We are introducing the new family of the Continuous Time Random Walks (CTRW) with long-term memory within consecutive waiting times. This memory is introduced to the model by the assumption that consecutive waiting times are the analog of CTRW themselves. This way, we obtain the 'Continuous' Time Random Walk in Continuous Time Random Walk. Surprisingly, this type of process , only with the long-term memory within waiting times, can successfully describe slowly decaying nonlinear autocorrelation function of the stock market return . The model achieves this result without any dependence between consecutive price changes. It proves the crucial role of inter-event times in  volatility clustering phenomenon  .
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 .
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We are introducing the new family of
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
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Walks
Walk
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with
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consecutive waiting times. This
the sequence of time intervals between observations are missing. Here, we fill this gap by introducing a new family of CTRWs. The
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consecutive waiting times are the analog of CTRW themselves. This way, we obtain the 'Continuous' Time Random Walk in Continuous Time Random Walk. Surprisingly, this type of process , only with the long-term memory within waiting times, can successfully describe
many consecutive time intervals can be the same. Surprisingly, in this process we can observe a
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of the stock market return
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
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without any
with no
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the
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observed in all stock markets