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A one-sided limit order book is modeled as a noncooperative game for several players. Agents offer various quantities of an asset at different prices, competing to fulfill an incoming order, whose size 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 incoming order. In a previous paper, 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 ordinary differential equations. Some additional assumptions are introduced, which yield a positive answer. In particular, this is true when there are exactly two players, or when all players assign the same exponential probability distribution to the incoming order. In both of these cases, we also prove that the Nash equilibrium is unique. A counterexample shows that these assumptions cannot be removed, in general.

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: u1=u1(t,x). The first main result provides the existence of a Stackelberg equilibrium solution, assuming that the family of feedback controls u1(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 u1 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 exist, but a minimizing sequence of strategies can be described.

The paper is concerned with problems of optimal feedback control with “non-classical” dynamics x ˙ = f ( t , x , u , D u ) , where the evolution of the state x depends also on the Jacobian matrix D u = ( ∂ u i / ∂ x j ) of the feedback control function u = u ( t , x ) . Given a probability measure μ on the set of initial states, we seek feedback controls u ( ⋅ ) which minimize the expected value of a cost functional. After introducing a basic framework for the study of these problems, this paper focuses on three main issues: (i) necessary conditions for optimality, (ii) equivalence with a relaxed feedback control problem in standard form, and (iii) dependence of the expected minimum cost on the probability measure μ .