STOCHASTIC

(a) Show that U obeys the functional equation
_1
U(i,t) = max F{i, t),-i- Y g,j U(J,t+1) -/ft0
1+P,T 1 4-/), A-i
for t = 1,..., T-1 and t/ft T) = 0 for all i.
It is useful to reformulate the problem as a pure entrance-fee problem in which the reward
for stopping is F'{i,t) = 0 and an entrance fee/'(',0 must be paid to continue. Note that
if/'((', 0 < 0, it is actually profitable to continue. In the present case, let
fV,t) = F(i,t)+f(i,t)- TT7,P' jFU ' t+1)
and F'(i,t) = 0. Let U'(i,t) be the maximum expected reward for the pure entrance-fee
problem, starting from state (i, t).
(b) Show that U'(i, t) obeys the equation
U'(J,t) = max
for t = 1,2,..., T-1, and U'(i, T) = 0 for all i.
(c) Show that U(i, t) = F(i, t) + U'(i, t). [Hint: Use backward induction.]
(d) Show that the optimal refunding policy for the original and the entrance-fee versions
is the same.
(e) Recalling the functional equation obeyed by the bond prices />,, t, show that
/'ftO = 2r + c, + 1-c,
l+Pt ! t= i,...,r-i.
(f) Recalling Pye's assumptions Pi < Pt + i, Ac, ^ 0, A 2 c, js 0, show that
f'(i+ 1,0 < /'(«,/) and /'(/,;*?,,+ i •
(b) Show that ysG.
(c) Assuming that x* is unique, show that its components satisfy the equation
MIND-EXPANDING EXERCISES
0,-A/,0 + 2>u**. + i
j
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