The Feynman-Kac Formula Of The Differential Equation Driven By A Time-changed Lévy Noise

The type of random motion like Brownian motion,is the only Lévy process with continuous path,and it has the property of semi-martingale.The Feynman-Kac formula corresponding to the solution of stochastic differential equation driven by it can be solved by the Markovian semigroup theory and the Ito formula of continuous semi-martingales,also known as the Kolmogorov equation.Moreover,we know that the transition probability of the solution process is the fundamental solution of the Feynman-Kac formula.This type of stochastic process is limited to continuous martingale-driven stochastic processes.Based on the generalization of this driving process(taking Brownian motion as a special case),we know that stochastic calculus can be driven by semi-martingales.Especially,for Lévy-driven stochastic differential equations,we can utilize the Ito formula with jumps to effectively derive its corresponding generator and discuss the partial differential equation corresponding to the solution of the equation.In this paper,starting from Lévy-driven stochastic differential equations,the existence of the solution Xt is guaranteed.The solution of XDlβ(t)corresponding to Feynman-Kac formula and its the existence and uniqueness of are discussed by using the semigroup theory and the idea of time-subordinated Xt.Since the time changed levy process is still a semi-martingale,we can discuss the stochastic differential equation satisfied by the process and try to establish the relationship between the present stochastic differential equation and the original stochastic differential equation.As an application of the Feynman-kac formula of time changed markov process,we also discuss the alternating motion between time-changed zero-mean Gaussian motion and time-changed α-stable motion,and its Feynman-Kac formula.