A one-sided limit order book is modeled as a noncooperative game for n players. Agents offer various quantities of an asset at different prices p ∈ [0, P ], competing to fulfill an incoming order, whose size X is not known a priori. Players can have different payoff functions, reflecting different beliefs about the fundamental value of the asset and probability distribution of the random variable X. In [4] the existence of a Nash equilibrium was established by means of a fixed point argument.The main issue discussed in the present paper is whether this equilibrium can be obtained from the unique solution to a two-point boundary value problem, for a suitable system of discontinuous ODEs. Some additional assumptions are introduced, which yield a positive answer. In particular, this is the case when there are exactly 2 players, or when all n players assign the same exponential probability distribution to the random variable X. A counterexample shows that these assumptions cannot be removed.
Abstract. We consider a control system with "nonclassical" dynamics: x, u, Dxu), where the right hand side depends also on the first order partial derivatives of the feedback control function. Given a probability distribution on the initial data, we seek a feedback u = u(t, x) which minimizes the expected value of a cost functional. Various relaxed formulations of this problem are introduced. In particular, three specific examples are studied, showing the equivalence or non-equivalence of these approximations.Mathematics Subject Classification. 49N25, 49N35.
The paper is concerned with Stackelberg solutions for a differential game with deterministic dynamics but random initial data, where the leading player can adopt a strategy in feedback form: u 1 = u 1 (t, x). The first main result provides the existence of a Stackelberg equilibrium solution, assuming that the family of feedback controls u 1 (t, ·) available to the leading player are constrained to a finite dimensional space. A second theorem provides necessary conditions for the optimality of a feedback strategy. Finally, in the case where the feedback u 1 is allowed to be an arbitrary function, an example illustrates a wide class of systems where the minimal cost for the leading player corresponds to an impulsive dynamics. In this case, a Stackelberg equilibrium solution does not exists, but a minimizing sequence of strategies can be described.
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