In this chapter, we study the theory of optimal pollution management in environmental economics. We consider a society consuming some good, which generates pollution as a byproduct of this consumption.

The pollution stock X(t) is only gradually degraded, and its growth rate incorporates a random shock with mean of zero and constant standard deviation r. The social welfare is defined by the utility U(c) of the consumption c net of the disutility D(x) of pollution x. The objective of the social planner is to choose time paths for consumption to maximize the social welfare with long-run average criteria.

By using the vanishing discount technique, we solve the HJB equation (10.6) associated with the long-run average problem as the limit equation when the discount rate β converges to zero. The optimal consumption policy is shown to exist in a feedback form, and the maximum value is independent of the initial condition X(0) > 0.

The Model

Consider a society consuming a homogeneous good and accumlating pollution. Define the following quantities:

1. X(t) = stock of pollution at time t.

2. r = the constant rate of pollution decay, r > 0.

3. L = the upper bound of the maximum flow of pollusion, L > 0.

4. c(t) = flow of pollution (or consumption) at time t.

5. B(t) = the standard Brownian motion.

6. σ = the nonzero diffusion constant. […]

In this chapter, we study how the viscosity solutions of HJB equations in Chapter 4 turn smooth. We first observe that the DPP holds for the value function v(x) with initial state x and v is a viscosity solution of the HJB equation associated with the problem. Next, the C2-regularity of v is shown by the following procedure:

1. (a) Consider the boundary value problem on any fixed ball, also called the Dirichlet problem, with the boundary condition given by v.

2. (b) In the Dirichlet problem, we have a classical solution w, which is also a viscosity solution.

3. (c) By the uniqueness of viscosity solutions, we obtain v = w and thus v is smooth.

This viscosity solutions technique is developed by D. Duffie et al. [51] in an optimal consumption/investment problem.

The plan of this chapter is as follows. We present a brief review of the Dirichlet problem for linear elliptic equations, and show the existence of classical solutions of the Dirichlet problem for HJB equations. Finally, we examine the important role of (a)–(c) in the stochastic linear-quadratic (LQ) control problems. As is well known in [40, 65] the control region for the stochastic linear regulator problem is unbounded and the associated HJB equation admits a classical solution. Based on the viscosity solutions technique, the optimal control policies of the LQ problems with bounded control regions are given. The technique (a)–(c) can be applied to economic problems in Part II.

In this chapter, we study the decision problem of investment rates and exit times for a private business in a stochastic environment. We consider a competitive firm that produces a capital good with the capital stock K(t).

The market price P(t) of the capital good is governed by a stochastic differential equation, and the market value of the capital given by X(t) ≔ P(t)K(t). The net cash flow of the firm coincides with the profit rX(t) for the profit rate r net the total cost ϕ(q(t))X(t) of investment q(t)K(t). The firm sells its whole business at any time τ, given a resale value function g(·), when the integration of capital goods is costly. The firm's fundamental value is the total of the net cash flow and its abandonment payoff g(X(τ)). The objective of the firm is to find the investment rate q*(t) and the exit time τ* simultaneously, which maximize the expected fundamental value.

We solve the nonlinear variational inequality associated with the firm's value maximizing problem. The optimal policies are shown to exist from the optimality conditions.

The Model

Consider the decision problem of investment rates and exit times. Define the following quantities:

1. P(t) = market price of the capital good at time t.

2. ν = expected decline rate of the price.

3. B(t) = the standard Brownian motion.

4. […]

We study the stochastic optimization problem of renewable resources. We consider planning authorities who want to determine the amount of harvest from publicly owned resources.

The authorities expoit at the rate c(t) from the stock of resource until the resource is totally exhaused. The stock X(t) is renewable if some positive level of stock X(t) can be maintaines by the rate of increase f(X(t)) of this stock indefinitely. The growth rate of stock X(t) coincides with the totality of f(X(t)) and the white noise process dB(t)/dt, which is interpreted as the uncertain fluctuation of the resource, when there is no exploitation of the stock. The natural resource allows us to exploit if its position remains to be nonnegative. The authorities' aim is to maximize the expected discounted utility of exploitation with a utility function U(c).

We analyze the associated HJB equation (8.7) and further show the C2-regularity of the viscosity solution. We present the existence of an optimal policy in a feedback form or a stochastic version of Hotelling's rule.

The Model

Consider a mathematical problem of the renewable natural resource with uncertainty. Define the following quantities:

1. X(t) = remaining stock of the renewable resource at time t.

2. f(X(t))) = rate of renewable change of the stock at time t.

3. c(t) = flow of exploitation (or consumption) at time t.

4. B(t) = the standard Brownian motion.

5. […]

The purpose of this book is to provide a fundamental description of stochastic control theory and its applications to dynamic optimization in economics. Its content is suitable particularly for graduate students and scientists in applied mathematics, economics, and engineering fields.

A stochastic control problem poses the question: what is the optimal magnitude of a choice variable at each time in a dynamical system under uncertainty? In stochastic control theory, the state variables and control variables, respectively, describe the random phenomena of dynamics and inputs. The state variable in the problem evolves according to stochastic differential equations (SDE) with control variables. By steering of such control variables, we aim to optimize some performance criteria as expressed by the objective functional. Stochastic control can be viewed as a problem of decision making in maximization or minimization. This subject has created a great deal of mathematics as well as a large variety of applications in economics, mathematical finance, and engineering.

This book provides the basic elements of stochastic differential equations and stochastic control theory in a simple and self-contained way. In particular, a key to the stochastic control problem is the dynamic programming principle (DPP), which leads to the notion of viscosity solutions of Hamilton–Jacobi–Bellman (HJB) equations. The study of viscosity solutions, originated by M. Crandall and P. L. Lions in the 1980s, provides a useful tool for dealing with the lack of smoothness of the value functions in stochastic control.