A Price-Based Framework for Stochastic Portfolio Theory: Strategy Generation and Benchmark Conversion under Split Adjustments

01 ABSTRACT

In a semimartingale market with predictable adjustment events, the paper writes share adjustments from splits into the self-financing condition and constructs additively and multiplicatively generated strategies with explicit jump-corrected wealth decompositions. The authors show that under additive generation an entropy-based example has relative wealth tending to negative infinity almost surely under repeated splits, and that advance adjustment information alone provides no model-free guarantee of improved performance. They also establish a correspondence with the capitalization-based framework that preserves absolute wealth and yields a strategy-independent benchmark conversion, illustrated with daily NYSE data.

02 KEY FINDINGS

1. Stock splits and reverse splits induce jumps in generating weights without changing the value of existing investments, requiring share adjustments to be included in the self-financing condition.
2. Under additive generation, an entropy-based example shows that repeated splits can drive relative wealth to negative infinity almost surely, so the long-horizon relative arbitrage argument cannot be applied directly.
3. Wealth-preserving restarts in multiplicative generation retain the classical portfolio allocation evaluated at current price weights; conversion to the capitalization-based framework preserves absolute wealth and is strategy-independent.