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For each person its' random walk.

Dollars received is additive process of 99 independent random variables from [0,1].

Dollars given is -1 or 0 if there is none.

The process has memory because givers can run out of money.

I think this is the correct way to model it. In particular you can model it as a markov chain (commonly studied in queueing theory):

Less than 99, because some have no dollars, but yes.