employ this method to analyze financial time
series from different stock markets. The result shows that while developed
markets evolve very much like an Ito process, the emergent markets are far from
efficient. Differences about the multifractal structures and leverage effects
between developed and emergent markets are discussed. The algorithm used here
can be applied in a similar fashion to study time series of other complex
systems.
