please find attached a start. I don't think that this handles all of your requirements correct, but let's do one step after the other.
I think in your sample, there was a selection on one Date value, which in case this really was should be avoided, since you want this date to draw against without any range limitations.
In your expression, I changed min(Date) to Date. You can't use an aggregation function like min() within another aggregation function like sum() without advanced aggregation (using aggr() function). I think you don't need min function here, so I just removed it (but am not 100% sure).
You now get results for the accumulated stock, I checked with some data and it looks reasonably to me. If you want to accumulate all values per Date, you might need to remove the DISTINCT from the sum and you might need to think about if same stock values in different periods could happen (thus they maybe won't be considered correctly using DISTINCT).
Also your dimension Week range is sensitive to your slider, but not sure if this is fully according to your requirements.
Just have a look and specify what is not correct.
111108 - Moving Inventory_2.qvw 180.8 K