Solutions of max-plus linear equations and large deviations

Large deviations for slow-fast stochastic partial differential equations

A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.

Max-plus Stochastic Control and Risk-sensitivity

In the Maslov idempotent probability calculus, expectations of random variables are defined so as to be linear with respect to max-plus addition and scalar multiplication. This paper considers control problems in which the objective is to minimize the max-plus expectation of some max-plus additive running cost. Such problems arise naturally as limits of some types of risk sensitive stochastic control problems. The value function is a viscosity solution to a quasivariational inequality (QVI) of dynamic programming. Equivalence of this QVI to a nonlinear parabolic PDE with discontinuous Hamilton…

Local Large Deviation Principle, Large Deviation Principle and Information theory for the Signal -to- Interference -Plus- Noise Ratio Graph Models

Given devices space $D$, an intensity measure $λm\in(0,\infty)$, a transition kernel $Q$ from the space $D$ to positive real numbers $(0,\infty,$ a path-loss function (which depends on the Euclidean distance between the devices and a positive constant $α$), we define a Marked Poisson Point process (MPPP). For a given MPPP and technical constants $τ_λ,γ_λ:(0,\,\infty)\to (0,\infty),$ we define a Marked Signal-to- Interference and Noise Ratio (SINR) graph, and associate with it two empirical measures; the \emph{empirical marked measure} and the \emph{empirical connectivity measure}. For a class …

关于一个离散的max-plus运输问题

On a discrete max-plus transportation problem

本文提出了解决热带最优运输问题的显式算法，指出在特定条件下最优热带计划可能不对应于完美匹配，并证明了解的唯一性罕见。

Max-plus objects to study the complexity of graphs

Given an undirected graph $G$, we define a new object $H_G$, called the mp-chart of $G$, in the max-plus algebra. We use it, together with the max-plus permanent, to describe the complexity of graphs. We show how to compute the mean and the variance of $H_G$ in terms of the adjacency matrix of $G$ and we give a central limit theorem for $H_G$. Finally, we show that the mp-chart is easily tractable also for the complement graph.

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…