Download A QUARTET OF SEMIGROUPS FOR MODEL

E. W. Anderson, L. P. Hansen, & T. J. Sargent Semigroups and Model Detection
93
state vector with jump dynamics.In particular,a choice a with instantaneous
in termsof
payoff a(y) if the statejumps to y has a price J Il(y, xt)a(y)r](dy\xt)
the consumptionnumeraire.This cost can be positiveor negative.Whena jump
does not take place, wealthevolves accordingto
dwt = p(xtJ)wt--
U(y, xtJ)a{y)r)(dy\xt-)- c,_ dt
where p(x) is the risk-freerate given state x and for any variablez, zt- =
limTf t zT.If the statex jumpsto y at datet, the new wealthis a(y). The Bellman
equationfor this problemis
8V(w,x) = max min U(c) + Vw(w,x)\ p(x)wtc,a /i£A
+ 0
+
II (j, x)a(y)ri(dy\x)- c
J
[
[1 - h(y, x) + h(y, *)log h(y, x)Mdy\x)
h(y,x)(V[a(y),y]
- V(w,
jc))i)(rfy|*)
The first-orderconditionfor c is the same as for the diffusion case and
equatesVwto the marginalutilityof consumption.The first-orderconditionfor
a requires
h(y, x)VJLa(y),
y] = Vw{w,
x)U(y, x),
and the first-orderconditionfor h requires
- 6 log h(y, x) = V[d(y), y] - V(w,x) .
Solving this second conditionfor fi gives the jump counterpartto the solution
assertedin Theorem5.1. Thus the robusta satisfies:
VJid(y),y]=
Vw(w,x)
II(y,s)
(~V[a(y),y] + V(x)\-
j
o
expl
case 8 = oo,ft is set to one. Since
In the limitingno-concern-about-robustness
Vwis equatedto the marginalutility for consumption,the first-ordercondition
for a equatesthe marginalrateof substitutionof consumptionbefore and after
thejumpto the priceTl(y,x). Introducingrobustnessscalesthe priceby thejump
distributiondistortion.
In this portrayal,the worst case h dependson the endogenousstate w, but
it is again possible to obtain an alternativerepresentationof the probability