Existence of non-CARA Meyer risk measures

Determine whether there exists a monetary risk measure on the space of bounded random variables that is consistent with the v-SD order for some threshold utility function v (i.e., is a v-Meyer risk measure) but is not consistent with any e_c-SD order generated by exponential (CARA) utilities e_c(x)=exp(cx) for c>0, e_0(x)=x, and e_c(x)=-exp(cx) for c<0.

Background

The paper studies monetary risk measures that are consistent with fractional stochastic orders (v-SD orders), called Meyer risk measures. For exponential (CARA) threshold utilities e_c, the authors provide constructive representations yielding many nontrivial examples. For general threshold utilities v with nonconstant absolute risk aversion, however, base risk measures typically fail to be v-SD-consistent, and additional impossibility results (e.g., under unbounded threshold risk aversion) severely limit existence.

After establishing that v-Meyer risk measures are naturally represented as lower envelopes of base risk measures but that these base measures are generally not v-consistent unless v is CARA, the authors pose the question whether any Meyer risk measures exist that are not simultaneously CARA-based (e_c-Meyer).