Rainer Buckdahn

Value function of differential games without Isaacs conditions. An approach with nonanticipative mixed strategies

In the present paper we investigate the problem of the existence of a value for differential games without Isaacs condition. For this we introduce a suitable concept of mixed strategies along a partition of the time interval, which are associated with classical nonanticipative strategies (with delay). Imposing on the underlying controls for both players a conditional independence property, we obtain the existence of the value in mixed strategies as the limit of the lower as well as of the upper value functions along a sequence of partitions which mesh tends to zero. Moreover, we characterize this value in mixed strategies as the unique viscosity solution of the corresponding Hamilton–Jacobi–Isaacs equation.

Differential games with asymmetric information and without Isaacs’ condition

We investigate a two-player zero-sum differential game with asymmetric information on the payoff and without Isaacs’ condition. The dynamics is an ordinary differential equation parametrized by two controls chosen by the players. Each player has a private information on the payoff of the game, while his opponent knows only the probability distribution on the information of the other player. We show that a suitable definition of random strategies allows to prove the existence of a value in mixed strategies. This value is taken in the sense of the limit of any time discretization, as the mesh of the time partition tends to zero. We characterize it in terms of the unique viscosity solution in some dual sense of a Hamilton–Jacobi–Isaacs equation. Here we do not suppose the Isaacs’ condition, which is usually assumed in differential games.

Stochastic variational inequalities on non-convex domains

Abstract The objective of this work is to prove in a first step the existence and the uniqueness of a solution of the following multivalued deterministic differential equation: { d x ( t ) + ∂ − φ ( x ( t ) ) ( d t ) ∋ d m ( t ) , t > 0 , x ( 0 ) = x 0 , where m : R + → R d is a continuous function and ∂ − φ is the Frechet subdifferential of a ( ρ , γ ) -semiconvex function φ; the domain of φ can be non-convex, but some regularities of the boundary are required. The continuity of the map m ↦ x : C ( [ 0 , T ] ; R d ) → C ( [ 0 , T ] ; R d ) associating to the input function m the solution x of the above equation, as well as tightness criteria allows to pass from the above deterministic case to the following stochastic variational inequality driven by a multi-dimensional Brownian motion: { X t + K t = ξ + ∫ 0 t F ( s , X s ) d s + ∫ 0 t G ( s , X s ) d B s , t ≥ 0 , d K t ( ω ) ∈ ∂ − φ ( X t ( ω ) ) ( d t ) .

Peng’s maximum principle for a stochastic control problem driven by a fractional and a standard Brownian motion

We study a stochastic control system involving both a standard and a fractional Brownian motion with Hurst parameter less than 1/2. We apply an anticipative Girsanov transformation to transform the system into another one, driven only by the standard Brownian motion with coefficients depending on both the fractional Brownian motion and the standard Brownian motion. We derive a maximum principle and the associated stochastic variational inequality, which both are generalizations of the classical case.