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.#### References

A. Alfonsi. On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl, pages 355–384, 2005. MR2186814

A. Alfonsi. High order discretization schemes for the CIR process: Application to affine term structure and Heston models. Mathematics of Computation, 79(269):306–237, 2010. MR2552224

S. Cheng. Differentiation under the integral sign with weak derivatives. Technical report, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.525.2529&rep=rep1&type=pdf, 2010.

S. Cerrai, Second Order PDE’s in Finite and Infinite Dimension: A Probabilistic Approach, Springer, 2001. MR1840644

J.C. Cox, J.E. Ingersoll, and S.A. Ross. A theory of the term structure of interest rates. Econometrica, 53:385–407, 1985. MR0785475

P.M.N. Feehan and C.A. Pop, A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients, J. Differential Equations, 254:4401–4445, 2013. MR3040945

M. Jeanblanc, M. Yor, and M. Chesney. Mathematical Methods for Financial Markets. Springer, 2009. MR2568861

P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992 (2nd corrected printing 1995). MR1214374