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We consider the existence of a classical smooth solution to the backwardKolmogorov equation \begin{align*} \begin{cases} \partial_t u(t,x)=Au(t,x),&x\ge0,\ t\in[0,T],\\ u(0,x)=f(x),& x\ge0, \end{cases} \end{align*} where $A$ isthe generator of the CIR process, the solution to the stochastic differentialequation \begin{equation*} X^x_t=x+\int_0^t\theta\bigl(\kappa-X^x_s\bigr)\,ds+\sigma\int _0^t\sqrt {X^x_s} \,dB_s, \quad x\ge0,\t\in[0,T], \end{equation*} that is,$Af(x)=\theta(\kappa-x)f'(x)+\frac{1}{2}\sigma^2xf''(x)$, $ x\ge0$($\theta,\kappa,\sigma>0$). Alfonsi \cite{Alfonsi} showed that the equation hasa smooth solution with partial derivatives of polynomial growth, provided thatthe initial function $f$ is smooth with derivatives of polynomial growth. Hisproof was mainly based on the analytical formula for the transition density ofthe CIR process in the form of a~rather complicated function series. In thispaper, for a CIR process satisfying the condition $\sigma^2\le4\theta\kappa$,we present a direct proof based on the representation of a CIR process in termsof a~squared Bessel process and its additivity property.

On backward Kolmogorov equation related to CIR process