Geodesic
Sharp maximal inequalities for stochastic processes
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 162-181

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This work is a survey of existing methods and results in the problem of estimating the mathematical expectation of the maximum of a random process up to an arbitrary Markov time. Both continuous-time (standard Brownian motion, skew Brownian motion, Bessel processes) and discrete-time (symmetric Bernoulli random walk and its modulus) processes are considered. 
@article{TM_2014_287_a9,
     author = {Ya. A. Lyulko and A. N. Shiryaev},
     title = {Sharp maximal inequalities for stochastic processes},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {162--181},
     publisher = {mathdoc},
     volume = {287},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2014_287_a9/}
}
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Ya. A. Lyulko; A. N. Shiryaev. Sharp maximal inequalities for stochastic processes. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Stochastic calculus, martingales, and their applications, Tome 287 (2014), pp. 162-181. http://geodesic.mathdoc.fr/item/TM_2014_287_a9/