# [R] estimation of drift of continuous random walk

Patrick Foley patfoley at csus.edu
Fri Jul 20 05:39:42 CEST 2001

```Vadim,

If I understand you correctly, the maximum likelihood estimator for the drift term
is

D =  (Y(T) -Y(0))/T

where you have data Y(t) t = 0 ... T and constant drift D and diffusion
coefficients.
Therefore if there is variable drift (but constant diffusion), one crude estimate
of D(X) would be to use this in chunks  in your regression and then smooth it at
the end. If you believe that D is a linear function of X then why not a regression
of (Y(t +dt)-Y(t))/dt on X(t)?

Patrick Foley
patfoley at csus.edu

> Dear R-Users,
>
> I have the following problem to solve and I wonder if there are means in R
> that can help me.
>
> At irregular time intervals I observe a random walk process, Y, with
> time-varying drift. I assume that the drift, D, is a (linear) function of
> some parameter X. The goal is to estimate D(X).
>
> I could regress Y_{t+dt} - Y_{t} ~ X, but it's probably not appropriate
> since Var(Y_{t+dt} - Y_{t}) is not constant (actually in theory it is
> proportional to dt).
>
> Any suggestions?
>
> Thank you,
>
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```