This paper derives a closed-form expansion for the conditional expectation of a continuous-time stochastic process, given by for , where evolves according to the extended Cox-Ingersoll-Ross process, for any functions f and g. We apply the Feynman-Kac theorem to state a Cauchy problem associated with and solve the problem by using the reduction method. Furthermore, we extend our method to any piecewise function f; demonstrating our method can be applied to price options in financial derivative markets. In numerical study, we employ Monte Carlo simulations to demonstrate the performance of the current method.