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Upper bound for $\mathbb{E^P}[\,Y_t^4\,]$

Question: Suppose $X_t$ is adapted stochastic process satisfying the condition $$\int_{a}^{b}\mathbb{E^P}[\,X_s^4\,]ds<\infty$$ and let $Y_t=\int_{a}^{t}X_s dW_s$, where $W_t$ is a Wiener process ...
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