Stochastic Calculus for Markov Processes Associated with Semi-Dirichlet Forms

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$F^{[u]}t:=\sum{0<s\leq t}(\tilde u(X_{s})-\tilde u(X_{s-}))1_{{|\tilde u(X_{s})-\tilde u(X_{s-})|>1}}$,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 $$A_t^{[u]}=Y_t^{[u]}+Z_t^{[u]}+F_t^{[u]}\mathrm{.}$$ 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 It^{o}'s formula.