Solving integral equation with CDF

Question

I would like to manipulate a functional equation where $F$ is a cumulative distribution function, which I would let Mathematica know.

Using a simple example, consider

$$(\rho + \lambda\overline{F}(w))V(w) = w + \lambda \int_w^\overline{w} V(x) \,\text{d}F(x)$$

where the integral is an expectation for the draws larger than $w$ (which is distributed according to $F$ over a support $[\underline{w}, \overline{w}]$) and $\overline{F}(w) = 1-F(w)$. I would like to manipulate this expression using the properties of the CDF, and eventually solve it for a function such as $V(w)$ —- in this case a simple task, but I would like Mathematica to do it.

I think RSolve should be the right tool to handle functional equations, but I can't manage to obtain an output. I am trying something like:

RSolve[(rho + l*(1-F[w]))*V[w] == w + l*Integrate[V[x], {F[x], w, w1}], V[w], w]

Answer 1

Under the assumption that the PDF exists we have $\int_w^{w1} V(x)\,dF(x)= \int_w^{w1} V(x)F'(x)\,dx$. Differentiating both sides of the equation under consideration by w, we reduce the integral equation under consideration to a usual ODE and solve it.

D[(rho + l*(1 - F[w]))*V[w], w] == D[w + l*Integrate[V[x]*F'[x], {x, w, w1}], w]
DSolve[%, V[w], w]
(*{{V[w] ->  C[1] + Inactive[Integrate][1/(l + rho - l F[K[1]]), {K[1], 1, w}]}}*)

No properties of the CDF are used in the above.