Estimates for multiple stochastic integrals and stochastic Hamilton-Jacobi equations

We study stochastic Hamilton-Jacobi-Bellman equations and the corresponding Hamiltonian systems driven by jump-type Lévy processes. The main objective of the present paper 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. An –in itself interesting– auxiliary result are pointwise a.s. estimates for iterated stochastic integrals driven by a vector of not necessarily independent jump-type semimartingales

Transience and non-explosion of certain stochastic Newtonian systems

We give sufficient conditions for non-explosion and transience of the solution (xt,pt) (in dimensions >= 3) to a stochastic Newtonian system of the form { dxdt = ptdt dpt = -δV(xt)/δx dt - δc(xt)/δx dξt where {ξt}t>=0 is a d-dimensional Lévy process, dξt is an Itô differential and c ∈ C2(Rd,Rd), V ∈ C2(Rd,R) such that V >= 0

Boundary-value problems for Hamiltonian systems and absolute minimizers in calculus of variations

We apply the method of Hamilton shooting to obtain the well-posedness of boundary value problems for certain Hamiltonian systems and some estimates for their solutions. The examples of Hamiltonian functions covered by the method include elliptic polynomials and exponentially growing functions. As a consequence we prove global existence, smoothness and almost everywhere uniqueness of absolute minimizers in the corresponding problem of calculus of variations and hence construct the global field of extremals