J. Math. Anal. Appl. 337 (2008) 480–492 www.elsevier.com/locate/jmaa
Optimal consumption models in economic growth Hiroaki Morimoto Department of Mathematical Sciences, Faculty of Science, Ehime University, Matsuyama 790-0826, Japan Received 5 October 2005 Available online 21 April 2007 Submitted by Goong Chen
Abstract We study the optimal consumption problem in the one-sector model of economic growth under uncertainty. We show the existence of a classical solution of the Hamilton–Jacobi–Bellman equation associated with the stochastic optimization problem, and then give an optimal consumption policy in terms of its solution. © 2007 Elsevier Inc. All rights reserved. Keywords: Hamilton–Jacobi–Bellman equation; Viscosity solutions; Economic growth
1. Introduction We are concerned with the one-sector model of optimal economic growth under uncertainty discussed by R.C. Merton [8]. Define the following quantities: y(t) = labour supply at time t 0, x(t) = capital stock at time t 0, λ = the constant rate of depreciation, λ 0, c(t) = consumption rate per person at time t 0, c(t)y(t) = the totality of consumption rate, F (x, y) = constant-returns-to-scale production function producing the commodity for the capital stock x 0 and the labour force y 0, U (c) = utility function for the consumption rate c 0. We now state the conditions of the model. The labour supply y(t) and the capital stock x(t) are governed by the stochastic differential equation dy(t) = ny(t) dt + σy(t) dW (t), y(0) = y > 0, n, σ = 0, x(t) ˙ = F x(t), y(t) − λx(t) − c(t)y(t), x(0) = x > 0, E-mail address:
[email protected]. 0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.04.024
(1) (2)
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
481
on a complete probability space (Ω, F, P ) carrying a standard Brownian motion {W (t)}. The objective function for c = {c(t)} is given by the discounted expected utilities with discount rate β > 0: τx −βt J (c) = E e U c(t) dt , (3) 0
where τx = inf{t 0: x(t) = 0}. The purpose of this paper is to present a synthesis of optimal consumption policy c∗ so as to maximize J (c) over the class A of non-negative consumption policies c = {c(t)} such that c(t) is progressively measurable w.r.t. the filtration Ft = σ (Ws , s t), t c(s) ds < ∞, ∀t 0, a.s.,
(4)
0
and (2) has a non-negative solution x(t) for x 0. This optimal consumption problem has been studied by [3,8] and [6, pp. 105–110]. But, the difficulty stems from the degeneracy in the associated Hamilton–Jacobi–Bellman (HJB, for short) equation. Our method consists in finding the viscosity solution V and further the regularity V of the HJB equation 1 βV (x, y) = σ 2 y 2 Vyy (x, y) + nyVy (x, y) + F (x, y) − λx Vx (x, y) + U˜ Vx (x, y)y , 2 V (0, y) = 0, x > 0, y > 0,
(5)
under the mild conditions on n, λ, σ , where U˜ (x) is the Legendre transform of −U (−x), i.e., U˜ (x) = maxc>0 {U (c) − cx}, and U (c) is assumed to have the following properties: U ∈ C[0, ∞) ∩ C 2 (0, ∞),
U (c):
U (c):
strictly decreasing,
strictly concave on [0, ∞),
U (∞) = U (0+) = 0,
U (0+) = U (∞) = ∞.
(6)
According to [8], we make the assumption on the production function F (x, y), F (γ x, γ y) = γ F (x, y) Fx > 0,
Fy > 0,
for γ > 0,
Fxx < 0,
F (0, y) = F (x, 0) = 0,
Fyy < 0,
Fx (0+, y) < ∞,
Fx (∞, y) = 0,
y > 0.
(7)
This paper is organized as follows. In Sections 2 and 3, we reduce (5) to the 1-dimensional HJB equation associated with the stochastic Ramsey problem [8], and we show the existence of viscosity solutions of the HJB equation. Section 4 is devoted to the C 2 -regularity of the viscosity solution. In Section 5, we give a synthesis of the optimal consumption policy. 2. Hamilton–Jacobi–Bellman equations We consider the HJB equation (5) and seek the solution V (x, y) of (5) of the form V (x, y) = v(z),
z=
x . y
(8)
Then, by (7), v(z) solves the 1-dimensional HJB equation 1 βv(z) = σ 2 z2 v (z) + f (z) − μz v (z) + U˜ v (z) , 2 v(0) = 0, z > 0, where
μ = n + λ − σ2
f (z):
(9)
and f (z) = F (z, 1). By (7), it is clear that
Lipschitz continuous, concave,
f (0) = 0.
(10)
482
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
We notice that (9) is the HJB equation associated with the stochastic Ramsey problem so as to maximize τz −βt ¯ J (c) = E e U c(t) dt ,
(11)
0
over the class A0 , subject to dz(t) = f z(t) − μz(t) − c(t) dt − σ z(t) dW (t),
z(0) = z 0,
(12)
where A0 denotes the class A with {z(t)} replacing {x(t)}. We rewrite (9) as
1 1 1 v(z) = σ 2 z2 v (z) + f (z) − μz v (z) + U˜ v (z) + v(z), β+ ε 2 ε v(0) = 0, z > 0,
(13)
for ε > 0 chosen later. We shall show that v is approximated by the solution u = uL for each L > 0 of
1 1 1 u(z) = σ 2 z2 u (z) + f (z) − μz u (z) + U˜ L u (z) + u(z), β+ ε 2 ε u(0) = 0, z > 0,
(14)
where U˜ L (x) = max0
0, 2 1 βv(z) σ 2 z2 X + f (z) − μz p + U˜ (p), ∀(p, X) ∈ J 2,− v(z), ∀z > 0, 2 where J 2,+ and J 2,− are the second-order superjets and subjets defined by v(y) − v(z) − p(y − z) − 12 X|y − z|2 2,+ 2 J v(z) = (p, X) ∈ R ; lim sup 0 , |y − z|2 y→z v(y) − v(z) − p(y − z) − 12 X|y − z|2 2,− 2 J v(z) = (p, X) ∈ R ; lim inf 0 . y→z |y − z|2 3.1. Existence We note that (14) is the HJB equation associated with the optimization problem τz 1 1 uL (z) = sup E e−(β+ ε )t U c(t) + uL z(t) dt , L > 0, ε c∈AL
(15)
0
where AL denotes the class of all non-negative, integrable, Ft -progressively measurable processes c ∈ A0 such that 0 c(t) L for all t 0, and the supremum is taken over all admissible control systems [2]. By (12), we have z(t) = z(t ∧ τz ) 0 for each c ∈ AL , because c(t) is identified with c(t)1{tτz } in (15). We assume β + μ > 0. Taking
(16)
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
0 < α < β,
0 < κ < α + μ,
483
(17)
we can choose A > 0 by concavity such that f (z) − κz < A.
(18)
Furthermore, we observe by (6) and (18) that ϕ(z) := z + A¯ satisfies 1 −αϕ(z) + σ 2 z2 ϕ (z) + f (z) − μz ϕ (z) + U˜ ϕ (z) −α A¯ + A + U˜ (1) < 0, z 0, (19) 2 for some constant A¯ > 0. Let B denote the Banach space of all functions h on [0, ∞), which fulfills the relation There exists Cρ > 0, for any ρ > 0, such that
h(z) − h(˜z) Cρ |z − z˜ | + ρ ϕ(z) + ϕ(˜z) ,
z, z˜ ∈ [0, ∞),
(20)
with norm h = supz0 |h(z)|/ϕ(z) < ∞. Lemma 3.2. Under (6), (7) and (16), there exists a unique solution u = uL ∈ B of (15) for some ε > 0. Proof. We first show that τz 1 −(α+ 1ε )t U c(t) + ϕ z(t) dt ϕ(z), e sup E ε c∈AL
(21)
0
1 sup E e−(α+ ε )τ z(τ ) − z˜ (τ ) |z − z˜ |,
(22)
c∈AL
for any stopping time τ , where {˜z(t)} is the solution of (12) to c ∈ AL with z˜ (0) = z˜ . It is easily seen that t E
t 2 2 z(s) ds < ∞. z(s)ϕ z(s) ds = E
0
0
This yields that Mε (t) := gives
t 0
e
−(α+ 1ε )s
ϕ (z(s))σ z(s) dW (s) is a martingale. Hence, by (19) and (12), Ito’s formula
1 0 E e−(α+ ε )(t∧τ ) ϕ z(t ∧ τ ) t∧τ
1 1 ϕ z(s) + f z(s) − μz(s) − c(s) ϕ z(s) e−(α+ ε )s − α + = ϕ(z) + E ε 0 1 2 2 + σ z(s) ϕ z(s) ds + Mε (t ∧ τ ) 2 t∧τ 1 −(α+ 1ε )s U c(s) + ϕ z(s) ds . ϕ(z) − E e ε 0 1
Thus, we deduce (21). We set Z(t) = z(t) − z˜ (t) and hξ (z) = (z2 + ξ ) 2 for ξ > 0. It is clear that dZ(t) = f z(t) − f z˜ (t) − μZ(t) dt − σ Z(t) dW (t)
f (0) + |μ| Z(t) dt − σ Z(t) dW (t), Z(0) = z − z˜ . Then, by Ito’s formula
(23)
484
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
1 E e−(α+ ε )τ hξ Z(τ ) = hξ (z − z˜ ) + E
τ
1 1 hξ Z(s) e−(α+ ε )s − α + ε
0
+ hξ
1 2 2 Z(s) f z(s) − f z˜ (s) − μZ(s) + σ Z(s) hξ Z(s) ds 2
hξ (z − z˜ ), where we take ε > 0 such that
1 1 − α+ hξ (z) + f (0) + |μ| zhξ (z) + σ 2 z2 hξ (z) ε 2
1 2 1 1 + f (0) + |μ| + σ 2 < 0, z +ξ 2 − α+ ε 2
∀z ∈ R.
Letting ξ → 0, we get
1 E e−(α+ ε )τ Z(τ ) |z − z˜ |, which implies (22). Next, we define τz TL h(z) = sup E c∈AL
e
−(β+ 1ε )t
1 U c(t) + h z(t) dt for h ∈ B, ε
(24)
0
and show that TL : B ϕ → B ϕ ,
(25)
where Bϕ is the closed subset of B defined by Bϕ = {h ∈ B: 0 h ϕ, h(0) = 0}. By (21), it is easy to see that TL h(0) = 0 TL h ϕ and TL h < ∞ for h ∈ Bϕ . Since z(t) 0, we note by (12) that z(t) = 0 if t > τz . Hence, by (20), we have τz 1 1 TL h(z) − TL h(˜z) sup E e−(β+ ε )t U c(t) + h z(t) dt ε c 0
τz˜ −
e 0
−(β+ 1ε )t
τz
sup E c
e
sup E
+
−(β+ 1ε )t
τz ∧τz˜
τz c
1 U c(t) + h z˜ (t) dt ε
e
−(β+ 1ε )t
τz ∧τz˜
ρ sup E ε c
U c(t) dt +
∞
∞ e 0
Cρ sup E U c(t) dt + ε c
0
∞
1 e−(β+ ε )t ϕ z(t) + ϕ z˜ (t) dt say.
By (12), we can take sufficiently small ε > 0 such that ∞ 1 2
−(α+ 1ε )s 2
E sup Mε (t) 2|σ |E e z(s) ds < ∞. t
1
h z(t) − h z˜ (t) dt ε
e 0
0
≡ J1 + J2 + J3 ,
−(β+ 1ε )t
−(β+ 1ε )t
z(t) − z˜ (t) dt
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
485
By the same line as (23), we have 1 1 J1 E e−(α+ ε )(τz ∧τz˜ ) ϕ z(τz ∧ τz˜ ) − e−(α+ ε )τz ϕ z(τz ) 1 1 = E e−(α+ ε )(τz ∧τz˜ ) ϕ z(τz ∧ τz˜ ) − e−(α+ ε )τz ϕ z(τz ) 1{τz˜ <τz } 1 1 E e−(α+ ε )τz˜ ϕ z(τz˜ ) − e−(α+ ε )τz˜ ϕ z˜ (τz˜ ) 1{τz˜ <τz }
1 sup E e−(α+ ε )τz˜ z(τz˜ ) − z˜ (τz˜ ) 1{τz˜ <τz } c
|z − z˜ |. Moreover, by (22) J2 Cρ |z − z˜ |/(β − α)ε. Also, we recall that z˜ (t) = 0 if t > τz˜ . Hence, by (21) J3 ρ ϕ(z) + ϕ(˜z) . Therefore, we get TL h ∈ B, which implies (25). Now, by (23), we have E e−α(t∧τz ) ϕ z(t ∧ τz ) ϕ(z). Hence
TL h1 (z) − TL h2 (z) = sup E c∈AL
τz
1
h1 z(t) − h2 z(t) dt ε
e
−(β+ 1ε )t
1 h1 − h2 ϕ z(t) dt ε
0 τz
sup E c∈AL
e
−(β+ 1ε )t
0
1 h1 − h2 ϕ(z). (β − α)ε + 1 Therefore, by the contraction mapping theorem, TL has a fixed point uL ∈ Bϕ . This completes the proof.
2
Theorem 3.3. We assume (6), (7), (16). Then u = uL ∈ B of (15) is a concave viscosity solution of (14). Proof. We recall (20) and also (23) to obtain 1 E e−(β+ ε )τ ϕ z(τ ) ϕ(z). Then, by a slight modification of the proof of [7, Theorem 2.3], we can show that the dynamic programming principle holds for u, i.e., τz ∧τ 1 −(β+ 1ε )t −(β+ 1ε )τz ∧τ u(z) = sup E e u z(τz ∧ τ ) U c(t) + u z(t) dt + e ε c∈AL 0
for any bounded stopping time τ . Thus, by the standard theory of viscosity solutions [2], we deduce that u is a viscosity solution of (14). To see the concavity of u, we also recall TL h(x) of (24). By the same line as [2, p. 204, Lemma 10.6], we observe that TL 0 is concave. Moreover, by induction, TLn 0 is concave for any n 1. By Lemma 3.2, we have TLn 0 → u
as n → ∞.
Therefore, this yields that u is concave.
2
486
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
Theorem 3.4. We assume (6), (7), (16). Then there exists a concave viscosity solution v of (9) such that 0 v ϕ. Proof. Let L < L . By (24), we have TL h TL h for h ∈ B. Hence uL has a limit, denoted by v uL ↑ v
as L → ∞.
It is clear that v is concave and then continuous on (0, ∞). It follows from (15) and (21) that τz 1 −(β+ 1ε )t U c(t) + v z(t) dt ϕ(z), z 0. e v(z) sup E ε c∈A0 0
For any ρ > 0, there exists c ∈ A0 such that τz τz −(β+ 1ε )t −(β+ 1ε )t 1 v z(t) dt . e U c(t) dt + E e v(z) − ρ < E ε 0
0
By the comparison theorem [5], we have τz ↓ θ,
z(t) q(t) ↓ 0,
a.s.
as z ↓ 0,
where q(t) is the solution of dq(t) = f (0) + |μ| q(t) dt − σ q(t) dW (t),
q(0) = z > 0.
Since E[sup0ts q(t)2 ] < ∞ for each s > 0, it is clear by (12) that E z(τz ∧ s) = z + E
τz ∧s f z(t) − μz(t) − c(t) dt . 0
Letting z ↓ 0 and then s → 0, we get θ θ E f (0) − c(t) dt 0, −c(t) dt = E 0
0
so that θ e
E
−(β+ 1ε )t
U c(t) dt = 0.
0
Passing to the limit, we obtain ∞ v(0+) − ρ E
e 0
−(β+ 1ε )t
1 1 v(0+) dt = v(0+), ε βε + 1
which implies v(0+) = 0. Thus v ∈ C[0, ∞). By Dini’s theorem, uL converges to v locally uniformly on [0, ∞). Therefore, by the standard stability results [2], we deduce that v is a viscosity solution of (13) and then (9). 2 3.2. Comparison We give a comparison theorem for the viscosity solution v of (9). Theorem 3.5. Let fi , i = 1, 2, satisfy (10) and let vi ∈ C[0, ∞) be the concave viscosity solution of (9) for fi replacing f such that 0 vi ϕ. Suppose f1 f2 .
(26)
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
487
Then, under (6), (7) and (16), we have v1 v2 . Proof. By (17), we first note that there exists 1 < ν < 2 such that 1 −β + σ 2 ν(ν − 1) + κν − μν < 0. 2 Hence, by (18), ϕν (z) := zν + B satisfies 1 −βϕν (z) + σ 2 z2 ϕν (z) + f2 (z) − μz ϕν (z) 2
1 2 −β + σ ν(ν − 1) + κν − μν zν + Aνzν−1 − βB < 0, 2
z 0,
(27)
for a suitable choice of B > 0. Suppose that v1 (z0 ) − v2 (z0 ) > 0 for some z0 ∈ (0, ∞). Then there exists η > 0 such that sup v1 (z) − v2 (z) − 2ηϕν (z) > 0. z0
Since v1 (z) − v2 (z) − 2ηϕν (z) ϕ(z) − 2ηϕν (z) → −∞ as z → ∞, we find z¯ ∈ (0, ∞) such that sup v1 (z) − v2 (z) − 2ηϕν (z) = v1 (¯z) − v2 (¯z) − 2ηϕν (¯z) > 0. z0
Define n Ψn (z, y) = v1 (z) − v2 (y) − |z − y|2 − η ϕν (z) + ϕν (y) 2 for any n > 0. It is clear that Ψn (z, y) ϕ(z) + ϕ(y) − η ϕν (z) + ϕν (y) → −∞ as z + y → ∞. Hence we find (zn , yn ) ∈ [0, ∞)2 such that Ψn (zn , yn ) = sup Φ(z, y): (z, y) ∈ [0, ∞)2 n = v1 (zn ) − v2 (yn ) − |zn − yn |2 − η ϕν (zn ) + ϕν (yn ) 2 v1 (¯z) − v2 (¯z) − 2ηϕν (¯z) > 0, from which n |zn − yn |2 v1 (zn ) − v2 (yn ) − η ϕν (zn ) + ϕν (yn ) 2 ϕ(zn ) + ϕ(yn ) − η ϕν (zn ) + ϕν (yn ) . Thus we deduce that the sequences {zn + yn } and {n|zn − yn |2 } are bounded by some constant C > 0, and √ |zn − yn | C/ n → 0 as n → ∞. Moreover, zn , yn → zˆ ∈ [0, ∞)
as n → ∞,
(28)
488
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
taking a subsequence if necessary. By (28) we recall that n Ψn (zn , yn ) = v1 (zn ) − v2 (yn ) − |zn − yn |2 − η ϕν (zn ) + ϕν (yn ) 2 v1 (zn ) − v2 (zn ) − 2ηϕν (zn ), and hence n |zn − yn |2 v2 (zn ) − v2 (yn ) + η ϕν (zn ) − ϕν (yn ) → 0 as n → ∞. 2 Passing to the limit in (28), we get v1 (ˆz) − v2 (ˆz) − 2ηϕν (ˆz) > 0 and zˆ = 0.
(30)
Next, let V1 (z) = v1 (z) − ηϕν (z) and V2 (y) = v2 (y) + ηϕν (y). Applying Ishii’s lemma [1,2] to n Ψn (z, y) = V1 (z) − V2 (y) − |z − y|2 , 2 we obtain X1 , X2 ∈ R such that n(zn − yn ), X1 ∈ J¯2,+ V1 (zn ), n(zn − yn ), X2 ∈ J¯2,− V2 (yn ),
0 1 −1 1 0 X1 3n , −3n 0 −X2 −1 1 0 1 where
¯2,±
J
∃zr → z, ∃(pr , Xr ) ∈ J 2,± Vi (zr ), Vi (z) = (p, X): Vi (zr ), pr , Xr → Vi (z), p, X
(29)
(31)
,
i = 1, 2.
Recall that p + ηϕν (z), X + ηϕν (z) : (p, X) ∈ J 2,+ V1 (z) , J 2,− v2 (y) = p − ηϕν (y), X − ηϕν (y) : (p, X) ∈ J 2,− V2 (y) . J 2,+ v1 (z) =
Hence (p1 , X¯ 1 ) := n(zn − yn ) + ηϕν (zn ), X1 + ηϕν (zn ) ∈ J¯2,+ v1 (zn ), (p2 , X¯ 2 ) := n(zn − yn ) − ηϕν (yn ), X2 − ηϕν (yn ) ∈ J¯2,− v2 (yn ). By the definition of viscosity solutions, we have 1 −βv1 (zn ) + σ 2 zn2 X¯ 1 + f1 (zn ) − μzn p1 + U˜ (p1 ) 0, 2 1 −βv2 (yn ) + σ 2 yn2 X¯ 2 + f2 (yn ) − μyn p2 + U˜ (p2 ) 0. 2 Putting these inequalities together, we get 1 β v1 (zn ) − v2 (yn ) σ 2 zn2 X¯ 1 − yn2 X¯ 2 + f1 (zn ) − μzn p1 − f2 (yn ) − μyn p2 + U˜ (p1 ) − U˜ (p2 ) 2 ≡ I1 + I2 + I3 , say. We consider the case when there are infinitely many zn yn 0. By (29) and (31), it is easy to see that 1 I1 σ 2 3n|zn − yn |2 + η zn2 ϕν (zn ) + yn2 ϕν (yn ) → σ 2 ηˆz2 ϕν (ˆz) 2 By monotonicity and p1 p2 , I3 0.
as n → ∞.
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
489
By (26), we have f1 (zn )(zn − yn ) f2 (zn )(zn − yn ). Hence I2 f2 (zn ) − f2 (yn ) n(zn − yn ) − μn(zn − yn )2 + η f1 (zn ) − μzn ϕν (zn ) + f2 (yn ) − μyn ϕν (yn ) → η f1 (ˆz) + f2 (ˆz) − 2μˆz ϕν (ˆz) as n → ∞. Thus, by (26) and (27)
1 2 2 β v1 (ˆz) − v2 (ˆz) 2η σ zˆ ϕν (ˆz) + f2 (ˆz) − μˆz ϕν (ˆz) 2 2ηβϕν (ˆz).
This is contrary with (30). Suppose that there are infinitely many yn zn 0. By concavity, we have vi (yn ) vi (zn ), i = 1, 2, so that v1 (yn ) − v2 (zn ) v1 (zn ) − v2 (yn ). Hence the maximum of Ψn (z, y) is attained at (yn , zn ). Interchanging yn and zn in the above argument, we get a contradiction. Thus the proof is complete. 2 4. Classical solutions In this section, we shall show the smoothness of the viscosity solution v of (9). Theorem 4.1. Under (6), (7) and (16), we have v ∈ C 2 (0, ∞) and v (0+) = ∞. Proof. Let [a, b] ⊂ (0, ∞) be arbitrary and we consider the boundary value problem 1 βw(z) = σ 2 z2 w (z) + f (z) − μz w (z) + U˜ w (z) ∨ v+ (b) 2 w(a) = v(a), w(b) = v(b),
in (a, b), (32)
(x) v+
where denotes the right-hand derivative of the concave viscosity solution v. By concavity, we see that p (b) > 0 for any (p, X) ∈ J 2,+ v(z) with a < z < b. Hence v is a viscosity solution of (32). v+ (b)), it is known in [4] that (32) has a Now, by the uniform ellipticity and the Lipschitz continuity of U˜ (· ∨ v+ smooth solution w. By the same line as the proof of Theorem 3.5, we can show the uniqueness of the viscosity solution of (32). Therefore we deduce that v = w ∈ C 2 (a, b), and hence v ∈ C 2 (0, ∞). Next, suppose v (0+) < ∞. By (9), we have v (z) 0 for all z > 0, since U˜ (v (z)) = ∞ if v (z) < 0. Moreover, by L’Hospital’s rule z2 v (z) = 0. z→0 z
lim z2 v (z) = lim z2 v (z) + 2zv (z) = lim
z→0
z→0
Passing to the limit in (9) as z → 0, we have that U˜ (v (0+)) = 0. This is contrary with (6). Therefore v (0+) = ∞.
2
Corollary 4.2. Under the assumptions of Theorem 4.1, we have the solution V (x, y) ∈ C 2 ((0, ∞)2 ) of (5). Proof. We define V (x, y) by (8). Then the proof is immediate from Theorem 4.1.
2
5. Optimal consumption In this section we give a synthesis of optimal consumption policy c∗ for the optimization problem (3). Lemma 5.1. Under (6), (7) and (16), we have lim inf E e−βt V x(t), y(t) = 0 t→∞
for every c ∈ A.
(33)
490
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
Proof. We set Φ(x, y) = ϕ( xy ). By (19), we have 1 −αΦ(x, y) + σ 2 y 2 Φyy (x, y) + nyΦy (x, y) + F (x, y) − λx Φx + U˜ Φx (x, y)y < 0, 2
x 0, y > 0. (34)
Hence, Ito’s formula gives E e−βt Φ x(t), y(t) Φ(x, y) + E
t e
−βs
(−β + α)Φ x(s), y(s) ds ,
0
from which
∞
(β − α)E
e
−βs
Φ x(s), y(s) ds < ∞.
0
Thus lim inf E e−βt Φ x(t), y(t) = 0. t→∞
By (8) V (x, y) Φ(x, y), which implies (33).
2
Now we consider the equation of the form: x˙ ∗ (t) = F x ∗ (t), y(t) − λx ∗ (t) − c∗ (t)y(t),
x ∗ (0) = x > 0,
(35)
where c∗ (t) = (U )−1 Vx x ∗ (t), y(t) y(t) 1{tτx ∗ } .
(36)
Lemma 5.2. Under (6), (7) and (16), there exists a unique solution x ∗ (t) 0 of (35). Proof. Let G(x, y) = F (x + , y) − λx − (U )−1 (Vx (x + , y)y). Since G(x, y) is continuous and G(0, y) = 0, there exists an Ft -progressively measurable solution χ(t) of dχ(t) = G χ(t), y(t) dt, χ(0) = x > 0. (37) Define x ∗ (t) = χ(t ∧ τχ ) 0. We note by the concavity of f that G(x, y) C1 x + + C2 y for some C1 , C2 > 0. Then we apply the comparison theorem to (37) and d x(t) ˆ = C1 x(t) ˆ + + C2 y(t) dt, x(0) ˆ = x > 0, (38) ˆ for all t 0. Further, it is easy to see that τx ∗ = τχ , and hence to obtain 0 x ∗ (t) x(t) x˙ ∗ (t) = 1{tτχ } G χ(t), y(t) = 1{tτx ∗ } F x ∗ (t), y(t) − λx ∗ (t) − c∗ (t)y(t) . ˜ be another solution of (35). We notice that the function Therefore x ∗ (t) solves (35). To prove uniqueness, let x(t) x → G(x, y) is locally Lipschitz continuous on (0, ∞). By using a standard technique, we can get x ∗ (t ∧ τx ∗ ∧ τx˜ ) = x(t ˜ ∧ τx ∗ ∧ τx˜ ).
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
491
Hence τx ∗ = τx˜ , This implies
x ∗ (t) = x(t) ˜
˜ x ∗ (t) = x(t)
for t τx ∗ .
for all t 0.
2
Theorem 5.3. We make the assumptions of Theorem 4.1. Then the optimal consumption c∗ = {c∗ (t)} is given by (36). Proof. We set ζ = s ∧ τx ∗ for any s > 0. By (5), (35) and Ito’s formula, we have ζn −βζ ∗ −βt n −βV (x, y) + Vx (x, y)1{t<τx ∗ } F (x, y) − λx − c∗ (t)y E e V x (ζn ), y(ζn ) = V (x, y) + E e 0
1 2 2
+ nyVy (x, y) + σ y Vyy (x, y) ∗ dt + M(ζn ) (x (t),y(t)) 2 ζn ∗ −βt = V (x, y) − E e U c (t) dt , 0
where ζn = ζ ∧ τn for a localizing sequence of stopping times τn ↑ ∞ of the local martingale M(t) with M(0) = 0. From (34), (38) and Doob’s inequalities for martingales it follows that E sup e−βζn V x ∗ (ζn ), y(ζn ) E sup e−αr Φ x ∗ (r), y(r) n
0rs
r
−αt ∗ Φ(x, y) + E sup e σ x (t) dW (t)
0rs
s Φ(x, y) + 2|σ |E
0
2 e−αt x ∗ (t) dt
0
s Φ(x, y) + 2|σ |E
1 2
1 2
x(t) ˆ dt 2
.
0
Therefore, letting n → ∞, we get by the dominated convergence theorem ζ −βζ ∗ ∗ −βt E e V x (ζ ), y(ζ ) = V (x, y) − E e U c (t) dt . 0
By (33), we have lim inf E e−β(s∧τx ∗ ) V x ∗ (s ∧ τx ∗ ), y(s ∧ τx ∗ ) = lim inf E e−βs V x ∗ (s), y(s) : τx ∗ s = 0. s→∞
s→∞
Passing to the limit, we deduce τx ∗ ∗ J c =E e−βt U c∗ (t) dt = V (x, y). 0
By the same calculation as above, we can obtain τx −βt J (c) = E e U c(t) dt V (x, y),
c ∈ A.
0
Since V (x, y) = supc∈A J (c), we remark that the solution V (x, y) of (5) is unique. The proof is complete.
2
492
H. Morimoto / J. Math. Anal. Appl. 337 (2008) 480–492
References [1] M.G. Crandall, H. Ishii, P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992) 1–67. [2] W.H. Fleming, H.M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 1993. [3] L. Foldes, The optimal consumption function in a Brownian model of accumulation Part A: The consumption function as solution of a boundary value problem, J. Econom. Dynam. Control 25 (2001) 1951–1971. [4] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. [5] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981. [6] M.I. Kamien, N.L. Schwartz, Dynamic Optimization, second ed., North-Holland, Amsterdam, 1991. [7] S. Koike, H. Morimoto, Variational inequalities for leavable bounded-velocity control, Appl. Math. Optim. 48 (2003) 1–20. [8] R.C. Merton, An asymptotic theory of growth under uncertainty, Rev. Econom. Stud. 42 (1975) 375–393.