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gaj


Total Posts: 23
Joined: Apr 2018
 
Posted: 2018-06-30 10:06
Is there a simple, perhaps well-known process that is "mean-reverting" but also martingale? I.e., I'm looking for a process x_t where:
1) The expected value of x_(t+1) at time t is equal to x_t.
2) If x_t is below the mean then x_(t+1) is more likely to go up, and vice versa.

Basically if x_t is below the mean, the distribution of x_(t+1) should be left-skewed. It has a probability of >50% to go up. But in the unlikely event that it continues to go down, the magnitude is bigger, i.e. fat left tail. And vice versa if x_t is above the mean.

OU process doesn't work because it's not martingale. The noise term should somehow be skewed depending on the level of x_t. I have some ideas but wonder if there's anything in the literature.

ronin


Total Posts: 314
Joined: May 2006
 
Posted: 2018-07-01 10:53
@gaj,

The process you are describing wouldn't satisfy 1) and 2). Skew doesn't affect the mean. Left skewed just means lots of small upticks, few big downticks.

You can try sub-Brownian processes, where and .

Depending on what you are looking for, they may be sufficiently "mean reverting".


"There is a SIX am?" -- Arthur

katastrofa


Total Posts: 448
Joined: Jul 2008
 
Posted: 2018-07-01 11:14
What do you mean by "mean" in 2)? If your "mean" = lim_{T -> infinity} E[X_T|x_0], then 2) is incompatible with 1).

pj


Total Posts: 3396
Joined: Jun 2004
 
Posted: 2018-07-01 14:46
IMHO, the requirements are feasible.

As far as mean reverting means bigger probability
of moving to the right direction
and martingale means the conditional expectation equal
to the current price.


The gaj martingales are feasible. And he himself proposes how to construct them.
Ar they interesting?
That I don't know.

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

katastrofa


Total Posts: 448
Joined: Jul 2008
 
Posted: 2018-07-01 16:28
Yes, but what is the "right direction"? If gaj means that mean = some parameter Theta, then I agree, it probably is feasible.
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