# Law of an integrated CIR Process as sum of Independent Random Variables

It is known (see for example Joshi-Chan "Fast and Accureate Long Stepping Simulation of the Heston SV Model" available at SSRN) that for a CIR process defined as :

$$dY_t= \kappa(\theta -Y_t)dt+ \omega \sqrt{Y_t}dW_t$$ $Y_0=Y$ together with a correct set of constraints on parameters' value.

Then the law of $\int_0^T Y_t dt$ conditionaly on $Y_0,Y_T$ can be seen as the sum of three (rather complicated) independent random variables (see proposition 4 eq 2.10 in Joshi Chan article)

NB : The original result is coming from Glassermann and Kim "Gamma Expansion of the Heston Stochastic Volatility Model" available at SSRN, but I'm more used to Joshi Chan's expression.

So here is my question :

Does the integrated CIR process itself by any chance has such a representation in the form of the sum of independent random variables ?

PS: The Laplace transform has a known closed-form expression but I couldn't infer directly from this such a representation.

Edit : As Tal has opened a bounty on this here is the Laplace transform of the integrated CIR process :

$$\mathcal{L}\left\{\int_0^t Y_s ds\right\}(\lambda)=\mathbb{E}\left[e^{-\lambda\int_0^t Y_s ds}\right]=e^{-A_\lambda(t)-Y_0.G_\lambda(t)}$$ with $A_\lambda(t)=-\frac{2\kappa.\theta }{\omega^2}. \mathrm{Ln}\left[\frac{2\gamma.e^{(\gamma+\kappa).s/2}}{\gamma.(e^{s.\gamma}+1)+\kappa.(e^{s.\gamma}-1)}\right]$ and $G_\lambda(t)=\frac{2.\lambda.(e^{s.\gamma}-1)}{\gamma.(e^{s.\gamma}+1)+\kappa.(e^{s.\gamma}-1)}$ where $\gamma=\sqrt{\kappa^2+\omega^2.\lambda}$

This is coming from Chesnay, Jeanblanc-Picqué, Yor "Mathematical Methods for Financial Markets" Proposition 6.3.4.1

Best regards

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Well the CIR can be written as non central chi-squared random variable if $\nu=\frac{4\kappa \theta}{\sigma^2} \in \mathbb{N}$. So a finite sum of such process sould also be a non central Chi-square. Now if we could find stabiling transformation such that $\nu_n% tend to$\vu\$. We could get a converging series. However, this remind me alot of the cramer-von-mises statistics series, which a sum of chi-squared with variance equal to the the reciprocal of the square of an odd number. That distribution is unknown. – Drmanifold Dec 13 '13 at 23:15