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  • curprev 14:48, 10 March 2023‎ Kevin Wang talk contribs‎ 494 bytes +494‎ Given: [$](\Omega,\mathcal{F},\mathbb{P})[/$] is a probability space [$]I[/$] is an index set with total order [$]\leq[/$] [$]\mathbb{F}=(\mathcal{F}_i)_{i\in I}[/$] is a filtration of the sigma algebra [$]\mathbb{F}[/$] [$](S,\Sigma)[/$] is a measurable space [$]X:I\times\Omega\rightarrow S[/$] is a stochastic process [$]X[/$] is adapted to the filtration [$](\mathcal{F}_i)_{i\in I}[/$] if [$]X_i:\Omega\rightarrow S[/$] is a [$](\mathcal{F}_i,\Sigma)[/$]-measurable function
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