So I have the following optimization problem:
min. $-E^Q[u(h(x))]$
s.t $\int h(x)q(x)dx \leq \frac{V_0}{B_0}$
Where $Q$ is the subjective probability which then gives:
$E^Q[u(h(x))]=\int u(h(x))p(x)dx$
Thus the lagrangian becomes:
$L(X,\lambda)=-\int u(h(x))p(x)dx+\lambda \int h(x)q(x)dx + \lambda \frac{V_0}{B_0}$
So If I remember correctly one takes the derivative of, in this case, $X$ and $\lambda$ then equal the derivative to $0$.
I would obviously get $2$ equations and from there solve the optimization problem however I do get one equation correctly after taking the derivative but I don't manage with the derivative with respect to $X$.
$\frac{dL}{d\lambda}=\int h(x)q(x)dx +\frac{V_0}{B_0}=0$
The derivative with respect to $X$ however I don't get it to be correct. The correct outcome should be, according to the book.
$\frac{dL}{dX}=\int(u'(h(x))p(x)-\lambda q(x))(g(x)-h(x))dx=0$
The $g(x)$ isn't defined which kinda makes me think that I might have missed something...can't see what though.