Focusing on gains & losses relative to a risk-free benchmark instead of terminal wealth, we consider an asset allocation problem to maximize time-consistently a mean-risk reward function with a general risk measure which is (i) law-invariant, (ii) cash- or shift-invariant, and (iii) positively homogeneous, and possibly plugged into a general function. Examples include (relative) Value at Risk, coherent risk measures, variance, and generalized deviation risk measures. We model the market via a generalized version of the multi-dimensional Black–Scholes model using α-stable Lévy processes and give supplementary results for the classical Black–Scholes model. The optimal solution to this problem is a Nash subgame equilibrium given by the solution of an extended Hamilton–Jacobi–Bellman equation. Moreover, we show that the optimal solution is deterministic under appropriate assumptions.

中文翻译：

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Lévy 市场中风险指标的时间一致资产配置


我们关注相对于无风险基准的收益和损失，而不是终端财富，我们认为一个资产配置问题，以最大化时间一致地是一个平均风险回报函数，其通用风险度量是（i）定律不变，（ii）现金或转移不变，以及（iii）正同质，并可能插入到通用函数中。示例包括 （相对） 风险值、连贯风险度量、方差和广义偏差风险度量。我们使用α稳定的 Lévy 过程通过多维 Black-Scholes 模型的广义版本对市场进行建模，并给出了经典 Black-Scholes 模型的补充结果。此问题的最佳解是由扩展的 Hamilton-Jacobi-Bellman 方程的解给出的 Nash 子博弈均衡。此外，我们表明，在适当的假设下，最优解是确定性的。