Stochastic Stackelberg Games with a Jump-Diffusion Follower: Existence, Verification, and a Fixed-Point Algorithm

We study a continuous-time Stackelberg game in which a leader sets the drift of a one-dimensional jump-diffusion through a feedback rule and a follower, who observes the rule, chooses when to exit. In the feedback sense, each rule a best response to the other, the equilibria are the solutions of a coupled system: the leader’s Hamilton–Jacobi–Bellman (HJB) equation off the follower’s exit region, and the follower’s obstacle problem under the leader’s drift. We prove existence of an equilibrium by a Schauder–Tychonoff argument, a half-line structure of the exit region for a put-type exit payoff under a non-increasing drift, and a comparison theorem for the leader’s equation when the exit region is a half-line. For the alternating fixed-point algorithm we show that each half-step, computed by splitting off the jump term, is a sup-norm contraction with rate at most $\lambda/(q+\lambda)$, and that in the half-line case the outer iteration is a scalar map on the follower’s threshold whose rate is not fixed by $q$ and $\lambda$. For double-exponential (Kou) jumps the follower’s best response to a constant drift is in closed form through the Wiener–Hopf factors of the resolvent paper of the Stochastic Analysis Desk: the threshold is $K – 1/\beta_1 – 1/\beta_2 + 1/\eta_-$. A grid implementation of the two half-steps reproduces this closed form at second order, and the follower sweeps contract at measured rates $0.899$, $0.947$ and $0.973$ against the bounds $0.909$, $0.952$ and $0.976$.