Dynamic portfolio selection under range value-at-risk with logarithmic returns

This paper studies continuous-time portfolio selection under Range Value-at-Risk (RVaR), with both portfolio performance and risk evaluated in terms of logarithmic returns. We formulate a mean--RVaR problem in a continuous-time financial market and reduce the dynamic optimization to a static quantile problem. Using quantile representation, Lagrangian optimization, and convexification, we characterize the optimal terminal wealth through the convex envelope of an RVaR-specific transformed weighting function. The RVaR-specific convex envelope exhibits two distinct geometric configurations. Under Scenario~1, the terminal wealth has a VaR-type structure: the minimum-RVaR payoff is digital-option-like, whereas for $0 0$, pure minimum-RVaR optimization may assign zero terminal wealth in sufficiently bad states because the extreme quantile region $[0,\alpha)$ is excluded from the RVaR criterion. These results clarify how interval-based quantile risk control shapes state-dependent portfolio payoffs and connects RVaR with its VaR and ES limiting cases.