On stochastic differential games with impulse controls and applications

De Santis, Davide (2020) On stochastic differential games with impulse controls and applications. PhD thesis, London School of Economics and Political Science.

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Abstract

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.

Item Type:	Thesis (PhD)
Additional Information:	© 2020 Davide De Santis
Library of Congress subject classification:	Q Science > QA Mathematics
Sets:	Departments > Sociology
Supervisor:	Campi, Luciano
URI:	http://etheses.lse.ac.uk/id/eprint/4310

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