PDE for Finance, Spring 2011 – Homework 4
Distributed 3/21/11, due 4/4/11
.
These problems concern deterministic optimal control (Section 4 material). Warning: some
of the problems here are a bit laborious (though they are not necessarily difficult).
1) Consider the finitehorizon utility maximization problem with discount rate
ρ
.
The
dynamical law is thus
dy/ds
=
f
(
y
(
s
)
, α
(
s
))
,
y
(
t
) =
x,
and the optimal utility discounted to time 0 is
u
(
x, t
) = max
α
∈
A
(
Z
T
t
e

ρs
h
(
y
(
s
)
, α
(
s
))
ds
+
e

ρT
g
(
y
(
T
))
)
.
It is often more convenient to consider, instead of
u
, the optimal utility discounted to time
t
; this is
v
(
x, t
) =
e
ρt
u
(
x, t
) = max
α
∈
A
(
Z
T
t
e

ρ
(
s

t
)
h
(
y
(
s
)
, α
(
s
))
ds
+
e

ρ
(
T

t
)
g
(
y
(
T
))
)
.
(a) Show (by a heuristic argument similar to those in the Section 4 notes) that
v
satisfies
v
t

ρv
+
H
(
x,
∇
v
) = 0
with Hamiltonian
H
(
x, p
) = max
a
∈
A
{
f
(
x, a
)
·
p
+
h
(
x, a
)
}
and finaltime data
v
(
x, T
) =
g
(
x
)
.
(Notice that the PDE for
v
is autonomous, i.e.
there is no explicit dependence on
time.)
(b) Now consider the analogous infinitehorizon problem, with the same equation of state,
and value function
¯
v
(
x, t
) = max
α
∈
A
Z
∞
t
e

ρ
(
s

t
)
h
(
y
(
s
)
, α
(
s
))
ds.
Show (by an elementary comparison argument) that ¯
v
is independent of
t
, i.e. ¯
v
= ¯
v
(
x
)
is a function of
x
alone.
Conclude using part (a) that if ¯
v
is finite, it solves the
stationary PDE

ρ
¯
v
+
H
(
x,
∇
¯
v
) = 0
.
2) Recall Example 1 of the Section 4 notes: the state equation is
dy/ds
=
ry

α
with
y
(
t
) =
x
, and the value function is
u
(
x, t
) = max
α
≥
0
Z
τ
t
e

ρs
h
(
α
(
s
))
ds
1
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with
h
(
a
) =
a
γ
for some 0
< γ <
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 Fall '11
 Bayou
 Finance, Dynamic Programming, Utility

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