On a Strategic Model of Pollution Control

Ferrari G, Koch T. On a Strategic Model of Pollution Control . Center for Mathematical Economics Working Papers. Vol 586. Bielefeld: Center for Mathematical Economics; 2017.This paper proposes a strategic model of pollution control. A firm, representative of the productive sector of a country, aims at maximizing its profits by expanding its production. Assuming that the output of production is proportional to the level of pollutants' emissions, the firm increases the level of pollution. The government of the country aims at minimizing the social costs due to the pollution, and introduces regulatory constraints on the emissions' level, which then effectively cap the output of production. Supposing that the firm and the government face both proportional and fixed costs in order to adopt their policies, we model the previous problem as a stochastic impulse two-person nonzero-sum game. The state variable of the game is the level of the output of production which evolves as a general linearly controlled one-dimensional Itô-diffusion. Following an educated guess, we first construct a pair of candidate equilibrium policies and of corresponding equilibrium values, and we then provide a set of sufficient conditions under which they indeed realize an equilibrium. Our results are complemented by a numerical study when the (uncontrolled) output of production evolves as a geometric Brownian motion, and the firm's operating prot and the government's running cost functions are of power type. An analysis of the dependency of the equilibrium policies and values on the model parameters yields interesting new behaviors that we explain as a consequence of the strategic interaction between the firm and the government

Nonzero-sum stochastic differential games between an impulse controller and a stopper

We study a two-player nonzero-sum stochastic differential game, where one player controls the state variable via additive impulses, while the other player can stop the game at any time. The main goal of this work is to characterize Nash equilibria through a verification theorem, which identifies a new system of quasivariational inequalities, whose solution gives equilibrium payoffs with the correspondent strategies. Moreover, we apply the verification theorem to a game with a one-dimensional state variable, evolving as a scaled Brownian motion, and with linear payoff and costs for both players. Two types of Nash equilibrium are fully characterized, i.e. semi-explicit expressions for the equilibrium strategies and associated payoffs are provided. Both equilibria are of threshold type: in one equilibrium players’ intervention are not simultaneous, while in the other one the first player induces her competitor to stop the game. Finally, we provide some numerical results describing the qualitative properties of both types of equilibrium

A fixed-point policy-iteration-type algorithm for symmetric nonzero-sum stochastic impulse control games

Nonzero-sum stochastic differential games with impulse controls offer a realistic and far-reaching modelling framework for applications within finance, energy markets, and other areas, but the difficulty in solving such problems has hindered their proliferation. Semi-analytical approaches make strong assumptions pertaining to very particular cases. To the author’s best knowledge, the only numerical method in the literature is the heuristic one we put forward in Aïd et al (ESAIM Proc Surv 65:27–45, 2019) to solve an underlying system of quasi-variational inequalities. Focusing on symmetric games, this paper presents a simpler, more precise and efficient fixed-point policy-iteration-type algorithm which removes the strong dependence on the initial guess and the relaxation scheme of the previous method. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. A novel provably convergent single-player impulse control solver is also provided. The main algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise very challenging problems, and even some which go beyond the scope of the currently available theory

Optimal market making under partial information and numerical methods for impulse control games with applications

The topics treated in this thesis are inherently two-fold. The first part considers the problem of a market maker who wants to optimally set bid/ask quotes over a finite time horizon, to maximize her expected utility. The intensities of the orders she receives depend not only on the spreads she quotes, but also on unobservable factors modelled by a hidden Markov chain. This stochastic control problem under partial information is solved by means of stochastic filtering, control and piecewise-deterministic Markov processes theory. The value function is characterized as the unique continuous viscosity solution of its dynamic programming equation. Afterwards, the analogous full information problem is solved and results are compared numerically through a concrete example. The optimal full information spreads are shown to be biased when the exact market regime is unknown, as the market maker needs to adjust for additional regime uncertainty in terms of P&L sensitivity and observable order ow volatility. The second part deals with numerically solving nonzero-sum stochastic differential games with impulse controls. These offer a realistic and far-reaching modelling framework for applications within finance, energy markets and other areas, but the diffculty in solving such problems has hindered their proliferation. Semi-analytical approaches make strong assumptions pertaining very particular cases. To the author's best knowledge, there are no numerical methods available in the literature. A policy-iteration-type solver is proposed to solve an underlying system of quasi-variational inequalities, and it is validated numerically with reassuring results. In particular, it is observed that the algorithm does not enjoy global convergence and a heuristic methodology is proposed to construct initial guesses. Eventually, the focus is put on games with a symmetric structure and a substantially improved version of the former algorithm is put forward. A rigorous convergence analysis is undertaken with natural assumptions on the players strategies, which admit graph-theoretic interpretations in the context of weakly chained diagonally dominant matrices. A provably convergent single-player impulse control solver, often outperforming classical policy iteration, is also provided. The main algorithm is used to compute with high precision equilibrium payoffs and Nash equilibria of otherwise too challenging problems, and even some for which results go beyond the scope of all the currently available theory

On some Stochastic Control Problems arising in Environmental Economics and Commodity Markets

Koch T. On some Stochastic Control Problems arising in Environmental Economics and Commodity Markets. Bielefeld: Universität Bielefeld; 2020

On stochastic differential games with impulse controls and applications

The thesis explores general stochastic differential games involving impulse controls and ultimately investigates competition in dealer markets. The work begins with the first chapter on general non-zero stochastic differential games between an impulse controller and a stopper, providing the first model of such class of games using impulse controls. Nash equilibria are characterised through a verification theorem, which identifies a new system of quasi-variational inequalities whose solution gives equilibrium payoffs with the correspondent strategies. Then, in order to show how the verification theorem is meant to be applied, an example is shown and two different types of Nash equilibrium are fully characterised. To conclude, some numerical results describing the qualitative properties of both types of equilibrium are provided. The dissertation continues with the second chapter on general zero-sum stochastic differential games with impulse controls. Here, two agents play feedback impulse control strategies instead of strategies defined in an Elliot-Kalton fashion, as commonly done in the literature, and are not allowed to apply impulses simultaneously, resulting in the upper value and lower value functions of the game being naturally associated with the cases in which either player has priority. The main objective is to apply the stochastic Perron's method in order to have the game value function as the viscosity solution to the double obstacle partial differential equation arising from the problem after a viscosity comparison result. The third and final chapter is about the study of competition in dealer markets. The setting consists in two dealers trading at discrete times via market orders with price impact, resulting in one of the first nonzero-sum game with impulse controls applied to optimal trading. Similarly to the first chapter, a verification theorem identifying the system of quasi-variational inequalities providing the equilibrium payoff functions and strategies is given. Furthermore, a framework to look for equilibria where both players apply impulses simultaneously is introduced. This is very important as it is not possible to find equilibria when only one dealer trades at a time, whereas there exists at least a Nash equilibrium when both dealers trade simultaneously

On a strategic model of pollution control

Ferrari G, Koch T. On a strategic model of pollution control. Annals of Operations Research. 2019;275(2):297-319.This paper proposes a strategic model of pollution control. A firm, representative of the productive sector of a country, aims at maximizing its profits by expanding its production. Assuming that the output of production is proportional to the level of pollutants' emissions, the firm increases the level of pollution. The government of the country aims at minimizing the social costs due to the pollution, and introduces regulatory constraints on the emissions' level, which then effectively cap the output of production. Supposing that the firm and the government face both proportional and fixed costs in order to adopt their policies, we model the previous problem as a stochastic impulse two-person nonzero-sum game. The state variable of the game is the level of the output of production which evolves as a general linearly controlled one-dimensional Ito-diffusion. Following an educated guess, we first construct a pair of candidate equilibrium policies and of corresponding equilibrium values, and we then provide a set of sufficient conditions under which they indeed realize an equilibrium. Our results are complemented by a numerical study when the (uncontrolled) output of production evolves as a geometric Brownian motion, and the firm's operating profit and the government's running cost functions are of power type. An analysis of the dependency of the equilibrium policies and values on the model parameters yields interesting new behaviors that we explain as a consequence of the strategic interaction between the firm and the government