Let ( , F , F , P ) ( , F , F , P ) be a filtered probability space and denote by ( L t , t 0 ) ( L t , t 0 ) the Radon Nikodm density of Q Q with respect to P P Then, if F F is a subfiltration of F F , prove that Q F t L t P F t Q F t L t P F t , where L t E P ( L t F t ) ...

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Let ( , F , F , P ) ( , F , F ,

Question:

Let $(Ω, F, F, P)$ be a filtered probability space and denote by $(L_{t}, t \geq 0)$ the Radon-Nikodým density of $Q$ with respect to $P$ . Then, if $\tilde{F}$ is a subfiltration of $F$ , prove that ${Q |}_{{\tilde{F}}_{t}} = {{\tilde{L}}_{t} P |}_{{\tilde{F}}_{t}}$ , where ${\tilde{L}}_{t} = E_{P} (L_{t} ∣ {\tilde{F}}_{t}) . ◃$