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- W3102041806 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),& xge0, tin[0,T], u(0,x)=f(x),& xge0, 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^ttheta bigl(kappa-X^x_sbigr),ds+sigmaint _0^tsqrt {X^x_s} ,dB_s, quad xge0, tin[0,T], end{equation*} that is, $Af(x)=theta(kappa-x)f'(x)+frac{1}{2}sigma^2xf''(x)$, $ xge0$ ($theta,kappa,sigma>0$). 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 $sigma^2le4thetakappa$, we present a direct proof based on the representation of a CIR process in terms of a~squared Bessel process and its additivity property." @default.
- W3102041806 created "2020-11-23" @default.
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- W3102041806 date "2018-03-06" @default.
- W3102041806 modified "2023-09-26" @default.
- W3102041806 title "On backward Kolmogorov equation related to CIR process" @default.
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- W3102041806 doi "https://doi.org/10.15559/18-vmsta98" @default.
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