Stochastic Calculus via Stopping Derivatives
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910012385787904 |
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| author | Simpson, Alex |
| author_facet | Simpson, Alex |
| contents | We show that a substantial portion of stochastic calculus can be developed along similar lines to ordinary calculus, with derivative-based concepts driving the development. We define a notion of stopping derivative, which is a form of right derivative with respect to stopping times. Using this, we define the drift and variance rate of a process as stopping derivatives for (generalised) conditional expectation and conditional variance respectively. Applying elementary, derivative-based methods, we derive a calculus of rules describing how drift and variance rate transform under operations on processes, culminating in a version of the multi-dimensional Itô formula. Our approach connects with the standard machinery of stochastic calculus via a theorem establishing that continuous processes with zero drift coincide with random translations of continuous local martingales. This equivalence allows us to derive a fundamental theorem of calculus for stopping derivatives, which relates the quantities of drift and variance rate, defined as stopping derivatives, to parameters used in the description of a process as a stochastic integral. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02862 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stochastic Calculus via Stopping Derivatives Simpson, Alex Probability 60H99 (Primary) 60G07, 60G48, 60H05 (Secondary) We show that a substantial portion of stochastic calculus can be developed along similar lines to ordinary calculus, with derivative-based concepts driving the development. We define a notion of stopping derivative, which is a form of right derivative with respect to stopping times. Using this, we define the drift and variance rate of a process as stopping derivatives for (generalised) conditional expectation and conditional variance respectively. Applying elementary, derivative-based methods, we derive a calculus of rules describing how drift and variance rate transform under operations on processes, culminating in a version of the multi-dimensional Itô formula. Our approach connects with the standard machinery of stochastic calculus via a theorem establishing that continuous processes with zero drift coincide with random translations of continuous local martingales. This equivalence allows us to derive a fundamental theorem of calculus for stopping derivatives, which relates the quantities of drift and variance rate, defined as stopping derivatives, to parameters used in the description of a process as a stochastic integral. |
| title | Stochastic Calculus via Stopping Derivatives |
| topic | Probability 60H99 (Primary) 60G07, 60G48, 60H05 (Secondary) |
| url | https://arxiv.org/abs/2501.02862 |