Consider 2 related aspects of a process for prices in a financial market:

- time &
- return.

**Time**

Say I've identified a distribution that reasonably models the occurrence of the lengths of price trends (how long in time the price moves up or down). Given the length of a given trend up to some point in time (i.e., now), one can readily calculate the mean residual life of the trend (the average residual lifetime of the trend given that the trend has survived up to a given time point)-- essentially conditional expectation.

I can then take the next step and calculate the conditional probability of reaching that mean residual life.

**Return**

Separately, say I have another distribution, which models returns (I do appreciate the challenges of actually identifying such a distribution).

One could then do a similar set of calculations for returns as for the length of the trend. Given the current return within the current trend one can calculate the mean residual life for the return and the conditional probability that the current move will reach that target.

**What to do with the above?**

Lots of extenuating issues and questions (i.i.d?, stable distribution?,...etc.) but just keeping it simple for now…

One could use this probability of reaching the *return's* mean residual life to scale the size of an investment with something like the Kelly criteria (See John Kelly's "A New Interpretation of Information Rate" or google "Kelly formula" for other references).

**What about the "time" information?**

But, I also have this interesting information about "time" and the probability that a trend will last a certain amount of time.

Intuitively, longer trends ought to have greater returns than shorter tends. Trend length and return give different views of the same process or perhaps describe different but related aspects of the same process. Again, intuitively the two distributions should have some relationship.

How can I define or identify the relationship between time and returns or would this even prove useful?

Can I combine the probabilities of reaching the mean residual lives of both returns and trends to get something more reliable or more robust than either alone?

How would one go about doing this or even thinking about it?

The time analysis seems a bit tricky to use because while related to returns, its not measured in the same units.

Just seems like an interesting question. Looking forward to the community's thoughts.

...

A quick footnote, from wikipedia's entry for Volatility:

For a financial instrument whose price follows a Gaussian random walk, or Wiener process, the width of the distribution increases as time increases. This is because there is an increasing probability that the instrument's price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero.

Ah, but prices don't follow a Gaussian or Wiener process.

So, how do I establish the relationship for my distributions?