(nh1/2)KI2i(fθθ,fθ> ) →p 0
the entire expression for (nh1/2)KI can be approximated in the following way
(105)
(nh1/2)
(nh1/2) 1√2σιNι - (nh1/2) 1√2σ2N2 - 2⅛ -
Jn1
- 2 Jn2)
(106)
Thus, if h к n β with β > ∣, cn ` C(nh) 1
(nh1/2) (kI + 2eə - √2σ1N1 - √2σ2N2 + αC
(107)
then,
(nh1/2) Çk+ + 2eə →d N (αC, 2 (σ2 - σ2))
(108)
9.4 Formula of MGF and expected utility
It can be shown that the moment generating function for the double Gamma distribution is:
MR(t)=exp(tγ)[pMGF (t)+(1- p)MGF (-t)] =
exp(tγ)[p(1 - φ1t)-ζ1 +(1- p)(1 + φ2t)-ζ2
hence E(U(R)) where t = aδ and φi = 1∕λi is given by the following expression:
Egr|s [U (W (ret, a))] = -MR(-t) = - exp(-aδγ)[p(1 - φ1aδ)-ζ1 +(1- p)(1 + φ2aδ)-ζ2]
For the Gumbel distribution we have the following expression:
MR(t) = exp(αt)Γ(1 - βt)
Egr|s [U (W (ret, a))] = -MR(-aδ)=-exp(-αaδ)Γ(1 + βaδ)
For the Normal we have the well known result:
MR(t) = exp(tμ - 2t2σ2)
Egrls [U(W(et,a))] = -Mfi(-aδ) = exp(-aδμ + 1 a2δ2σ2).
33
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