Stochastic Calculus for Markov Processes Associated with Semi-Dirichlet Forms

TL;DR

This paper develops stochastic calculus tools, including martingale decompositions and Itô's formula, for Markov processes associated with semi-Dirichlet forms, extending classical stochastic calculus to a broader class of processes.

Contribution

It introduces a unique decomposition of additive functionals and defines stochastic integrals for semi-Dirichlet form-associated processes, advancing stochastic calculus in this setting.

Findings

Existence and uniqueness of martingale and zero quadratic variation additive functionals.

Definition of stochastic integrals for semi-Dirichlet form processes.

Derivation of Itô's formula for these processes.

Abstract

Let $(\mathcal{E},D(\mathcal{E}))$ be a quasi-regular semi-Dirichlet form and $(X_t)_{t\geq0}$ 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 $$A_t^{[u]}=Y_t^{[u]}+Z_t^{[u]}+F_t^{[u]}.$$ Further, we define the stochastic integral $\int_0^t\tilde v(X_{s-})dA_s^{[u]}$ for $v\in D(\mathcal{E})_{loc}$ and derive the related It\^{o}'s formula.