On backward Kolmogorov equation related to CIR process

Abstract

We consider the existence of a classical smooth solution to the backward Kolmogorov 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 $A$ is the generator of the CIR process, the solution to the stochastic differential equation \begin{equation*} X^x_t=x+\int_0^tθ\bigl(κ-X^x_s\bigr)\,ds+σ\int _0^t\sqrt {X^x_s} \,dB_s, \quad x\ge0,\ t\in[0,T], \end{equation*} that is, $Af(x)=\theta (\kappa -x)f^{\prime }(x)+\frac{1}{2}{\sigma }^{2}x{f}^{\prime \prime }(x)$, $x\ge 0$ ($\theta ,\kappa ,\sigma >0$). Alfonsi \cite{Alfonsi} 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.

Published at https://doi.org/10.15559/18-VMSTA98 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)

References in corpus (1)

• A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients