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Modeling and Computation of Transboundary Industrial Pollution with Emission Permits Trading by Stochastic Differential Game.

Chang S, Wang X, Wang Z - PLoS ONE (2015)

Bottom Line: The two regions' cooperative and noncooperative optimal emission paths, which maximize the regions' discounted streams of the net revenues, together with the value functions, are obtained.The effects of parameters in the established model on the results have been also examined.All the results demonstrate that the stochastic emission permits prices can motivate the players to make more flexible strategic decisions in the games.

View Article: PubMed Central - PubMed

Affiliation: Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin 300222, China.

ABSTRACT
Transboundary industrial pollution requires international actions to control its formation and effects. In this paper, we present a stochastic differential game to model the transboundary industrial pollution problems with emission permits trading. More generally, the process of emission permits price is assumed to be stochastic and to follow a geometric Brownian motion (GBM). We make use of stochastic optimal control theory to derive the system of Hamilton-Jacobi-Bellman (HJB) equations satisfied by the value functions for the cooperative and the noncooperative games, respectively, and then propose a so-called fitted finite volume method to solve it. The efficiency and the usefulness of this method are illustrated by the numerical experiments. The two regions' cooperative and noncooperative optimal emission paths, which maximize the regions' discounted streams of the net revenues, together with the value functions, are obtained. Additionally, we can also obtain the threshold conditions for the two regions to decide whether they cooperate or not in different cases. The effects of parameters in the established model on the results have been also examined. All the results demonstrate that the stochastic emission permits prices can motivate the players to make more flexible strategic decisions in the games.

No MeSH data available.


Related in: MedlinePlus

Computed errors in the L∞-norm at t = 0.
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pone.0138641.g001: Computed errors in the L∞-norm at t = 0.

Mentions: First of all, we consider the convergence rate of our discretization method to show its accuracy and efficiency. Owing to the limitation of space, we only test region 1’s value function VN1 under the noncooperative game. Additionally, since the closed-form solution of the HJB equation cannot be found, we regard the solution of the NP = 256 = NS and M = 256 mesh in both space and time, respectively, as the “exact” solution VN1. We compute the errors in the discrete L∞-norm at the computational final time step t = 0 on a sequence of meshes with NP = NS = M = 2n for a positive integer n from n = 2 to a maximum n = 7. The discrete L∞-norm is defined as:∥VN1h(P,S,0)-VN1(P,S,0)∥∞=max1<i<NP,1<j<NS/VN1h(Pi,Sj,0)-VN1(Pi,Sj,0)/,where denotes the numerical solution. The log-log plots of the computed maximum errors, along with the linear fitting, are depicted in Fig 1. From the figure we see that the rate of convergence of in the discrete L∞ norm is of the order 𝒪(h0.6353), where . Note that this result is reasonable because of the coupling in the HJB equations. Moreover, it numerically demonstrates that our numerical methods for the HJB equations governing the differential game in transboundary industrial pollution is useful and efficient. Some theoretical analysis about convergence rates should be discussed in the future works.


Modeling and Computation of Transboundary Industrial Pollution with Emission Permits Trading by Stochastic Differential Game.

Chang S, Wang X, Wang Z - PLoS ONE (2015)

Computed errors in the L∞-norm at t = 0.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4581629&req=5

pone.0138641.g001: Computed errors in the L∞-norm at t = 0.
Mentions: First of all, we consider the convergence rate of our discretization method to show its accuracy and efficiency. Owing to the limitation of space, we only test region 1’s value function VN1 under the noncooperative game. Additionally, since the closed-form solution of the HJB equation cannot be found, we regard the solution of the NP = 256 = NS and M = 256 mesh in both space and time, respectively, as the “exact” solution VN1. We compute the errors in the discrete L∞-norm at the computational final time step t = 0 on a sequence of meshes with NP = NS = M = 2n for a positive integer n from n = 2 to a maximum n = 7. The discrete L∞-norm is defined as:∥VN1h(P,S,0)-VN1(P,S,0)∥∞=max1<i<NP,1<j<NS/VN1h(Pi,Sj,0)-VN1(Pi,Sj,0)/,where denotes the numerical solution. The log-log plots of the computed maximum errors, along with the linear fitting, are depicted in Fig 1. From the figure we see that the rate of convergence of in the discrete L∞ norm is of the order 𝒪(h0.6353), where . Note that this result is reasonable because of the coupling in the HJB equations. Moreover, it numerically demonstrates that our numerical methods for the HJB equations governing the differential game in transboundary industrial pollution is useful and efficient. Some theoretical analysis about convergence rates should be discussed in the future works.

Bottom Line: The two regions' cooperative and noncooperative optimal emission paths, which maximize the regions' discounted streams of the net revenues, together with the value functions, are obtained.The effects of parameters in the established model on the results have been also examined.All the results demonstrate that the stochastic emission permits prices can motivate the players to make more flexible strategic decisions in the games.

View Article: PubMed Central - PubMed

Affiliation: Research Center for Mathematics and Economics, Tianjin University of Finance and Economics, Tianjin 300222, China.

ABSTRACT
Transboundary industrial pollution requires international actions to control its formation and effects. In this paper, we present a stochastic differential game to model the transboundary industrial pollution problems with emission permits trading. More generally, the process of emission permits price is assumed to be stochastic and to follow a geometric Brownian motion (GBM). We make use of stochastic optimal control theory to derive the system of Hamilton-Jacobi-Bellman (HJB) equations satisfied by the value functions for the cooperative and the noncooperative games, respectively, and then propose a so-called fitted finite volume method to solve it. The efficiency and the usefulness of this method are illustrated by the numerical experiments. The two regions' cooperative and noncooperative optimal emission paths, which maximize the regions' discounted streams of the net revenues, together with the value functions, are obtained. Additionally, we can also obtain the threshold conditions for the two regions to decide whether they cooperate or not in different cases. The effects of parameters in the established model on the results have been also examined. All the results demonstrate that the stochastic emission permits prices can motivate the players to make more flexible strategic decisions in the games.

No MeSH data available.


Related in: MedlinePlus