# 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

• 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$. We could get a converging series. However, this remind me alot of the cramer-von-mises statistics series, which a sum of chi-squared with variance equal to the the reciprocal of the square of an odd number. That distribution is unknown. – Drmanifold Dec 13 '13 at 23:15 • Can I ask you, what use case do you have in mind for the determining some law of integrated exponential martingales? Asian commodity options? Perpetuities and annuities? – David Addison Apr 17 '18 at 5:36 • Very old post no use today for me but I was interested at the time by stochastic volatility models where volatilty would be driven by a CIR process. Inverting Fourier transform could give many interesting computationnal possibilities for such dynamics in pricing applications – TheBridge Apr 17 '18 at 8:01 • Very interesting. Being on the equities side, I had not thought of that use case. The problem looks related to the Yor process, which is defined as the sum of lognormals, which is difficult to evaluate since its distribution is not lognormal. Also, I doubt you’ll be able to express the law of integrated CIR as the sum of independent variables since, like an O-U, it’s increments are not independent. Dassios and Jayalaxschmi explore integrated CIR processes as they relate to arithmetic Asian options: eprints.lse.ac.uk/2851/1/…. – David Addison Apr 17 '18 at 16:34 • I'm actually very interested in this, but not in the realm of finance. I study bioinformatics (namely phylogenetics), and Lepage et al 2006 used CIR to model mutation rates over time. I am now trying to sample mutation rates for each branch of an evolutionary tree, which requires integrating$dY_t$(given$Y_0$) over some$t\$ – Niema Moshiri Jun 22 '18 at 17:50