Occupied Processes: Going with the Flow
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914514657607680 |
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| author | Tissot-Daguette, Valentin |
| author_facet | Tissot-Daguette, Valentin |
| contents | A stochastic process $X$ becomes occupied when it is enlarged with its occupation flow $\mathcal{O}$ that tracks the time spent by the path at each level. When $X$ is Markov, the occupied process $(\mathcal{O},X)$ 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 $\mathcal{O}$ 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. The framework's benefits are illustrated via an optimal stopping problem involving local times, followed by financial applications. For the latter, we show how occupation flows provide unified Markovian lifts for exotic options and variance instruments, allowing financial institutions to price derivatives books with a single numerical solver. We finally explore an extension of forward variance models so as to leverage the entire forward occupation surface. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07936 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Occupied Processes: Going with the Flow Tissot-Daguette, Valentin Probability Mathematical Finance Pricing of Securities 60J55, 60J25, 60H30, 60G40, 91G20 A stochastic process $X$ becomes occupied when it is enlarged with its occupation flow $\mathcal{O}$ that tracks the time spent by the path at each level. When $X$ is Markov, the occupied process $(\mathcal{O},X)$ 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 $\mathcal{O}$ 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. The framework's benefits are illustrated via an optimal stopping problem involving local times, followed by financial applications. For the latter, we show how occupation flows provide unified Markovian lifts for exotic options and variance instruments, allowing financial institutions to price derivatives books with a single numerical solver. We finally explore an extension of forward variance models so as to leverage the entire forward occupation surface. |
| title | Occupied Processes: Going with the Flow |
| topic | Probability Mathematical Finance Pricing of Securities 60J55, 60J25, 60H30, 60G40, 91G20 |
| url | https://arxiv.org/abs/2311.07936 |