A Wealth Realizability Framework Based on Liquidity Constraints and Optimal Stopping Non-linear Conversion of Realized and Unrealized P&L

Traditional investment performance evaluation metrics (e.g., IRR, TWR, Total Return) linearly aggregate realized profit and loss (R) and unrealized profit and loss (U), implicitly assuming both are equivalent in wealth measurement and economic substance. We demonstrate that this approach lacks rigor in both mathematical and microeconomic contexts: R is a deterministic scalar with zero conditional entropy, whereas U is fundamentally a probability distribution modulated by liquidity shocks and future uncertainty, representing a non-collapsed stochastic state. Treating them as linearly additive systematically overestimates the realizable liquidation value of a portfolio and may induce adverse selection and intertemporal moral hazard among financial institutions. A unified nonlinear framework for wealth realizability is developed, distinguishing two performance measurement functions: Liquidation Perspective (CE-Liq): A cross-sectional state function governed jointly by the liquidity retention factor L_i(q_i, D_i, κ_i) and institutional frictions φ_i, excluding intertemporal diffusion penalties. Optimal Stopping Perspective (CEˢᵗᵒᵖ): Explicitly formalized as an Optimal Stopping Problem incorporating holding-period dividend streams and risk-adjusted discount rates. Through Hamilton-Jacobi-Bellman (HJB) variational inequalities and the Snell envelope, we establish that under given regularity conditions and a local continuation discriminant, the objective functional yields a finite interior stopping time. To isolate market-driven passive returns, a dimensionally closed Realizability Alpha (RA-α) is introduced. In a stylized event study of the March 2020 liquidity crisis, the framework characterizes the dynamic deterioration of wealth realizability under the sudden withdrawal of market depth and surging volatility, demonstrating the capability of CEˢᵗᵒᵖ to identify nominal book wealth illusions in stress regimes.