Max-Plus decomposition of supermartingales and convex order. Application to American options and portfolio insurance

We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any supermartingale of class $(\mathcal{D})$ as a conditional expectation of some running supremum process. As an application, we show how the Max-Plus supermartingale decomposition allows, in particular, to solve the American optimal stopping problem without having to compute the option price. Some illustrative examples based on one-dimensional diffusion processes are then provided. Another interesting application concerns the portfolio insurance. Hence, based on the ``Max-Plus martingale,'' we solve in the paper an optimization problem whose aim is to find the best martingale dominating a given floor process (on every intermediate date), w.r.t. the convex order on terminal values.Comment: Published in at http://dx.doi.org/10.1214/009117907000000222 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems

Many problems in Physics are described by dynamical systems that are conformally symplectic (e.g., mechanical systems with a friction proportional to the velocity, variational problems with a small discount or thermostated systems). Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting. We provide all details for maps, but we present also the modifications needed to obtain a direct proof for the case of differential equations. We consider a family of conformally symplectic maps $f_{\mu, \epsilon}$ defined on a $2d$-dimensional symplectic manifold $\mathcal M$ with exact symplectic form $\Omega$; we assume that $f_{\mu,\epsilon}$ satisfies $f_{\mu,\epsilon}^*\Omega=\lambda(\epsilon) \Omega$. We assume that the family depends on a $d$-dimensional parameter $\mu$ (called drift) and also on a small scalar parameter $\epsilon$. Furthermore, we assume that the conformal factor $\lambda$ depends on $\epsilon$, in such a way that for $\epsilon=0$ we have $\lambda(0)=1$ (the symplectic case). We study the domains of analyticity in $\epsilon$ near $\epsilon=0$ of perturbative expansions (Lindstedt series) of the parameterization of the quasi--periodic orbits of frequency $\omega$ (assumed to be Diophantine) and of the parameter $\mu$. Notice that this is a singular perturbation, since any friction (no matter how small) reduces the set of quasi-periodic solutions in the system. We prove that the Lindstedt series are analytic in a domain in the complex $\epsilon$ plane, which is obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The radii of the excluded balls decrease faster than any power of the distance of the center to the origin