Dept. of Math./CMA University of Oslo Pure Mathematics No 24 ISSN 0806–2439 December 2010 A DYNKIN GAME WITH ASYMMETRIC INFORMATION JUKKA LEMPA AND PEKKA MATOM ¨ AKI Abstract. We study a Dynkin game with asymmetric information. The game has a random expiry time, which is exponentially distributed and independent of the underlying process. The players have asymmetric information on the expiry time, namely only one of the players is able to observe its occurrence. We propose a set of conditions under which we solve the saddle point equilibrium and study the implications of the information asymmetry. Results are illustrated with an explicit example. 1. Introduction Dynkin games are game variants of optimal stopping problems, for the seminal study see [6]. Such a game has two players, ”buyer” and ”issuer”, and both of them can stop the underlying process prior the terminal time. In this paper we study the following formulation of the game. First, we assume that the underlying process X is a time homogenous diffusion; we will elaborate the assumptions on X in the next section. At the initial time t = 0, the players choose their own stopping times τ (buyer) and γ (issuer) and at the time of the first exercise, i.e. at τ ∧ γ , the issuer pays the buyer the amount (1.1) g 1 (X τ )1 {τ<γ} + g 2 (X γ )1 {τ>γ} + g 3 (X γ )1 {τ =γ} ; we will pose assumptions on the payoff functions g i in the next section. An interpretation of this is that, at any stopping time γ , the issuer can cancel the buyer’s right to exercise, but she has to pay the cost g 2 (X γ ) to do so. Now, it is the buyers (issuers) objective to choose the stopping time τ (γ ) such that the expected 2010 Mathematics Subject Classification. 60G40,60J60. Key words and phrases. Dynkin game, Nash equilibrium, linear diffusion, resolvent operator, Poisson process. Address. Jukka Lempa, Centre of Mathematics for Applications, University of Oslo, PO Box 1053 Blindern, NO – 0316 Oslo, e-mail: jlempa@cma.uio.no Pekka Matom¨ aki, Department of Accounting and Finance, Turku School of Economics, University of Turku, Rehtorinpellonkatu 3, FIN – 20500 Turku, e-mail: pjsila@utu.fi. 1