The stationary distributions of state-dependent diffusions reflected at one and two sides
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909351012204544 |
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| author | Miyazawa, Masakiyo |
| author_facet | Miyazawa, Masakiyo |
| contents | Consider a one-dimensional diffusion process which has state-dependent drift and deviation and is reflected at the origin, which is called a one-side reflected diffusion or simply reflected diffusion. We are particularly interested in the case that its drift and deviation are discontinuous. We define this reflected diffusion as the solution of a stochastic integral equation, and find conditions for its positive recurrence, We then derive its stationary distribution under these conditions. As a related problem, we also consider the case that it is reflected at two sides, which is called a two-sides reflected diffusion. Its existence, positive recurrence and stationary distribution are similarly studied.
In the literature, these problems are studied through a state-dependent diffusion on the whole line particularly when the drift and deviation are discontinuous. However, the reflected process itself is not defined in such a study. Thus, the stationary distribution has not been fully studied for a general state-dependent reflected diffusion. We aim to fills this insufficiency and to make the stationary distributions of reflected diffusions widely available in application. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11496 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The stationary distributions of state-dependent diffusions reflected at one and two sides Miyazawa, Masakiyo Probability 60H20, 60J25, 60J55, 60H30, 60K25 Consider a one-dimensional diffusion process which has state-dependent drift and deviation and is reflected at the origin, which is called a one-side reflected diffusion or simply reflected diffusion. We are particularly interested in the case that its drift and deviation are discontinuous. We define this reflected diffusion as the solution of a stochastic integral equation, and find conditions for its positive recurrence, We then derive its stationary distribution under these conditions. As a related problem, we also consider the case that it is reflected at two sides, which is called a two-sides reflected diffusion. Its existence, positive recurrence and stationary distribution are similarly studied. In the literature, these problems are studied through a state-dependent diffusion on the whole line particularly when the drift and deviation are discontinuous. However, the reflected process itself is not defined in such a study. Thus, the stationary distribution has not been fully studied for a general state-dependent reflected diffusion. We aim to fills this insufficiency and to make the stationary distributions of reflected diffusions widely available in application. |
| title | The stationary distributions of state-dependent diffusions reflected at one and two sides |
| topic | Probability 60H20, 60J25, 60J55, 60H30, 60K25 |
| url | https://arxiv.org/abs/2410.11496 |