In the setting of Proposition 6.32, prove the representation (6.62) for the conditional probabilities associated to the risk neutral probability measure \(\mathbb{P}^{*}\) defined by (6.61) (compare also with Huang \& Litzenberger [971, Section 8.5]).

Data From Proposition 6.32

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Proposition 6.32 For any i = 1,..., I, let u': R+ R be strictly increasing and concave and denote by c = (c)=0.1... the corresponding consumption process solving problem (6.6) with respect to the couple (S, D) of price-dividend processes. Then a risk neutral probability measure P* can be defined by letting :=(8r;)Tu =(877) Tu (cf (w)) u (c) - for all w 22. (6.61) Furthermore, the conditional probabilities associated to P* are given by TA,As = P* (A,|A,) = (8r)- u" (c* (A)) u (c (As)) -A,\A, (6.62) for all s, t = {0,1,..., T3 with t>s and A, E, A, EF, such that A, CA. Proof If c* is an optimal consumption process associated to the utility function u', then Corollary 6.5 (in the case where a risk free security is traded) together with (6.14) implies that u (c) = 8r+E[u (c++1)F] for all t=0,1,..., T-1. (6.63)