We review recent results on the new concept of worst-case portfolio optimization, i.e. we consider the determination of portfolio processes which yield the highest worst-case expected utility bound if the stock price may have uncertain (down) jumps. The optimal portfolios are derived as solutions of non-linear differential equations which itself are consequences of a Bellman principle for worst...

In this paper we provide estimates of the rates of convergence of monotone approximation schemes for non-convex equations in one spacedimension. The equations under consideration are the degenerate elliptic Isaacs equations with x-depending coefficients, and the results applies in particular to finite difference methods and control schemes based on the dynamic programming principle. Recently, K...

We study a class of stochastic target games where one player tries to find a strategy such that the state process almost-surely reaches a given target, no matter which action is chosen by the opponent. Our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non-linear partial differential equation. Because ab...

The evolution of wireless and mobile networks becomes faster and faster, so the optimal allocation of radio resources is a problem which is imperative. This development of telecommunication networks is accompanied with an efficient deployment of wireless networks such Wireless Fidelity and networks mobile as LongTerm Evolution (LTE). In this paper, we propose an algorithm improving the global a...

Optimization of decision problems in stochastic environments is usually concerned with maximizing the probability of achieving the goal and minimizing the expected episode length. For interacting agents in time-critical applications, learning of the possibility of scheduling of subtasks (events) or the full task is an additional relevant issue. Besides, there exist highly stochastic problems wh...

We present a new method for the numerical solution of the Hamilton Jacobi Bellman PDE that arises in an infinite time optimal control problem. The method can be of higher order to reduce ”the curse of dimensionality”. It proceeds in two stages. First the HJB PDE is solved in a neighborhod of the origin using the power series method of Al’brecht. From a boundary point of this neighborhood, an ex...

We study a class of controlled differential equations driven by rough paths (or rough path realizations of Brownian motion) in the sense of T. Lyons. It is shown that the value function satisfies a HJB type equation; we also establish a form of the Pontryagin maximum principle. Deterministic problems of this type arise in the duality theory for controlled diffusion processes and typically invol...

This paper is concerned with Sobolev weak solution of Hamilton-Jacobi-Bellman (HJB) equation. This equation is derived from the dynamic programming principle in the study of the stochastic optimal control problem. Adopting Doob-Meyer decomposition theorem as one of main tool, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. For the ...

This paper is concerned with Sobolev weak solution of Hamilton-Jacobi-Bellman (HJB) equation. This equation is derived from the dynamic programming principle in the study of the stochastic optimal control problem. Adopting Doob-Meyer decomposition theorem as one of main tool, we prove that the optimal value function is the unique Sobolev weak solution of the corresponding HJB equation. For the ...

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group SE(2) = R o S with a metric tensor depending on a smooth external cost C : SE(2) → [δ, 1], δ > 0, computed from image data. The method consists of a first step where geodesically equidistant surfaces are computed as a viscosity solution of a Ha...