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

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.

Modeling and computation of water management by real options

“…The finite volume method possesses a special feature of the local conservativity of the numerical fluxes, and is becoming more and more popular. See, for instance, [27] for degenerate parabolic problems, [20] for hyperbolic problems, [21] for elliptic problems, and [37,9] for HJB equations.…”

Numerical Modelling of Cooperative and Noncooperative Three Transboundary Pollution Problems Under Learning by Doing in Three Gorges Reservoir Area

“…Li and Pan [15] constructed a dynamic general equilibrium model of pollution to derive the steady-state equilibrium properties and optimal levels of emission permits and pollution treatment. Similar to [8], we have used the fitted finite volume method to study three transboundary pollution of Three Gorges Reservoir Area with the emission permits trading by cooperative stochastic differential game in [17]. To improve the emission reduction and accumulate the emission reduction experience, the thought of learning by doing has been more and more used in the water pollution emissions trading, and this idea has also been widely used in several other industries, such as clothing manufacturing, instrument manufacturing, automotive assembly, semiconductor manufacturing, has attracted a lot of scholars [2,3,10].…”

“…Based on Yeung's model, Li [13] took emission permits trading into the game, in which the revenues were influenced by emission permits prices and initial quotas. Chang et al [8] extended Li's work to a stochastic version and presented a numerical method to solve the model. Above all literatures about transboundary pollution game took emission permits trading and abatement into consideration, and the two mechanisms affected the optimal strategies to some extent.…”

An empirical study of transboundary air pollution of the Beijing-Tianjin region

“…Moreover, Bernard [2] has tested the impact of the strategic interactions between Russia and China in the international carbon emission permits market. Zhang [6] generalized the emission permits price model obeying a geometric Brownian motion, which is commonly used to described the path of an underlying asset in financial markets. Our goal is to make use of optimal control theory to seek the cooperative optimal emission paths of the Beijing and Tianjing regions.…”

“…According to the derivation of the air transboundary pollution model and references [12,14,15], the parameters are chosen as θ P = 0.06, β = 1.2, ρ = 0.5, and r = 0.08. In compliance with other areas and considering reference [6], the other parameters are chosen to be A 1 = 5, D 1 = 0.1, σ S = 0.3, µ S = 0.2, g i = 3, g j = 2,P i = 1100, andP j = 1200. For the empirical analysis, we set T = 10, E i0 = 5, E j0 = 6, P max = 2000, S min = 20, S max = 50, and σ P = 0.3, which are reasonable values and come from certain authoritative bodies 1 .…”