Stochastic Calculus for Markov Processes Associated with Semi-Dirichlet Forms

Abstract

Let $(\mathcal{E},D(\mathcal{E}))$ be a quasi-regular semi-Dirichlet form and $(X_t{)}_{t\ge 0}$ be the associated Markov process. For $u\in D(\mathcal{E}{)}_{loc}$, denote $A_t^{[u]}:=\tilde{u}(X_t)-\tilde{u}(X_0)$ and $F_t^{[u]}:={\sum }_{0<s\le t}(\tilde{u}(X_s)-\tilde{u}(X_{s-}))1_{\{|\tilde{u}(X_s)-\tilde{u}(X_{s-})|>1\}}$, where $\tilde{u}$ is a quasi-continuous version of $u$. We show that there exist a unique locally square integrable martingale additive functional $Y^{[u]}$ and a unique continuous local additive functional $Z^{[u]}$ of zero quadratic variation such that $$A_t^{[u]}=Y_t^{[u]}+Z_t^{[u]}+F_t^{[u]}.$$ Further, we define the stochastic integral ${\int }_0^t\tilde{v}(X_{s-})d{A}_s^{[u]}$ for $v\in D(\mathcal{E}{)}_{loc}$ and derive the related Itô's formula.