## Abstract

For an L _{2}-bounded martingale starting at 0 and having final variance σ^{2}, the expected local time at a∈ R is at most σ2+a2-|a|. This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval (a-σ2+a2,a+σ2+a2). In particular, the maximal expected local time anywhere is at most σ, and this bound is sharp. Sharp bounds for the expected maximum, maximal absolute value, maximal diameter and maximal number of upcrossings of intervals have been established by Dubins and Schwarz (Societé Mathématique de France, Astérisque 157(8), 129–145 1988), by Dubins et al. (Ann Probab 37(1), 393–402 2009) and by the authors (2018).

Original language | English |
---|---|

Pages (from-to) | 1952-1955 |

Number of pages | 4 |

Journal | Journal of Theoretical Probability |

Volume | 35 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2022 |

### Funding

Funders | Funder number |
---|---|

Rice University | |

Istituto Nazionale di Alta Matematica "Francesco Severi" | 217/16 |

Israel Science Foundation |

## Keywords

- Brownian motion
- Local time
- Martingale
- Upcrossings

## Fingerprint

Dive into the research topics of 'A note on the maximal expected local time of L_{2}-bounded martingales'. Together they form a unique fingerprint.