Examples of causal integration

The integration operator has a pole at Z = 1, which is exactly on the unit circle |Z|=1. The implied zero division has paradoxical implications (page ) that are easy to avoid by introducing a small positive number $\epsilon$and defining $\rho\ =\ 1\,-\, \epsilon$.The integration operator becomes

$$\begin{eqnarray} I(Z) &=& {1 \over 2 }\ {1\ +\ \rho Z \over 1\ -\ \rho Z} \ I(Z... ...\over 2 }\ +\ \rho Z\ +\ ( \rho Z )^2\ +\ ( \rho Z )^3 \ +\ \cdots\end{eqnarray}$$ (23) (24)

Because $\rho$ is less than one, this series converges for any value of Z on the unit circle. If $\epsilon$ had been slightly negative instead of positive, a converging expansion could have been carried out in negative powers of Z. A plot of I(Z) is found in Figure 4.

 

cint
Figure 4 A leaky causal-integration operator I.

Just for fun I put random noise into an integrator to see an economic simulation, shown in Figure 5. With $\rho=1$, the difference between today's price and tomorrow's price is a random number. Thus the future price cannot be predicted from the past. This curve is called a ``random walk."

 

price
Figure 5 Random numbers into an integrator.