[R] estimation of drift of continuous random walk

[Vadim Ogranovich]

Thank you to Patrick Foley and Adrian Trapletti. Their compiled answer is:

a) fix model specification and regress
(Y(t+dt) - Y(t))/dt ~ X(t) // originally I forgot to devide by dt

b) use weighted regression with weights = 1/Var((Y(t+dt) - Y(t))/dt) = dt.

Thanks,
Vadim

-----Original Message-----
From: Patrick Foley [mailto:patfoley at csus.edu]
Sent: Thursday, July 19, 2001 8:40 PM
To: Vadim Ogranovich
Cc: r-help at stat.math.ethz.ch
Subject: Re: [R] estimation of drift of continuous random walk


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

Vadim Ogranovich wrote:

> 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,
> Vadim
>
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