Occupied Processes: Going with the Flow

A stochastic process $X$ becomes \textit{occupied} when it is enlarged with its occupation flow $\mathcal{O}$, which tracks the time spent by the path at each level. Crucially, the occupied process $(\mathcal{O},X)$ enjoys a Markov structure when $X$ is Markov. We develop a novel It\^o calculus for occupied processes that lies midway between Dupire's functional It\^o calculus and the classical version. We derive a surprisingly simple It\^o formula and, through Feynman-Kac, unveil a broad class of path-dependent PDEs where $\mathcal{O}$ plays the role of time. The space variable, given by the current value of $X$, remains finite-dimensional, thereby paving the way for standard elliptic PDE techniques and numerical methods. In the financial applications, we demonstrate that occupation flows provide unified Markovian lifts for exotic options and variance instruments, allowing financial institutions to price and manage derivatives books with a single numerical solver. We then explore avenues in financial modeling where volatility is driven by the occupied process. In particular, we propose the local occupied volatility (LOV) model which not only calibrates to European vanilla options but also offers the flexibility to capture stylized facts of volatility or fit other instruments. We also present an extension of forward variance models that leverages the entire forward occupation surface.