Re: Memory Via Volterra Equations, Volterra Series, Delay Equations

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 11/19/04 

Date: Fri, 19 Nov 2004 19:00:51 +0000 (UTC)

On 19 Nov 04 10:06:31 -0500 (EST), Osher Doctorow wrote:
>Anna Karzewska of U. Zielona Gora Poland in "Stochastic Volterra
>convolution with Levy process," arXiv:math.PR/0411148 v1 7 Nov 2004
>uses Volterra equations driven by Levy processes rather than the
>usual stochastic differential equations (SDEs) driven by semimartin-
>gales, which change has not only the advantage of simplicity but
>the advantage of being motivated by the increasing applied interest
>in Levy processes when empirical observations can't be explained
>easily by the normal/Gaussian distribution. She considers a
>stochastic verison of a linear, scalar type Volterra equation in

Here is an interesting theorem of Anna Karczewska (2004) which shows
us how very applicable to the "real world" her paper is:

Th. Let R(t) be the family of resolvent operators of the Volterra
equation X(t) = I[a(t-tau)AX(tau)dtau + Xo + Z(t) and Z(t) be a
Levy process, then I[R(t-tau)]dZ(tau) = (def.) Plim I[Rn(-tau)]dZ(
tau) is well defined random variable with infinitely divisible
distribution, where I...dtau or dZ(tau) is the integral from to t.

The Rn here are step functions sum R(si)1_[ti-1, ti] where 1_ is an
indicator function on the interval [ti-1, ti] and si is in this
interval for i = 1 to n (also i = 1 to n in sum).

How exactly anybody could conclude that Volterra equations and memory
and their relationships with PI can have no important practical
applications from this escape me, but undoubtedly some critics will
either not read it or find that it lacks the simplicity of telling
them how many computer buttons or keys to press!

Osher Doctorow