Ok, I hope this question makes sense...
Suppose the the variable y has a distribution of F(y|x).
Suppose after some work I have the following equation:
(1) $\int D dD = \int_{-\infty}^\infty[\int h(x,y,g)dg]d(F(y|x)) $
Clearly, we have that $ \int D dD = 0.5*D^2 +C_1$. Suppose that $ \int h(x,y,g)dg = g(x,y)+C_2 $
where $C_1$ & $C_2$ are both constants of integration.
My question is does $ C_2 $ necessarily depend on x and y? Put another way, can I rewrite equation (1) as:
$0.5*D^2 +C = \int_{-\infty}^\infty g(x,y)d(F(y|x)) $
or must I leave it as:
$0.5*D^2 +C_1 = \int_{-\infty}^\infty [g(x,y)+C_2]d(F(y|x)) $
?
Hope, I managed to make some sense here.
I do have an initial condition that I would satisfy to find the constants, its that D(x)=0 when x=0. But, I'm still confused as to what that would mean. Would that imply that we have:
$ C=\int_{-\infty}^\infty g(0,y)dF(y|x=0) $
I wouldn't worry so much about the infinite limits, its just there as a substitute for the range of y, if you prefer, you can think about the problem as being from $a$ to $b$ instead of the infinite limits.