CIR process characteristic function

Question

what is the characteristic function of a CIR process given by $dv_t = \kappa (\theta - v_t)dt + \sigma \sqrt{v_t}dW_t$ Unfortunately, I could not find the answer in the literature. I know it is in the class of affine diffusion processes, but how can we find the characteristic function?

Answer 1

The ODEs you posted can be written as following (I will use my notation, which is more common in the finance literature): $$\frac{d}{dt}[C(i\omega,t)]=\theta \mu D(i\omega,t)$$ $$\frac{d}{dt}[D(i\omega,t)]=-\theta D(i\omega,t)+\frac{1}{2}\eta^2D(i\omega,t)^2$$ with initial conditions $D(i\omega,0)=i\omega$ and $C(i\omega,0)=0$. We solve $y:=D(i\omega,t)$ first. Divide by $y^2$ and change to the variable $u=1/y$. You get $$\frac{du}{dt}=\theta u -\frac{1}{2}\eta^2$$ This is a common first order ODE with solution $$u=u_0e^{\theta t}-\frac{\eta^2}{2\theta}(e^{\theta t}-1)$$ By changing back to $D(i\omega,t)$ (take into account the initial value) you get $$D(i\omega,t)=\frac{i\omega e^{-\theta t}}{1-\frac{i\omega}{2\theta}\eta^2(1-e^{-\theta t})}$$ By integration in time (from $0$ to $t$) you find the solution to the other ODE $$C(i\omega, t)=-\frac{2\theta \mu}{\eta^2}\ln\bigg(1-\frac{i\omega}{2\theta}\eta^2(1-e^{-\theta t})\bigg)$$ Since $$\phi(i\omega,t)=\exp\{C(i\omega, t)+D(i\omega,t)v_0\}$$ you get $$\phi(i\omega,t)=\bigg(1-\frac{i\omega}{2\theta}\eta^2(1-e^{-\theta t})\bigg)^{-\frac{2\theta \mu}{\eta^2}}\exp\bigg\{\frac{i\omega v_0e^{-\theta t}}{1-\frac{i\omega}{2\theta}\eta^2(1-e^{-\theta t})}\bigg\}$$