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Occupied processes: Going with the flow

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  • Tissot-Daguette, Valentin

Abstract

A stochastic process X becomes occupied when it is enlarged with its occupation flow O that tracks the time spent by the path at each level. When X is Markov, the occupied process (O,X) enjoys a Markov structure as well. We develop an Itô calculus for occupied processes that lies midway between Dupire’s functional Itô calculus and the classical version. We derive Itô formulae and, through Feynman-Kac, unveil a broad class of path-dependent PDEs where 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.

Suggested Citation

  • Tissot-Daguette, Valentin, 2026. "Occupied processes: Going with the flow," Stochastic Processes and their Applications, Elsevier, vol. 195(C).
    • Handle: RePEc:eee:spapps:v:195:y:2026:i:c:s0304414926000220
    DOI: 10.1016/j.spa.2026.104890
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