Best Bounds in Doob's Maximal Inequality for Bessel Processes
✍ Scribed by Jesper Lund Pedersen
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 115 KB
- Volume
- 75
- Category
- Article
- ISSN
- 0047-259X
No coin nor oath required. For personal study only.
✦ Synopsis
Let ((Z t ), P z ) be a Bessel process of dimension :>0 started at z under P z for z 0. Then the maximal inequality
is shown to be satisfied for all stopping times { for (Z t ) with E z ({ pÂ2 )< , and all p>(2&:) 6 0. The constants ( pÂ( p&(2&:))) pÂ(2&:) and pÂ( p&(2&:)) are the best possible. If * is the greater root of the equation * 1&(2&:)Â p &*=(2&:)Â (cp&c(2&:)), the equality is attained in the limit through the stopping times
when c tends to the best constant ( pÂ( p&(2&:))) pÂ(2&:) from above. Moreover we show that E z ({ qÂ2 *, p )< if and only if *>((1&(2&:)Âq) 6 0) pÂ(2&:) . The proof of the inequality is based upon solving the optimal stopping problem
by applying the principle of smooth fit and the maximality principle. In addition, the exact formula for the expected waiting time of the optimal strategy is derived by applying the minimality principle. The main emphasis of the paper is on the explicit expressions obtained.