Suppose that $T>0$, $(\alpha_t)_{0 \leq t \leq T}$ is a stochastic process such that there exists a constant $a>0$ so that for all $t\in[0, T], \alpha_t \geq a$. Let $(W_t)_t$ a Brownian motion. Is it true that $$\sup_{0\leq t \leq T}\int_0^t a dW_s \leq \sup_{0\leq t\leq T}\int_0^t \alpha_s dW_s \quad a.s. \quad?$$
2026-03-27 17:04:54.1774631094
Comparison of stochastic integrals
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