Portfolio Optimization Strategy Under the Semi-Martingale Financial Model

This paper develops a comprehensive methodology for optimal portfolio selection in semi-martingale and non-stationary financial markets. Employing rigorous statistical tests, we first establish that real market data exhibit both non-stationary behavior and long-memory properties, challenging the validity of traditional stationary models. Motivated by these empirical findings, we propose an innovative asset price model based on sub-mixed fractional Brownian motion (smfBm). The core theoretical contribution of this work is the derivation of a novel, closed-form analytical solution for geometric Asian option pricing within this smfBm framework. This formula is a significant extension of existing results, as it explicitly incorporates the non-stationary, time-dependent volatility structure of the smfBm for [Formula: see text], providing an exact and arbitrage-free valuation tool not previously available in the literature. Our primary practical contribution lies in the novel integration of these smfBm-priced options into a Cardinality-Constrained Mean-Variance (CCMV) portfolio model. We demonstrate that this hybrid strategy uniquely mitigates risk and enhances returns, with numerical results showing that a concentrated 5-asset portfolio can achieve a 37.52% return. Thus, this work makes distinct contributions through its original pricing formula and its pioneering application to constructing efficient, non-stationary portfolios.