What does $\mathbb{E}^x((X_t)_{t \in T})$ for some stochastic process $(X_t)_{t \in T}$ mean? I heard that it is some kind of conditional expectation given that $X(0)=x$, but I can't find a precise definiton. Is it the stochastic process $t\mapsto\mathbb{E}(X_t|\{X(0)=x\})$, where $0$ is the first element in $T$? If yes, is there a simpler way to rewrite this?
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Related question; see this question for some material on Brownian motion – saz Jan 22 '20 at 05:35