Doob’s Inequality and Lower Estimation of The Maximum of Martingales

Li Zhichan
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Li Zhichan: Hebei University of Technology, Department of Mathematics

Chapter Chapter 20 in Markov Processes and Controlled Markov Chains, 2002, pp 341-349 from Springer

Abstract: Abstract For estimation of the maximum of submartingales, there are classical Doob’s inequalities 0.1 $$ E\mathop{{\sup }}\limits_{t} {{\left| {{{\chi }_{t}}} \right|}^{p}}{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ 1,\frac{1}{p} + \frac{1}{q} = 1, $$ 0.2 $$ E\mathop{{\sup }}\limits_{t} \left| {{{\chi }_{t}}} \right|{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ }}E{{\left| {{{\chi }_{\infty }}} \right|}^{p}},(p{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ > }}1). $$ [1] and [4] have given respectively a non-trivial estimation to the non-negative continuous martingales for p = 1 and 2. This paper considered lower estimation for all cases of p ≥ 1, and got the corresponding inequalities.

Keywords: martingale; inequality of martingales; Dubins’ and Gilat’s conjectures (search for similar items in EconPapers)
Date: 2002
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4613-0265-0_20

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DOI: 10.1007/978-1-4613-0265-0_20

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