Time-consistent asset allocation for risk measures in a Lévy market

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.