# dynamic programming with serially independent returns

Book suggests that "asset returns are assumed to be serially independent, so wealth is a single state connecting one period to the next". I understand path dependency is lost in case of serial independence but how do all states at a stage collapse to a single state?

• Your question is not quite clear to me, would you care to elaborate, please? – Ulysses Nov 28 '14 at 15:39
• I am unable to understand the statement made in the 2nd paragraph of page 215 of the Book. My understanding is that returns must have a distribution at each stage and so wealth must have a distribution. Hence, there must be a fan like structure connected to another fan like structure. I am struggling to understand the shape of the scenario tree. – Kumar Nov 29 '14 at 2:39
• I guess it just says that if returns $(r_k)_{k\geq 0}$ are independent, then the wealth $S_n = S_0 \mathrm e^{\sum_k r_k}$ is a Markov process: distribution of $S_{n+1}$ given $S_n$ is same as distribution of $S_{n+1}$ given the whole history $S_0,\dots,S_n$ – Ulysses Dec 1 '14 at 9:16
• Can you explain that in terms of the shape of the scenario tree in dynamic programming context with stages and states – Kumar Dec 1 '14 at 21:40
• Not sure what is the scenario tree - can you give an available reference? – Ulysses Dec 2 '14 at 6:53