A note on the maximal expected local time of $${\text {L}}_2$$-bounded martingales
| Autor: | Isaac Meilijson, Laura Sacerdote, David Gilat |
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| Rok vydání: | 2021 |
| Předmět: |
Statistics and Probability
Local time Local time Brownian Motion Martingale Upcrossings General Mathematics First exit time Sigma Local time Martingale Absolute value (algebra) Combinatorics symbols.namesake Martingale Wiener process Bounded function Brownian Motion Upcrossings symbols Interval (graph theory) Statistics Probability and Uncertainty Martingale (probability theory) Mathematics |
| Zdroj: | Journal of Theoretical Probability. 35:1952-1955 |
| ISSN: | 1572-9230 0894-9840 |
| DOI: | 10.1007/s10959-021-01118-0 |
| Popis: | For an $${\text {L}}_2$$ -bounded martingale starting at 0 and having final variance $$\sigma ^2$$ , the expected local time at $$a \in \text {R}$$ is at most $$\sqrt{\sigma ^2+a^2}-|a|$$ . This sharp bound is attained by Standard Brownian Motion stopped at the first exit time from the interval $$(a-\sqrt{\sigma ^2+a^2},a+\sqrt{\sigma ^2+a^2})$$ . In particular, the maximal expected local time anywhere is at most $$\sigma $$ , 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 (Societe Mathematique de France, Asterisque 157(8), 129–145 1988), by Dubins et al. (Ann Probab 37(1), 393–402 2009) and by the authors (2018). |
| Databáze: | OpenAIRE |
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