An integral representation of the local time of the Brownian motion via the Clark–Ocone formula
Let (LB(t,x),t≥0,x∈R) be the local time of (Bt,t≥0), the real-valued one-dimensional Brownian motion. In this paper, in case of g, a strictly increasing and bijective function, we propose some integral representations of Lg(B)(t,x), of the form: R(t,x)+∫0tK(t,x,Bs)dBs, where R(t,x) is a deterministi...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Elsevier
2025-05-01
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| Series: | Results in Applied Mathematics |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2590037425000275 |
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| Summary: | Let (LB(t,x),t≥0,x∈R) be the local time of (Bt,t≥0), the real-valued one-dimensional Brownian motion. In this paper, in case of g, a strictly increasing and bijective function, we propose some integral representations of Lg(B)(t,x), of the form: R(t,x)+∫0tK(t,x,Bs)dBs, where R(t,x) is a deterministic function and K(t,x,Bs) is a random function depending on t and F, the cumulative distribution function of the standard normal distribution N(0,1) and some Brownian functionals with no Malliavin derivative. Our study is based on the case LB(t,x). An exact formula of the expectation E[LB(t,x)] is given in this paper. |
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| ISSN: | 2590-0374 |