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The asymptotics of utility from terminal wealth is studied. First, a finite horizon problem for any utility function is considered. To study a long run infinite horizon problem, a certain positive homogeneity (PH) assumption is imposed. It is then shown that assumption (PH) is practically satisfied only by power and logarithmic utility functions. 1. Finite horizon problem. Consider a market with d assets, the prices of which are modelled in discrete time by the formula (1.1) Si(n+ 1) Si(n) = ζi(z(n), ξ(n)) := ζi(n), where Si(n) is the price of the ith asset at time n, (z(n)) is a Markov process of economic factors taking values in a measurable space D, and (ξ(n)) is a sequence of i.i.d. random variables. We denote by ζ(n) the vector of ratios Si(n+ 1)/Si(n). Starting with an initial wealth W we invest in assets, denoting by πn i the portion of capital invested at time n in the ith asset. It is clear that the vector πn consisting of the πn i for i = 1, . . . , d is our investment strategy at time n and πn ∈ S = {η ∈ Rd : ηi ≥ 0, ∑d i=1 ηi = 1}. The wealth process W (n) at time n is then given by the formula (1.2) W (n) = W (n− 1)(πn−1 · ζ(n− 1)) = W (0) n−1 ∏

On the Bellman equation for asymptotics of utility from terminal wealth