Lemma 4 ∀ δ > O sufficiently small ∀ d ∈ Rn with ∣∣d∣∣ = δ 3 ε > O indepen-
dent of d and a state i ∈ Ωmax s.t.
L(x* + d, S{) — L(x*,S{) > ε
Proof of lemma 4:. The difference can be expressed as
L(x* + d,si) — L(x*r,si) = VL(x*r,si)d + d'V2L(x)i, si)d + 0(3) (15)
where 0(3) is a third order approximation error. Consider the first order
term: From the optimality of x*r follows that
VL(x*r,si)d ≥ O (16)
for some i ∈ Ωmax. Next, fix such an i and consider the second order term.
Since V2L(xr,si) is normal and positive definite, we have
V2L(x*r,si) = U,iDi Ui
where Ui is unitary and
Di = diag(λi,ι... λi,n)
with λij > O being the eigenvalues of V2L(xtr,, si). Then defining λijmin =
minj χi,j
d'V2L(x*,s)d = d'U'iDi Uid
≥ λi,mind'U'Uid (17)
= λi,mind'd (18)
= λi,minδ (19)
Letting λmin = mini∈Ωmax λijmin it follows from (15), (??), and (19) that
L(xr + d, si) — L(xr,si) ≥ λminδ2 + 0(3)
Choosing δ sufficiently small the third order approximation error can be
made arbitrarily small, e.g. smaller than λm2inδ, then choosing ε = λm2"δ
establishes the claim. ■
Next, normalize the transformed objective of the Bayesian decision maker
(6) as follows
—k. 1 ʌ , , z 4
l(∙c) = - ∑√≈bfe⅛i (20)
i=l
12
More intriguing information
1. How to do things without words: Infants, utterance-activity and distributed cognition.2. How do investors' expectations drive asset prices?
3. MICROWORLDS BASED ON LINEAR EQUATION SYSTEMS: A NEW APPROACH TO COMPLEX PROBLEM SOLVING AND EXPERIMENTAL RESULTS
4. The name is absent
5. The name is absent
6. An Empirical Analysis of the Curvature Factor of the Term Structure of Interest Rates
7. The name is absent
8. ‘Goodwill is not enough’
9. The name is absent
10. The name is absent