Smooth random functions, random ODEs, and Gaussian processes
The usual way in which mathematicians work with randomness is by a rigorous formulation of the idea of Brownian motion, which is the limit of a random walk as the step length goes to zero. A Brownian path is continuous but nowhere differentiable, and this nonsmoothness is associated with technical c...
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| Format: | Journal article |
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Society for Industrial and Applied Mathematics
2019
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| _version_ | 1826295958269329408 |
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| author | Filip, S Javeed, A Trefethen, L |
| author_facet | Filip, S Javeed, A Trefethen, L |
| author_sort | Filip, S |
| collection | OXFORD |
| description | The usual way in which mathematicians work with randomness is by a rigorous formulation of the idea of Brownian motion, which is the limit of a random walk as the step length goes to zero. A Brownian path is continuous but nowhere differentiable, and this nonsmoothness is associated with technical complications that can be daunting. However, there is another approach to random processes that is more elementary, involving smooth random functions defined by finite Fourier series with random coefficients or, equivalently, by trigonometric polynomial interpolation through random data values. We show here how smooth random functions can provide a very practical way to explore random effects. For example, one can solve smooth random ordinary differential equations using standard mathematical definitions and numerical algorithms, rather than having to develop new definitions and algorithms of stochastic differential equations. In the limit as the number of Fourier coefficients defining a smooth random function goes to $\infty$, one obtains the usual stochastic objects in what is known as their Stratonovich interpretation. |
| first_indexed | 2024-03-07T04:08:58Z |
| format | Journal article |
| id | oxford-uuid:c72dbe69-b1de-48bf-83f8-838ad356e44d |
| institution | University of Oxford |
| last_indexed | 2024-03-07T04:08:58Z |
| publishDate | 2019 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:c72dbe69-b1de-48bf-83f8-838ad356e44d2022-03-27T06:43:08ZSmooth random functions, random ODEs, and Gaussian processesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:c72dbe69-b1de-48bf-83f8-838ad356e44dSymplectic Elements at OxfordSociety for Industrial and Applied Mathematics2019Filip, SJaveed, ATrefethen, LThe usual way in which mathematicians work with randomness is by a rigorous formulation of the idea of Brownian motion, which is the limit of a random walk as the step length goes to zero. A Brownian path is continuous but nowhere differentiable, and this nonsmoothness is associated with technical complications that can be daunting. However, there is another approach to random processes that is more elementary, involving smooth random functions defined by finite Fourier series with random coefficients or, equivalently, by trigonometric polynomial interpolation through random data values. We show here how smooth random functions can provide a very practical way to explore random effects. For example, one can solve smooth random ordinary differential equations using standard mathematical definitions and numerical algorithms, rather than having to develop new definitions and algorithms of stochastic differential equations. In the limit as the number of Fourier coefficients defining a smooth random function goes to $\infty$, one obtains the usual stochastic objects in what is known as their Stratonovich interpretation. |
| spellingShingle | Filip, S Javeed, A Trefethen, L Smooth random functions, random ODEs, and Gaussian processes |
| title | Smooth random functions, random ODEs, and Gaussian processes |
| title_full | Smooth random functions, random ODEs, and Gaussian processes |
| title_fullStr | Smooth random functions, random ODEs, and Gaussian processes |
| title_full_unstemmed | Smooth random functions, random ODEs, and Gaussian processes |
| title_short | Smooth random functions, random ODEs, and Gaussian processes |
| title_sort | smooth random functions random odes and gaussian processes |
| work_keys_str_mv | AT filips smoothrandomfunctionsrandomodesandgaussianprocesses AT javeeda smoothrandomfunctionsrandomodesandgaussianprocesses AT trefethenl smoothrandomfunctionsrandomodesandgaussianprocesses |