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Optimal consumption of the stochastic Ramsey problem for non-Lipschitz diffusion
Journal of Inequalities and Applications volume 2014, Article number: 391 (2014)
The stochastic Ramsey problem is considered in a growth model with the production function of a Cobb-Douglas form. The existence of a unique classical solution is proved for the Hamilton-Jacobi-Bellman equation associated with the optimization problem. A synthesis of the optimal consumption policy in terms of its solution is proposed.
MSC:49L20, 49L25, 91B62.
We are concerned with the stochastic Ramsey problem in a growth model discussed by Merton . For recent contribution in this direction, we refer to . A firm produces goods according to the Cobb-Douglas production function for capital x, where (cf. Barro and Sala-i-Martin ). The stock of capital at time t is modeled by the stochastic differential equation
on a complete probability space carrying a standard Brownian motion endowed with the natural filtration generated by .
The capital stock can be consumed and the flow of consumption at time t is assumed to be written as . The rate of consumption per capital stock is called an admissible policy if is an -progressively measurable process such that
We denote by the set of admissible policies. Given a policy , the capital stock process obeys the equation
The objective is to find an optimal policy so as to maximize the expected discounted utility of consumption
over , where is a discount rate and is a utility function in , which is assumed to have the following properties:
The Hamilton-Jacobi-Bellman (HJB for short) equation associated with this problem is given by
This kind of economic growth problem has been studied by Kamien and Schwartz  and Sethi and Thompson [, Chapter 11]. However, the optimization problem is unsolved. It is not guaranteed that (1.2) admits a unique positive solution and the value function is a viscosity solution of the HJB equation. The main difficulty stems from the fact that (1.5) is a degenerate nonlinear equation of elliptic type with the non-Lipschitz coefficient in . It is also analytically studied by , nevertheless in the finite time horizon. The resulting HJB equation is a parabolic partial differential equation (PDE, for short), which is very different from the elliptic PDE dealt with in the present paper.
In this paper, we provide the existence results on a unique positive solution to (1.2) and a classical solution u of (1.5) by the theory of viscosity solutions. For the detail of the theory of viscosity solutions, we mention the works [7, 8] and . An optimal policy is shown to exist in terms of u.
This paper is organized as follows. In Section 2, we show that (1.2) admits a unique positive solution. In Section 3, we show the existence of a viscosity solution u of the HJB equation (1.5). Section 4 is devoted to the -regularity of its solution. In Section 5, we present a synthesis of the optimal consumption policy.
In this section, we show the existence of a unique solution to (1.2).
Proposition 2.1 There exists a unique positive solution to (1.2), for each , such that
Proof We set . Then, by Ito’s formula and (1.2),
By linearity, (2.4) has a unique solution . Since
has a positive solution , we see by the comparison theorem [, Chapter 6, Theorem 1.1] that holds almost surely (a.s.). Therefore, (1.2) admits a unique positive solution of the form , which satisfies for each .
Let be the right-hand side of (2.1) and . Obviously, we see that is a unique solution of
By (1.2) and Jensen’s inequality,
Since , we deduce , which implies (2.1).
Similarly, let be the right-hand side of (2.2) and . By substitution, it is easy to see that is a unique solution of
Furthermore, by (1.2), Ito’s formula and Jensen’s inequality,
Thus, we deduce and , which implies (2.2).
Next, let denote the solution of (1.2) with and . Then, by (2.4)
Setting , we have
By Young’s inequality , for any ,
Hence, for any , we choose such that
Therefore, by (2.1) and (2.6), we have
which implies (2.3). □
Remark 2.1 The uniqueness for (1.2) is violated if and is deterministic since 0 and the limit process satisfy (1.2) with , and
3 Viscosity solutions of the HJB equation
Definition 3.1 Let . Then u is called a viscosity solution of (1.5) if the following relations are satisfied:
where and are the second-order superjets and subjets .
Define the value function u by
where the supremum is taken over all systems .
In this section, we study the viscosity solution u of the HJB equation (1.5). Due to Proposition 2.1, we can show the value function u with the following properties.
Lemma 3.1 We assume (1.4). Then we have
for some constant , and there exists for any such that
Proof Clearly, u is nonnegative. By concavity, there is such that
By (1.1) and (2.1), we have
which implies (3.2).
Now, let be arbitrary. By (1.4), there is such that for all . Furthermore,
Thus, we obtain a constant , depending on , such that
By (1.1), (2.3) and (3.4), we get
where the constant is independent of ε, . Replacing ρ by and choosing sufficiently small , we deduce (3.3). □
Remark 3.1 The continuity of u is immediate from (3.3).
Theorem 3.1 We assume (1.4). Then the value function u is a viscosity solution of (1.5).
Proof According to , the viscosity property of u follows from the dynamic programming principle for u, that is,
for any bounded stopping time , where the supremum is taken over all systems . Let be the right-hand side of (3.6). Let and , when is non-random. Then we have
for the shifted process of c by r, i.e., . It is easy to see that
with respect to the conditional probability . We take such that and sufficiently large to obtain
By (3.2) in Lemma 3.1, Ito’s formula and Doob’s inequality, we have
for some constant . Hence, approximating τ by a sequence of countably valued stopping times, we see that
Taking the supremum, we deduce .
We shall show the reverse inequality in the case that is constant. For any , we consider a sequence of disjoint subsets of such that
for chosen later. We take and such that
By the same argument as (3.5), we note that
for some constant . We choose such that . Then we have
Hence, by (3.7) and (3.8),
By definition, we find such that
In view of [, Chapter 6, Theorem 1.1], we can take c, on the same probability space. Define
where denotes the indicator function. Then belongs to . Let be the solution of
Clearly, a.s. if . Further, for each , we have on
Hence, coincides with the solution of (1.2) for a.s. on with . Thus, we get
where denotes the expectation with respect to .
Now, we fix and choose , by (2.1), (2.2) and (3.1), such that
where the constant depends only on r, . By (3.9), (3.10) and (3.11), we have
for some constant independent of ε. Thus
Letting , we get .
In the general case, by the above argument, we note that
Hence is a supermartingale. By the optional sampling theorem,
Taking the expectation and then the supremum over , we conclude that . Noting the continuity of u, we obtain (3.6). □
4 Classical solutions
In this section, using the viscosity solutions technique, we show the -regularity of the viscosity solution u of (1.5). For any fixed , we consider the boundary value problem
with boundary condition
given by u.
Proposition 4.1 Let , , be two viscosity solutions of (3.1), (4.2). Then, under (1.4), we have
Proof It is sufficient to show that . Suppose that there exists such that . Clearly, by (4.2), , and we find such that
for . Then there exists such that
Furthermore, by the definition of ,
Hence, by uniform continuity
By (4.3), (4.4) and (4.5), extracting a subsequence, we have
Now, we may consider that for sufficiently large k. Applying Ishii’s lemma  to , we obtain such that
By Definition 3.1,
where . Putting these inequalities together, we get
By (4.5) and (4.7), it is clear that
Also, by (4.5)
By (1.6), (3.4), (4.5) and (4.6), we have
Consequently, by (4.6), we deduce that
which is a contradiction. □
Theorem 4.1 We assume (1.4). Then there exists a solution of (1.5).
Proof For any , we recall the boundary value problem (4.1), (4.2). Since
Hence, by (1.4)
Also, by (1.6)
Thus the nonlinear term of (4.1) is Lipschitz. By uniform ellipticity, a standard theory of nonlinear elliptic equations yields that there exists a unique solution of (4.1), (4.2). For details, we refer to [, Theorem 17.18] and [, Chapter 5, Theorem 3.7]. Clearly, by Theorem 3.1, u is a viscosity solution of (4.1), (4.2). Therefore, by Proposition 4.1, we have and u is smooth. Since a, b are arbitrary, we obtain the assertion. □
5 Optimal consumption
In this section, we give a synthesis of the optimal policy for the optimization problem (1.4) subject to (1.2). We consider the stochastic differential equation
where and denotes the maximizer of (1.6) for , i.e.,
Our objective is to prove the following.
Theorem 5.1 We assume (1.4). Then the optimal consumption policy is given by
To obtain the optimal consumption policy , we should study the properties of the value function u and the existence of strong solution of (5.1). We need the following lemmas.
Lemma 5.1 Under (1.4), the value function u is concave. In addition, we have
Proof Let , . For any , there exists such that
where is the solution of (1.2) corresponding to with . Let , and we set
which belongs to . Define and by
By the comparison theorem, we have
Thus, by (1.4)
Therefore, letting , we obtain the concavity of u.
To prove (5.4), by Theorem 4.1, we recall that u is smooth. Furthermore, we get for . If not, then for some . By concavity,
which is a contradiction. Suppose that for some . Then, by concavity, we have for all . Hence, by (1.5) and (1.6),
This is contrary to (1.4). Thus, we obtain (5.4).
Next, by definition, we have
where is the solution of (1.2) corresponding to . As in (2.7), the limit process is different from 0. Hence
Suppose that . By (1.5) and concavity, we get , which is a contradiction. This implies (5.5). □
Lemma 5.2 Under (1.4), there exists a unique positive strong solution of (5.1).
Proof Let be the solution of (1.2) corresponding to . We can take the Brownian motion on the canonical probability space [, p.71]. Since , the probability measure is defined by
for every . Girsanov’s theorem yields that
Thus, (5.1) admits a weak solution.
Now, by (5.2), we have
Hence, by (1.4) and concavity,
Thus, is nondecreasing on . We rewrite (5.1) as the form of (2.4) to obtain a.s. Then we see that the pathwise uniqueness holds for (5.1). Therefore, by the Yamada-Watanabe theorem , we deduce that (5.1) admits a unique strong solution . □
Proof of Theorem 5.1 Since satisfies (1.1), it belongs to . By Lemma 5.2, we note that
Hence, by (2.2) and (3.2),
This yields that is a martingale. By (1.6), (5.3) and Ito’s formula,
By (2.1) and (3.2), it is clear that
Letting , we deduce
By the same calculation as above, we obtain
for any . The proof is complete. □
Remark 5.1 From the proof of Theorem 5.1, it follows that the solution u of the HJB equation (1.5) coincides with the value function. This implies that the uniqueness holds for (1.5).
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I would like to thank Professor H Morimoto for his useful help. The research was supported by the National Natural Science Foundation of China (11171275) and the Fundamental Research Funds for the Central Universities (XDJK2012C045).
The author declares that they have no competing interests.
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Liu, C. Optimal consumption of the stochastic Ramsey problem for non-Lipschitz diffusion. J Inequal Appl 2014, 391 (2014). https://doi.org/10.1186/1029-242X-2014-391
- Hamilton-Jacobi-Bellman equation
- viscosity solutions
- Ramsey problem
- Cobb-Douglas production function