Say I receive or lose $\Delta N$ dollars with equal probability, but if I have zero dollars, I can't lose any (i.e. I can only gain when I have no dollars), then how do I describe statistically the increase in the amount of money I have? I am guessing the money would have to increase with time. Does this problem have a name and can it be generalized to the case when probabilities are not equal? (I adapted this from a physics problem)
2026-04-08 00:46:20.1775609180
How much do I gain if a dollar is given or taken away repeatedly (so long as I have *some* money)?
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This problem can be studied using discrete-time Markov chains/processes.
The problem can be generalized by considerating different amounts of money with different probabilities, for example: say you start with m dollars, and every hour you either receive 1 dollar, 3 dollars or lose 2 dollars with probabilities 1/4, 1/4 and 1/2 respectively; if you have 1 dollar and you lose 2, you only get to 0; if you have 0 dollars then you can only gain 1 or 3 with probabilities 1/2 and 1/2. If you model this using a Markov chain you can calculate, for example, the probability that you will have 100 dollars after 12 hours.
Note that the probability of having certain amount of money after n hours only depends on the amount of money you had after n-1 hours. This is what characterizes Markov processes. You can find more information about Markov processes in any introductory book to stochastic processes.