Dynamic minimisation of the commute time for a one-dimensional diffusion
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914709445279744 |
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| author | Hernández-Hernández, Ma. Elena Jacka, Saul |
| author_facet | Hernández-Hernández, Ma. Elena Jacka, Saul |
| contents | Motivated in part by a problem in simulated tempering (a form of Markov chain Monte Carlo) we seek to minimise, in a suitable sense, the time it takes a (regular) diffusion with instantaneous reflection at 0 and 1 to travel to $1$ and then return to the origin (the so-called commute time from 0 to 1). Substantially extending results in a previous paper, we consider a dynamic version of this problem where the control mechanism is related to the diffusion's drift via the corresponding scale function. We are only able to choose the drift at each point at the time of first visiting that point and the drift is constrained on a set of the form $[0,\ell)\cup(i,1]$. This leads to a type of stochastic control problem with infinite dimensional state. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06715 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dynamic minimisation of the commute time for a one-dimensional diffusion Hernández-Hernández, Ma. Elena Jacka, Saul Probability Primary 60J25, secondary 60J27, 60J60, 93E20 Motivated in part by a problem in simulated tempering (a form of Markov chain Monte Carlo) we seek to minimise, in a suitable sense, the time it takes a (regular) diffusion with instantaneous reflection at 0 and 1 to travel to $1$ and then return to the origin (the so-called commute time from 0 to 1). Substantially extending results in a previous paper, we consider a dynamic version of this problem where the control mechanism is related to the diffusion's drift via the corresponding scale function. We are only able to choose the drift at each point at the time of first visiting that point and the drift is constrained on a set of the form $[0,\ell)\cup(i,1]$. This leads to a type of stochastic control problem with infinite dimensional state. |
| title | Dynamic minimisation of the commute time for a one-dimensional diffusion |
| topic | Probability Primary 60J25, secondary 60J27, 60J60, 93E20 |
| url | https://arxiv.org/abs/2403.06715 |