Oct

4

With the Dow's returning to its prior all time high, the issue of a Markov Chain in the form of a random walk's returning to its starting point is a good research topic. Probability and Random Processes by Geoffrey Grimmett has a discussion of transient and persistent Markov chains. One way to measure periods is the distribution of time it takes to return to a starting point for the first time. The question arises: will price ever return to the starting point? If so, when? A random walk, a form of Markov chain, is transient if the probability of returning to its starting point is less than 1 and persistent if p=1. This analysis might apply to a number of market situations other than whether the Dow will reach its all time high, such as:

The transition point is the boundary number which might yield some answers to the issue of whether we are in a range or the range is breaking out. It has been shown that the market drift in the long term results in long transient epochs and has an upward drift that makes the probability of a return to its start low. This has broad implications in day to day operations. Over shorter terms the market can be transient or persistent. Transient phase is a trending phase, and a persistent state travels through the same point over and over. Last week and this week price persistently went through 1347. The '60s - '82 epoch of persistence and the 80's bull market transient phase are commonly cited examples. We are in a persistent market phase now, with a 6 year lag of attempting to return to new highs.

Grimmett speaks of epochs of persistence. This is a cycle of range based price action. The states formulas are quantified on page 222. A symmetric random walk is persistent. A random walk with drift will tend towards transience. The mean recurrence time can be measured for drift. This allows classification of epochs by periods as the greatest common divisor of the epoch. Under the Central Limit Theorem, sampling of a random walk with the appropriate drift should give the distribution of times to return to price and the mean time. This might give the operator a measure of time expectation for trade durations and be a useful measure. A hypothesis is that on the short and intermediate term, the S&P tends to be persistent. These provide quantification for a definition of trend that has been problematic.


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