An elementary Burkholder's inequality for simple stochastic integral says that given nonnegative martingale $M$ and simple bounded predictable process $H$, it holds that for all $c>0$, the tail probability of the running max of the stochastic integral is bounded by $$ cP[|H\cdot M|_t^*\ge c]\le9\|H\|_\infty \|M_t\|_1$$ where $^*$ refers to running max. I need to generalize to arbitrary martingale $M$ by using the decomposition $M=U-V$ where $U,V$ are nonnegative martingales, and prove the same inequality, with the constant $9$ replaced by $18$. However, I am stuck at $$cP[|H\cdot (U-V)|_t^*\ge c]\le cP[|H\cdot U|_t^*+|H\cdot V|_t^*\ge c]$$ and cannot proceed. I don't think I am following the right direction. Any suggestions? Thank you very much.
2026-04-02 19:34:45.1775158485
Burkholder's inequality for elementary stochastic integral
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By assumption, $$P[|H\cdot M|^*_t\ge c]\le P[|H\cdot U|^*_t\ge c/2]+P[|H\cdot V|^*_t\ge c/2]\le 18/c(\|U_t\|_1+\|V_t\|_1)$$ The $M=U-V$ decomposition as described by Writing a martingale as the difference of two non-negative martingales satisfies $\|M_t\|_1=\|U_t\|_1+\|V_t\|_1$.