Occupied Processes: Going with the Flow
Modeling strongly path-dependent financial derivatives—such as exotic options and variance instruments—remains challenging due to the non-Markovian nature of path-dependent functionals. Method: This paper introduces the “occupied process” framework, augmenting the original process $X$ with its occupation measure flow $O$ to form a Markovian lifted system $(O,X)$. It defines the novel “occupation derivative”, unifying functional Itô calculus and mean-field derivatives, and recasts a broad class of path-dependent PDEs as parabolic equations in the occupation measure time variable. Contribution/Results: The framework enables an Itô calculus tailored to path occupation-time functionals and extends the Feynman–Kac formula accordingly. It yields closed-form solutions to local-time-driven optimal stopping problems, with direct applications to corridor variance swap pricing and path-dependent volatility modeling. By bridging stochastic analysis, mean-field theory, and financial mathematics, this work substantially expands both the theoretical foundations and practical applicability of path-dependent stochastic modeling.