Stochastic Markets

The previous results were derived assuming i.i.d. markets, but can be extended to time-dependent markets. In stochastic markets, it has been shown, with increasing levels of generality on the stochastic process, that

$$\lim_{n \rightarrow \infty} \frac{1}{n} \ln \frac{S_n}{S_n^*} \leq 0,$$ (12)

for every sequential portfolio. This tells us that b^*(F) is asymptotically optimal in this sense, and W^*(F) is the highest possible exponent for the growth rate of wealth.