On backward Kolmogorov equation related to CIR process

Volume 5, Issue 1 (2018), pp. 113–127

Vigirdas Mackevičius

^{ }Gabrielė Mongirdaitė^{ }
Pub. online: 6 March 2018
Type: Research Article
Open Access

Received

20 November 2017

20 November 2017

Revised

31 January 2018

31 January 2018

Accepted

2 February 2018

2 February 2018

Published

6 March 2018

6 March 2018

#### Abstract

We consider the existence of a classical smooth solution to the backward Kolmogorov equation

\[ \left\{\begin{array}{l@{\hskip10.0pt}l}\partial _{t}u(t,x)=Au(t,x),\hspace{1em}& x\ge 0,\hspace{2.5pt}t\in [0,T],\\{} u(0,x)=f(x),\hspace{1em}& x\ge 0,\end{array}\right.\]

where *A*is the generator of the CIR process, the solution to the stochastic differential equation \[ {X_{t}^{x}}=x+{\int _{0}^{t}}\theta \big(\kappa -{X_{s}^{x}}\big)\hspace{0.1667em}ds+\sigma {\int _{0}^{t}}\sqrt{{X_{s}^{x}}}\hspace{0.1667em}dB_{s},\hspace{1em}x\ge 0,\hspace{2.5pt}t\in [0,T],\]

that is, $Af(x)=\theta (\kappa -x){f^{\prime }}(x)+\frac{1}{2}{\sigma }^{2}x{f^{\prime\prime }}(x)$, $x\ge 0$ ($\theta ,\kappa ,\sigma >0$). Alfonsi [1] showed that the equation has a smooth solution with partial derivatives of polynomial growth, provided that the initial function *f*is smooth with derivatives of polynomial growth. His proof was mainly based on the analytical formula for the transition density of the CIR process in the form of a rather complicated function series. In this paper, for a CIR process satisfying the condition ${\sigma }^{2}\le 4\theta \kappa $, we present a direct proof based on the representation of a CIR process in terms of a squared Bessel process and its additivity property.