convert confidence intervall from differneces in levels

[Gilbril]

Dear Tom,
I have a question concerning the topic of forecasting.
I estimate an AR(1) GARCH(1,1) model. As the time series has to be stationary, I take log differences of the level values. Then I estimate and forecast the model, which works pretty good and my estimates are significant. Now I would like to convert the forecast back to levels. This is easy for the series itself, but I would also like to convert the confidence intervals to levels. How would that be done, is there any special formula for the conversion of confidence intervals from log differences into levels?

Thanks in advance

[TomDoan]

Are you computing the confidence intervals taking into account that the multi-step forecast errors are moving averages of the error process? If you are, then the ARIMA(1,1,0) just generates a different moving average than the AR(1).