Asymptotic Invariance of Kelly Allocation Under Power-Law Asset Dynamics: Evidence from Bitcoin

We examine log-optimal portfolio allocation when the long-run price of an asset follows a power-law trajectory, $P(t)=At^\alpha$, and its instantaneous return variance decays as $\sigma^2(t)=\sigma_0^2 t^{-2\gamma}$. Under a zero risk-free-rate benchmark, the continuous-time Kelly fraction scales as $K^\ast(t)=(\alpha/\sigma_0^2)t^{2\gamma-1}$. Exact temporal invariance therefore occurs when $\gamma=1/2$, whereas deviations from this value produce systematic age dependence in the allocation. We propose a scaling hypothesis connecting growth in network participation, effective market liquidity, and declining volatility. Under a specified set of scaling assumptions, this model predicts the benchmark exponent $\gamma=1/2$. Using historical daily Bitcoin prices, we estimate the power-law price exponent and examine the sensitivity of the volatility exponent to the length of the rolling window. For windows of four to nine years, the estimated volatility exponents have an arithmetic mean of 0.53 and a cross-window standard deviation of approximately 0.03. Because these estimates are obtained from overlapping observations and the same underlying price history, this spread is interpreted as a measure of model sensitivity rather than a formal confidence interval. Finally, we show that a time-dependent multiplicative contribution to return variance generally breaks exact Kelly invariance. We illustrate this result using a scenario in which Bitcoin transaction-fee variability affects the effective variance process. The results identify the conditions under which log-optimal allocation can remain stable under non-stationary power-law asset dynamics and clarify the assumptions required when applying this result to Bitcoin.