AI 中文总结
本文针对CVaR约束下的连续时间动态投资组合优化问题，提出对偶嵌套二分-黄金分割搜索算法，证明策略收敛性，数值实验揭示风险约束紧时的状态依赖策略及非交易风险、价格冲击的影响。
AI 中文摘要
我们研究在投资者终端损失的条件风险价值（CVaR）约束下的连续时间动态投资组合优化问题。对于一类通用的凸交易目标，我们利用CVaR的辅助阈值表示，在不要求市场完备性的前提下，建立了最优策略的存在性和强对偶性。这些结果启发了一种基于对偶的嵌套二分法-黄金分割搜索算法，用于在阈值和拉格朗日乘子上进行优化，其中内层迭代可简化为标准的无约束随机控制问题。我们证明，随着迭代次数趋于无穷，所得策略收敛到最优控制。数值实验表明，当风险约束非紧时，该策略会恢复为默顿（Merton）策略；当约束紧时，最优策略与状态相关：投资者在不利结果后会降低风险敞口，但在有利结果后会维持风险敞口，且临近到期时可能会增加风险敞口。因此，终端CVaR约束会产生跨状态的不对称再分配，而非统一的去风险化。非交易性的禀赋风险会放大这种保守调整，而价格冲击则会降低期望头寸和调整速度。
英文摘要
We study continuous-time dynamic portfolio optimization under a Conditional Value-at-Risk (CVaR) constraint on the investor's terminal loss. For a general class of convex trading objectives, we exploit the auxiliary-threshold representation of CVaR to establish the existence of an optimal strategy and strong duality without requiring market completeness. These results motivate a dual-based nested bisection--golden-search algorithm over the threshold and Lagrangian multiplier, where the inner iterations reduce to standard unconstrained stochastic control problems. We prove that the resulting strategies converge to the optimal control as the number of iterations tends to infinity. Numerical experiments recover the Merton policy when the risk constraint is nonbinding. When the constraint is binding, the optimal strategy becomes state dependent: the investor reduces risky exposure following adverse outcomes but preserves, and near maturity may increase, exposure following favorable outcomes. Thus, a terminal CVaR constraint produces an asymmetric reallocation across states rather than uniform de-risking. Nontraded endowment risk amplifies the conservative adjustment, whereas price impact lowers desired positions and adjustment speeds.
