For example, if I had a set of intervals prices, and then we need to do a forecast for the next period time.
The simplest way to handle that is to express each interval using two numbers, and then forecast the resulting bivariate series. For example, you could use the length of the interval and the centre of the interval.
Once you have a bivariate series, you can forecast it using a VAR model, or do separate forecasting of each component using univariate models.
One complexity will be constraining the forecasts to lie in the correct sample space; i.e., the length of the interval must always be non-negative. For that reason, I suggest you model the log of the length of the interval. Then backtransform the forecast of the log-length to ensure it stays positive.