Stochastic integral - Quadrature formula

[elgiovo]

Given the random process $bbs(t)$, suppose one wants to estimate the r.v. $int_0^T bbs(t)"d"t$ with a linear combination of the r.v. $bbs(0)$ and $bbs(T)$ in the sense of mean squares. In other words, one should minimize the error $E[(int_0^T bbs(t)"d"t-a_0bbs(0)-a_1bbs(T))^2]$. Show the coefficients ot the combination satisfy $a_0=a_1=(int_0^T R_(bbs bbs)(t)"d"t)/(R(0)+R(T))$.

[elgiovo]

Here is a solution:

Seing random variables as elements of a vector space, one can easily show that the error of an estimate has to be orthogonal to the data to be minimum.
This is the well-known orthogonality principle.

In the geometry of random variables, the dot product between $bbx$ and $bby$ is given by $E[bbx * bby]$, and two such r.v. are orthogonal iff $E[bbx * bby]=0$.

So, in this case, we must obtain

${(E[(int_0^T bbs(t)"d"t -a_0 bbs(0) -a_1 bbs(T) ) * bbs(0)]=0),(E[(int_0^T bbs(t)"d"t -a_0 bbs(0) -a_1 bbs(T) ) * bbs(T)]=0):}$

in order to minimize

$e=E[(int_0^T bbs(t)"d"t -a_0 bbs(0) -a_1 bbs(T) )^2]$.

Calculating expected values, one obtains the system

${(int_0^T R_(bbsbbs)(t) "d"t =a_0 * R_(bbsbbs)(0) +a_1 * R_(bbsbbs)(T) ),(int_0^T R_(bbsbbs)(T-t) "d"t =a_0 * R_(bbsbbs)(T) +a_1 * R_(bbsbbs)(0) ):}$

Observing that

$int_0^T R_(bbsbbs)(t) "d"t =int_0^T R_(bbsbbs)(T-t) "d"t $

and solving for the coefficients,

$a_0=a_1=1/( R_(bbsbbs)(0) + R_(bbsbbs)(T) ) *int_0^T R_(bbsbbs)(t) "d"t $.