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Mar 8, 2023 at 9:00 comment added Pierre PC If $C$ is (continuous, finite) adapted, then the above will be one solution to the equation, and $\mathbb P(\{X_t\to1\}\Delta\{I=\infty\})=0$ for $\Delta$ the symmetric difference. I think the solution will be unique up to explosion, and by the explicit expression we know it actually does not blow up.
Mar 7, 2023 at 17:28 comment added Fawen90 Thanks for the answer. If my understanding is correct, when $C$ is an adapted stochastic process, i.e. $C=(C_t: t\ge 0)$, the above expression is still the solution, right?
Mar 7, 2023 at 17:27 vote accept Fawen90
Mar 7, 2023 at 12:06 history answered Pierre PC CC BY-SA 4.0