STOCHASTIC GAMES OF CONTROL AND STOPPING FOR A LINEAR DIFFUSION

We study three stochastic differential games. In each game, two players control a process X = {X_t, 0 ≤ t < ∞} which takes values in the interval I = (0,1), is absorbed at the endpoints of I, and satisfies a stochastic differential equation

In the first of our games, which is zero-sum, player 𝔄 has a continuous reward function u : [0,1] → ℝ. In addition to α(·), player 𝔄 chooses a stopping rule τ and seeks to maximize the expectation of u(X_τ); whereas player 𝔅 chooses β(·) and aims to minimize this expectation.

In the second game, players 𝔄 and 𝔅 each have continuous reward functions u(·) and v(·), choose stopping rules τ and ρ, and seek to maximize the expectations of u(X_τ) and v(X_ρ), respectively.

In the third game the two players again have continuous reward functions u(·) and v(·), now assumed to be unimodal, and choose stopping rules τ and ρ. This game terminates at the minimum τ∧ρ the stopping rules τ and ρ, and players 𝔄, 𝔅 want to maximize the expectations of u(X_τ∧ρ) and ν(X_τ∧ρ), respectively.

Under mild technical assumptions we show that the first game has a value, and find a saddle point of optimal strategies for the players. The other two games are not zero-sum, in general, and for them we construct Nash equilibria.