# 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

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Are you sure a solution exists? – Did Mar 9 '12 at 9:07

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.