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# Stochastic Models

Generally, we would like to solve systems of the form by finding a fix point of the operator H where .
One way of solving is to perform iterations of the form .
Note that if {rn} has a limit such that and H is continuous around r* then Hr*=r*.

Remark : if H is a contracting operator then we showed that such a limit always exists.

An equivalent way for iteration is :

which has the same convergence property.

Let us assume that H is not known, or hard to compute. We can replace H by a sample of the form S=Hr+W, where W is the sample's "noise" and E(W)=0.

Such an S can be given by simulation of the system or by a random experiment. We can use S instead of Hr and get the following iterative algorithm :

Such an algorithm is called Stochastic Approximation.

Yishay Mansour
1999-12-16