Hammerstein stochastic integral equation I'm in trouble with the following integral equation:
$$\phi(t)=\rho\int_0^1 t^2 s \phi(s)^2 ds+\nu(t)$$
where $\nu(t)$ is a white gaussian noise with variance $\sigma$ and mean value $\mu$.
Is it possible to solve this equation in a closed form? Alternatively, can you obtain some property of the spectrum of $\phi(t)$ without solve it?
Thanks for any suggestion

Let $\phi$ denote any solution. Then $\phi(t)=\rho t^2X+\nu(t)$ where $X=\int\limits_0^1s\phi(s)^2\mathrm ds$, hence
$$
X=\int\limits_0^1s(\rho s^2X+\nu(s))^2\mathrm ds=\int\limits_0^1(\rho^2 s^5X^2+2\rho s^3X\nu(s)+s\nu(s)^2)\mathrm ds.
$$
One sees that $X=\tfrac16\rho^2X^2+2\rho XY+Z$,
with
$$Y=\int\limits_0^1s^3\nu(s)\mathrm ds,\qquad Z=\int\limits_0^1s\nu(s)^2\mathrm ds.
$$
Thus $X$ is a root of a quadratic polynomial with (random) discriminant proportional to
$$
D=3(2\rho Y-1)^2-2\rho^2Z.
$$
Unless a reason I fail to see ensures that $D\geqslant0$ with full probability, it seems no solution $\phi$ exists.
