In this paper, we investigate Nash equilibrium payoffs for nonzero-sum stochasticdifferential games with reflection. We obtain an existence theorem and a characterizationtheorem of Nash equilibrium payoffs for nonzero-sum stochastic differential games withnonlinear cost functionals defined by doubly controlled reflected backward stochasticdifferential equations.

We provide a deterministic-control-based interpretation for a broad class of fullynonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in asmooth domain. We construct families of two-person games depending on a small parameterε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary conditionby introducing some specific rules near the boundary. We show that the value functionconverges, in the viscosity sense, to the solution of the PDE as ε tendsto zero. Moreover, our construction allows us to treat both the oblique and the mixed typeDirichlet–Neumann boundary conditions.

In this paper, we study one kind of stochastic recursive optimal control problem for thesystems described by stochastic differential equations with delay (SDDE). In ourframework, not only the dynamics of the systems but also the recursive utility depend onthe past path segment of the state process in a general form. We give the dynamicprogramming principle for this kind of optimal control problems and show that the valuefunction is the viscosity solution of the corresponding infinite dimensionalHamilton-Jacobi-Bellman partial differential equation.

The aim of the paper is to provide a linearization approach to the $\mathbb{L}^{\infty}$$\mathrm{See\; PDF}$-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the $\mathbb{L}^{p}$$\mathrm{See\; PDF}$ approach and the associated linear formulations. This seems to be the most appropriate tool for treating $\mathbb{L}^{\infty}$$\mathrm{See\; PDF}$ problems in continuous and lower semicontinuous setting.

We consider a two-player zero-sum-game in a bounded open domain Ωdescribed as follows: at a point x ∈ Ω, Players I and IIplay an ε-step tug-of-war game with probability α, andwith probability β (α + β = 1), arandom point in the ball of radius ε centered at x ischosen. Once the game position reaches the boundary, Player II pays Player I the amountgiven by a fixed payoff function F. We give a detailed proof of the factthat the value functions of this game satisfy the Dynamic Programming Principle

for x ∈ Ω withu(y) = F(y) wheny ∉ Ω. This principle implies the existence ofquasioptimal Markovian strategies.

We study a hybrid control system in which both discrete and continuous controls are involved. The discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jumpset A or a controlled jump set C where controller can choose to jump or not. At each jump, trajectory can move to a different Euclidean space. We allow the cost functionals to be unbounded with certain growth and hence the corresponding value function can be unbounded. We characterize the value function as the unique viscosity solution of the associated quasivariational inequality in a suitable function class. We also consider the evolutionary, finite horizon hybrid control problem with similar model and prove that the value function is the unique viscosity solution in the continuous function class while allowing cost functionals as well as the dynamics to be unbounded.