I am glad you are considering UCMs for your time series modeling. Since you already use ARIMA and ESM, transition to UCMs should be easy. I think you will like the rich output provided by PROC UCM. First let me suggest a book: Pelagatti, M. M. (2015). Time Series Modelling with Unobserved Components. Boca Raton, FL: CRC Press. This book a easier to follow compared to the classic by Harvey mentioned in the UCM doc. Additionally, I have provided links to several white papers on UCM modeling at the end of my reply to help you familiarize with PROC UCM and a related ETS procedure, PROC SSM. Now let us consider your questions one by one.
1. Can one specify damped trend model in PROC UCM? If you specify your trend using the LEVEL and SLOPE statements then the answer is no. However, since you can specify an ARIMA model in UCM, you can specify an ARIMA(1,1,2) model that is similar to ESM damped trend model. By the way, the trends specified using LEVEL and SLOPE statements can model quite general trend patterns: e.g., if the trend pattern damps during the historical period then the time-varying slope can adjust to such damping (if the trend does not damp during the historical period but one expects it to damp in the future, then a damped trend model could be useful but it's damping parameter will not have been well-estimated). Alternatively, you can specify a damped trend model in PROC SSM by using the TREND statement.
2. The trend specified using LEVEL and SLOPE statements can adjust to such changes in the historical period. By the way, PROC UCM provides a variety of ways to specify regression effects (usual regressors in the MODEL statement, time-varying regression coefficients and nonlinear regression in RANDOMREG and SPLINEREG statements, and transfer function in the TF statement (in the latest release ETS 15.1)).
3. You are correct that you can specify differencing using the DEPLAG statement (see the ARIMA modeling example in the UCM doc https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=etsug_ucm_examples08.htm&docsetVersion=15.1&locale=en to see how to specify the Airline model ARIMA(0,1,1)(0,1,1)12). Typically when you specify trend using differencing (DEPLAG statement), you don't use the LEVEL and SLOPE statements.
I know it might take a little bit of thought and going through the DOC examples to follow my answers. Anyway, here are some useful links on this subject:
Latest UCM and SSM chapters in the ETS/15.1 doc:
https://go.documentation.sas.com/?docsetId=etsug&docsetTarget=titlepage.htm&docsetVersion=15.1&locale=en
A conference proceeding: https://forecasters.org/wp-content/uploads/gravity_forms/7-621289a708af3e7af65a7cd487aee6eb/2016/07/Selukar_Rajesh_ISF2016.pdf
An SGF paper: http://support.sas.com/resources/papers/proceedings17/SAS0456-2017.pdf
Hope this helps.
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