Optimal Consumption and Retirement Time under Shortfall Risk Measure

Battle Count: 87

The paper is primarily relevant to portfolio management and risk management rather than algorithmic trading. The benchmark tracking framework and shortfall risk measure are directly applicable to institutional portfolio management and performance evaluation. The optimal portfolio allocation under benchmark constraints could inform systematic trading strategies that track indices. However, the focus on life-cycle decisions (retirement timing) and consumption makes it more relevant to wealth management than high-frequency or short-term trading.

The paper involves highly advanced mathematical techniques including hybrid stochastic control, convex duality theory, reflected state processes, variational inequalities with Neumann boundary conditions, and free-boundary problems. The transformation from the original problem to the dual stopping problem requires deep understanding of stochastic analysis. Numerical implementation would require solving PDEs/variational inequalities in two dimensions with state reflection, which is computationally demanding. The analytical solutions are only available for special cases (zero benchmark, constant parameters).

The paper provides complete analytical proofs in the electronic companion, explicit formulas for two examples (zero benchmark and GBM benchmark), and detailed parameter settings for numerical simulations (Figures 1-7). However, no code repository is provided, and the numerical implementation details are not fully specified. The theoretical framework is self-contained with all assumptions clearly stated.