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Markov Times

Let {X,} be an asset price observed by decision makers. Let {/,} be the sequence of information sets (sigma-algebras) generated by the X, and possibly by other data observed up to time /.

Definition. We say that a random variable С‚ is a Markov time if the event

Рђ, = {С‚<1)

is /,-measurable—that is, whether or not т is less than / can be decided given /,. According to this definition, Markov times are random time periods, the value of which can be determined by looking at the current information set. Thus, Markov times cannot depend on future informa �tion. In order to see the distinction between Markov times and non- Markov limes better, and to emphasize the importance of this concept in studying methods of technical analysis, two examples are discussed.

Example /. Let С‚, denote the date at which a process {A',}, ob �served continuously, shows a 10% jump for the first time during / 6

[0, oo);

С‚, = inf {/ G [0,   � : d{\nX,)Idt > .1}. (I)

Then С‚, is a Markov time since, by looking at the current information set, it is possible to tell whether such a jump in X, has occurred or not.

Example 2. Let тг denote the beginning date of a business cycle or a stock market uptrend. Then, т2 is not a Markov time since, in order to know whether тг = i, one needs to have access to lrts, s > 0. In fact, suppose one is at time / and that an uptrend started at time т2 = t - 2. In general, one has to wait more than 2 months to be sure that an upturn is under way. Thus, one needs 3, 4, . . . before one knows {t2 < /}—that is, future information is needed before deciding which value т2 has assumed.

Clearly, any well-defined technical analysis rule has to pass the test of being a Markov time since any buy or sell signal should, in principle, be an announcement based on data available at time Р» If a rule gener �ates a sequence of buy and sell orders that fail to be Markov times, then the procedure would be using future information in order to issue such signals. The procedure would anticipate the future. This implies a signaling decision based on considerations that are not part of the available information at time t. These are often the subjective feelings of the forecaster or information not available to the general public.

It is surprising that such infeasible rules of technical analysis may look perfectly reasonable when illustrated on a chart displaying past data. In using charts, an investigator may implicitly use "future" infor �mation while defining a procedure. For example, note that, on a chart displaying observed data, the beginning dates of any uptrend can easily be identified, yet these dates are not Markov times as example 2 dem �onstrates. Graphic methods are not the best ways of determining classes of Markov times that are useful in prediction. Yet, more often than not, this is how technical analysis rules are defined. Hence the importance of developing formal algorithms that can duplicate the buy and sell signals given by technicians.

This discussion suggests that any method that exploits the current inflection point of a series will fail to generate Markov times since these latter are not /,-measurable. At the same time, several popular forms of technical analysis use past local maxima (minima) and these are /r-measurable.

We now have a criterion to determine which rules of technical analy �sis can be quantified. Indeed, if one can show that signals generated by a rule of technical analysis are Markov times, then this would simul �taneously imply (I) that the method can be quantified, (2) that it is feasible, and (3) that one can investigate its predictive power using formal statistical models.

The following theorem is important in sorting out Markov times.

Theorem. Let {A',} be a random process assuming values on the real line R. Let Р’ be the set of all intervals belonging to R, and /, be the information set at time Р» Then the times

t'a = inf{f <$:лг,ел,д e£}, (2)

are Markov times (Shiryayev 1985).

Basically, this theorem states that the first entry of X, in an interval A is always a Markov time. The interval in question can, for example, be [0, =c) or (-< �, 0J; but it can also depend on the /, itself since, if X, G A„ we can define Y, = X, - ДА,) such that Y, G [0, *>], as long as A, is /, measurable. For example, suppose that a forecaster intends to issue a sell signal as soon as observed price X, crosses, from above, a trend line /(/,, /):

/(/„ I) = a,t + fe„

where a, and h, are /,-measurable slope and intercepts of the trend line. Then a signal is issued at:

T = inf {/:*,l% i

This signal deals with the first entry of X, in a time-dependent set A, = [0, /(/,, /)]. The time dependence of A, can easily be eliminated by redefining

Y, = X, - f(Ir t)

and issuing a signal at the first entry on Y, in (- �>, 0]:

С‚ = inf{/: K,<0}. i

Hence the above theorem can be applied to first entries of A", in /^mea �surable sets as well.

Also, the fact that the theorem deals with the first entry is not a real restriction. The same theorem can be proven for nth entry of X, into A. The important restriction is that n be known a priori by the fore �caster. In fact, I intend to show below that most methods of technical analysis arc ill defined precisely because they do not set this parameter n a priori. Below is shown which of the broad forms of technical analysis can be formulated as Markov times.