 Solution

Write $D=\mathbb{E}{\int }_0^{\infty }{e}^{-bt/R}{Z}_t^{1-1/R}dt$ and use the product measure $d\mathbb{P}dt$. For $0<R<1$, the conjugate exponents are $p=1/(1-R)$ and $q=1/R$. Factor the utility integrand asThe Holder inequality on this product measure givesDivide by $1-R$ to obtain the Hölder bound for discounted CRRA consumption:For a finite right side this is ordinary Hölder; on the infinite measure space one can first truncate time and then pass by monotone convergence theorem. If $D=\infty$ and $X_0>0$, the bound is valid but uninformative. If $X_0=0$, the budget forces $c=0$ almost everywhere, so the utility integral is zero; handle this separately instead of writing an undefined $0\cdot \infty$.

When $D$ is finite and the bound is attained, equality requires full budget use and $c_t=(X_0/D)e^{-bt/R}{Z}_t^{-1/R}$. This is the equality pattern of Hölder, not an assertion that every market can finance that process.