Geodesic
A sharp maximal inequality for continuous martingales and their differential subordinates
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 1001-1018.

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Assume that $X$, $Y$ are continuous-path martingales taking values in $\mathbb R^\nu $, $\nu \geq 1$, such that $Y$ is differentially subordinate to $X$. The paper contains the proof of the maximal inequality $$ \|\sup _{t\geq 0} |Y_t| \|_1\leq 2\|\sup _{t\geq 0} |X_t| \|_1. $$ The constant $2$ is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder's method and rests on the construction of an appropriate special function.
DOI : 10.1007/s10587-013-0068-3
Classification : 60G44, 60G46
Keywords: martingale; stochastic integral; maximal inequality; differential subordination 
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     title = {A sharp maximal inequality for continuous martingales and their differential subordinates},
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Osękowski, Adam. A sharp maximal inequality for continuous martingales and their differential subordinates. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 4, pp. 1001-1018. doi : 10.1007/s10587-013-0068-3. http://geodesic.mathdoc.fr/articles/10.1007/s10587-013-0068-3/

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