For weighted exponential utilities, the alpha-robust risk measure is law-invariant and the intractable-claim utility maximization reduces to a concave quantile optimization whose optimum satisfies a two-dimensional first-order ODE system with mixed boundary conditions.
Merton,Lifetime portfolio selection under uncertainty: The continuous-timecase, The Review of Economics and Statistics, 51 (1969), pp
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$\alpha$-robust utility maximization with intractable claims: A quantile optimization approach
For weighted exponential utilities, the alpha-robust risk measure is law-invariant and the intractable-claim utility maximization reduces to a concave quantile optimization whose optimum satisfies a two-dimensional first-order ODE system with mixed boundary conditions.