Financial Time Operator for random walk markets

Based on previous work on non-equilibrium statistical mechanics, and the recent extensions of Time Operators to observations and financial processes, we construct a general Time Operator for non-stationary Bernoulli Processes. The Age and the innovation probabilities are defined and discussed in detail and a formula is presented for the special case of random walks. The formulas reduce the computations to variance estimations. Assuming that a stock market price evolves according to a random walk, we illustrate a financial application. We provide an Age estimator from historical stock market data. As an illustration we compute the Age of Greek financial market during elections and we compare with the Age of another period with less irregular events. The Age of a process is a new statistical index, assessing the average level of innovations during the observation period, resulting from the underlying complexity of the system.