Continuous Time Random Walk with correlated waiting times. The crucial role of inter-trade times in volatility clustering.

In many physical, social or economical phenomena we observe changes of a studied quantity only in discrete, irregularly distributed points in time. The stochastic process used by physicists to describe this kind of variables is the Continuous Time Random Walk (CTRW). Despite the popularity of this type of stochastic processes and strong empirical motivation, models with a long-term memory within the sequence of time intervals between observations are missing. Here, we fill this gap by introducing a new family of CTRWs. The memory is introduced to the model by the assumption that many consecutive time intervals can be the same. Surprisingly, in this process we can observe a slowly decaying nonlinear autocorrelation function without a fat-tailed distribution of time intervals. Our model applied to high-frequency stock market data can successfully describe the slope of decay of nonlinear autocorrelation function of stock market returns. The model achieves this result with no dependence between consecutive price changes. It proves the crucial role of inter-event times in the volatility clustering phenomenon observed in all stock markets.