Random Walk

muzialis

Hello there,

I am wondering if somebody could help in an issue far from my expertise.

I have some data which is reasonable to conjecture could be modelled with a random walk with drift.
I am struggling though to understand how to estimate from the empriic data the most likely drift and variance value necessary to simulate the random walk.

So far I thought about this possible method.

1) From the empiric data estimate the hitting time to a conventional value for each available experimental path.
2) As hitting times are distributed according to a Inverse Gaussian distribution, I could use the data from 1) to estimate the Inverse Gaussian parameters using standard Maximum Likelihood estimators
3) From calcualtion at 2) I should be able to estimate drift and variance as theory tells us how they relate to the Inverse Gaussian parameters.



Any comment on this? Any suggestion? Many thanks in advance

Best Regards

alan2

This is a common problem in financial math. You want an ARIMA package which will do this for you. These are reasonably good lecture notes on the topic. It should point you in the right direction.

http://web.duke.edu/~rnau/411home.htm

Stephen Tashi

It's important to clarify whether you want to use a continuous time version of a random walk or a discrete time version.

If all your data is measured at a common time interval then the discrete time approach, using ARIMA models is adequate. If you are trying to work with continuous time "Wiener process", I think you can use the fact that a (constant) drift is directly proportional to the elapsed time between measurements and the random jumps in the process have a standard deviation that, as I recall, is proportional to the elapsed time. So it looks to me like you can do an analysis that uses every data point instead of relying on a property like hitting time. (The ARIMA models can also use all the data. )