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I'm trying to deal with the following problem:

There is a third-order differential equation,

$3G(p) + p G'(p) - 2s[3G''(p)+pG'''(p)] = 0$,

where $G(p)$ is an unknown cumulative distribution function (defined over some support $[\underline{p},\overline{p}]$, without any mass points or holes), and $s \in (0,1/2)$ is a parameter.

The boundary conditions are $G(\overline{p}) = 1$ (since $G(\cdot)$ is a CDF), $G'(\overline{p}) = 0$, and

$G''(\overline{p}) = \frac{-1+3\overline{p}-3\overline{p}^2-2s}{2s\overline{p}(1-2\overline{p})}$

(the latter two follow from a theoretical model).

Hence, conditional on any $\overline{p}$, the CDF is fully specified, and can quickly be computed numerically (there is even an analytical solution that can be obtained via Mathematica, but it seems too unwieldy to work with).

Anyway, I don't really care about the solution to the differential equation, but I want to find the value of $\overline{p}$ such that the resulting CDF, conditional on $\overline{p}$ and the boundary conditions, gives rise to an expected value of

$E(\tilde{p}) = \frac{-1-3\overline{p}^2+4\overline{p}+2s}{2-4\overline{p}}$

(again, this follows from the theoretical model).

I do not expect to get an analytical solution for $\overline{p}$, but at least it would be nice to have a fast numerical solution.

The best I can do so far is the following:

(1) Fix any (initally low) $\overline{p}$. (2) Compute $G(p|\overline{p})$ as the numerical solution to the differential equation. (3) Numerically compute the expected value of it. (4) Compare this to the "predicted" value of $E(\tilde{p})$ as indicated above. (5) If they are sufficiently close, take $\overline{p}$ as the solution to the problem. If not, increase $\overline{p}$ slightly and repeat from (1). (This can be automatized, but takes several seconds to solve for each $s$). E.g., for $s=0.1$, the unique solution seems to be at $0.302046$.

I thus wonder: Is there any way to find the fixed point more efficiently? Ideally, I'd like to have a single equation in $\overline{p}$ that can be solved for numerically (without first solving the differential equation + applying numerical integration).

Any ideas are appreciated! Many thanks in advance.

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  • $\begingroup$ In case it helps: The differential equation (also) follows from the condition $\frac{1}{2}p^3[G(p)-2sG''(p)] = k$, where $k$ is some constant. From theory, $k$ depends on $\overline{p}$ as follows: $k = \frac{\overline{p}^2 \left(\overline{p}^2-2 \overline{p}+2 s+1\right)}{2-4 \overline{p}}$. $\endgroup$ – Martin Jun 21 '17 at 15:43

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