### 4 Recherche d'orbites périodiques d'un champ hamiltonien associé à une structure symplectique non standard

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We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like $\sqrt{h}$ where h is the mesh step. Such...

In this paper we prove a comparison result between semicontinuous viscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form ${u}_{t}+H(x,Du)=0$ in $\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}{\mathrm{R}}^{n}\times (0,T)$ where the Hamiltonian H may be noncoercive in the gradient Du. As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.

In this paper we present a few results on convergence for the prime integrals equations connected with the bounce problem. This approach allows both to prove uniqueness for the one-dimensional bounce problem for almost all permissible Cauchy data (see also [6]) and to deepen previous results (see [3], [5], [7]).

We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup ${P}_{t}\phi $ does exist for any bounded $\phi $; and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.

We study stochastic Hamilton-Jacobi-Bellman equations and the corresponding Hamiltonian systems driven by jump-type Lévy processes. The main objective of the present papel is to show existence, uniqueness and a (locally in time) diffeomorphism property of the solution: the solution trajectory of the system is a diffeomorphism as a function of the initial momentum. This result enables us to implement a stochastic version of the classical method of characteristics for the Hamilton-Jacobi equations....

In this paper we investigate the expected terminal utility maximization approach for a dynamic stochastic portfolio optimization problem. We solve it numerically by solving an evolutionary Hamilton-Jacobi-Bellman equation which is transformed by means of the Riccati transformation. We examine the dependence of the results on the shape of a chosen utility function in regard to the associated risk aversion level. We define the Conditional value-at-risk deviation ($CVaRD$) based Sharpe ratio for measuring...