| Minnesota – p. 2/19 (2009) | |||||||||||||||||
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| • Risks with known probabilities • Risks with unknown probabilities Minnesota – p. 2/19 Risk and uncertainty • Risks with known probabilities • Risks with unknown probabilities • Control and model misspecification Minnesota – p. 2/19 Risk and uncertainty • Risks with known probabilities • Risks with unknown probabilities • Control and model misspecification • Control and estimation risk Minnesota – p. 2/19 Notation and setup • state vector {xt: t ≥ 0}. • control vector {at: t ≥ 0}. • joint distribution of xt+1 and a signal st+1 that depends on xt and at. • St denote an event history generated by current and past signals-constrains actions. • Xt denote the event history generated by current and past state vectors Xt ⊃ St. • Partition the state vector: xt = yt zt where yt is observed and zt is not observed. Minnesota – p. 3/19 More notation • let Z denote a space of admissible unobserved states, Z a corresponding sigma algebra of subsets of states λ a measure on the measurable space of hidden states (Z, Z). • let S denote the space of signals, S a corresponding sigma algebra, and η a measure on the measurable space (S, S) of signals. • law of motion for the evolution of the observable states: yt+1 = πy(st+1, yt, at). • density τ(z ∗ , s ∗ |xt, at); τ is a density relative to the product measure λ × η. • density qt relative to λ for zt conditioned on information St. Minnesota – p. 4/19 Bayes law Use τ to construct two densities for the signal, one of which is conditioned on a finer information set than the other: κ(s ∗ � |yt, zt, at) = τ(z ∗ , s ∗ |yt, zt, at)dλ(z ∗) ς(s ∗ � |yt, qt, at) = κ(s ∗ |yt, z, at)qt(z)dλ(z) | |||||||||||||||||
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