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In a paper I am reading, it is claimed that the SDE $$ dX_t = b X_t dt + (\sigma X_t + \beta_t) dB_t , \quad t \in [0,T],$$ satisfies $$ E\left[ \sup_{t \in [0,T]} |X_t|^p \right] < \infty, \forall p \in [2,\infty), $$ where $b, \sigma \in \mathbb{R}$ are constants and $\beta$ is a process satisfying $$ E \left[ \int_0^T | \beta_t |^2 dt \right] < \infty . $$ Is this true?

White
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  • What is the definition of $X_t$ ? – user619894 Jan 06 '19 at 12:16
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    Without additional assumptions on $\beta$ this is not true. Consider for instance $b=0$, $\sigma=0$ and $\beta_t := Z$ for some random variable $Z \in L^2(\mathbb{P})$ which is independent of $(B_t)_t$ and which fails to have a finite moment of order $p>2$ (i.e. $\mathbb{E}(|Z|^p)=\infty$) – saz Jan 06 '19 at 12:48
  • that's what I thought, thank you. – White Jan 06 '19 at 12:53

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