Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
Are you sure a solution exists? – Did Mar 9 '12 at 9:07
up vote 1 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.