Give a probability space $\left(\Omega,\mathscr{F},P\right)$ and a filtration $\left\{ \mathscr{F}_{t}:0\le t<\infty\right\} $ and a random variable $X$. Define the doob martingale $S_{t}=E\left[X|\mathscr{F}_{t}\right]$. Under what condition $\left\{ S_{t},\mathscr{F}_{t}:0\le t<\infty\right\}$ is a continuous martingale? Thanks in advance.
2026-03-30 08:06:24.1774857984
When doob martingale is continuous
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