# Mean in the context of Mean-Reversion

I want a running mean estimator that acts as a mode through mean reversion cycles of target amplitude or frequency.   The key characteristics should be:

• determination of diffusion related squared return
• determination of jump related squared return
• determination as to how much of the jump should be absorbed into the mean
• model of mean reversion
• calibrated to a desired long-run rate of reversion
• allowance for changes in reversion constant and reversion to long run
• model of mean
• autoregressive
• innovations scaled by sigma term (with MR component and jumps removed)
• recursive backward estimation of ML
• implicitly decide how innovation is distributed amongst mean, mean-reversion, and noise

A SDE-based Approach
The model is an expanded variant of the familiar OrnsteinUhlenbeck process, with specialized mean-reversion, mean, and volatility processes.   It also attempts to correct for jumps.    Let’s start with the following SDEs (in continuous time):

Variance
There are many approaches to modeling volatility (all with issues).   Initially I had though to use a predictive model based on:

• intensity process (based on “first exit” duration)
This is a very complex process.  First approximations have been to use ACD, a family of AR models for duration.   ACD models perform very poorly on HF data however.    It seems that a markov chain model recognizing the patterns will be most appropriate.
• amplitude process
The amplitudes of squared returns seem to follow a largely AR process.   This seems fairly well behaved.

Before fully committing to a complex volatility model thought its makes sense to first try with a non-predictive measure of realized variance.  I will use:

The choice of α determines the degree of smoothing with previous values based on how local (and noisy) we want this function to be.   For example, here is the estimate with a smoothing factor of 60 and a threshold of 3e-5:

Discretising
Using Ito’s lemma we discretise the processes as follows:

Simplifying the volatility term in S(t), we first determine the variance of the SDE:

We reorganize as follows:

Putting it together
We can now model this discretely as a state-space based filter, searching for parameters that fit a-posteriori idealized view on the mode and mean-reversion process.   Post-parameterization, the process can be used in real-time to provide an estimate of the mode.

Final Notes
As you may have seen I took a (useful) 2-3 week diversion before coming back to the SDE based approach.   This is not a final model by any means, but I think a a solid starting point.    The purpose of the above is as a one of a number of factors in a multi-factor  strategy that want to optimize further.