Let $X$ and $Y$ be two continuous-time martingales. If quadratic variation of $X$ minus that of $Y$ is a nondecreasing and nonnegative function of time, we say that $Y$ is differentially subordinate to $X$ and prove that $\|Y\|_p \leq (p^\ast - 1)\|X\|_p$ for $1 < p < \infty$, where $p^\ast = p \vee q$ and $q$ is the conjugate of $p$. This inequality contains Burkholder's $L^p$-inequality for stochastic integrals, which implies that the above inequality is sharp. We also extend his concept of strong differential subordination and several other of his inequalities, and sharpen an inequality of Banuelos.
"Differential Subordination and Strong Differential Subordination for Continuous-Time Martingales and Related Sharp Inequalities." Ann. Probab. 23 (2) 522 - 551, April, 1995. https://doi.org/10.1214/aop/1176988278