A closed-form expansion for the conditional expectations of the extended CIR process

This paper derives a closed-form expansion for the conditional expectation of a continuous-time stochastic process, given by V-t,V-T := e(-ftT g(vs)ds) f(v(T)) for 0 <= t <= T, where v(t) evolves according to the extended Cox-IngersollRoss process, for any C-infinity functions f and g. We apply the Feynman-Kac theorem to state a Cauchy problem associated with V-t,V-T and solve the problem by using the reduction method. Furthermore, we extend our method to any piecewise C-infinity 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.