Random variable

Using the martingale method of forecast evolution (MMFE) of heath and
Jackson and obtain a formula for the mean and variance of the lead time
demand and apply these formulas to specific demand models.

The Martingale Method of Forecast Evolutions (MMFE) can be represented
as follows:

for the demand to prevail

in period s for each s >= 1 in the planning horizon. Forecasts are
updated at the beginning

of every period as follows: For all s >= t



is the forecast at the beginning of period t for the demand to prevail
during

,random variable that becomes known

.Thus, for

1 =< t =< s + 1 we can write the actual demand for period s as



Ds at the beginning of period t1.

Let Et[Ds] and Vart[Ds] denote the expectation and the variance of Ds
given what is

known at the beginning of period t.

Thus,

is just the unbiased forecast

is a measure of the forecast error

Questions?

If we apply (MMFE) to Autoregressive Integrated Moving Average (ARIMA)
Demand Model.

for s >=2



is constant and is between 0 and 1.

, how can I prove that this model (ARIMA) can be fit into the framework
of (MMFE)?

?

I need to find the variance.

What is the variance (if it is too much work to find the
variance, it is okay to leave it)?