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. 
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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/

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