Applied Mathematics & Optimization

This paper presents a theory of stochastic evolution equations for nuclear-space-valued processes and provides a unified treatment of several examples from the field of applications. (C 0 , 1) reversed evolution systems on countably Hilbertian nuclear spaces are also investigated.

This work deals with a numerical method for solving a mean-field type control problem with congestion. It is the continuation of an article by the same authors, in which suitably defined weak solutions of the system of partial differential equations arising from the model were discussed and existence and uniqueness were proved. Here, the focus is put on numerical methods: a monotone finite difference scheme is proposed and shown to have a variational interpretation. Then an Alternating Direction Method of Multipliers for solving the variational problem is addressed. It is based on an augmented Lagrangian. Two kinds of boundary conditions are considered: periodic conditions and more realistic boundary conditions associated to state constrained problems. Various test cases and numerical results are presented.

A stochastic control representation for solution of the Schrödinger equation is obtained, utilizing complex-valued diffusion processes. The Maslov dequantization is employed, where the domain is complex-valued in the space variable. The notion of stationarity is utilized to relate the Hamilton–Jacobi form of the dequantized Schrödinger equation to its stochastic control representation. Convexity is not required, and consequently, there is no restriction on the duration of the problem. Additionally, existence is reduced to a real-valued domain case.

Abstract. We prove the existence of an invariant measure for the transition semigroup associated with a nonlinear damped stochastic wave equation in R n of the Klein—Gordon type. The uniqueness of the invariant measure and the structure of the corresponding Kolmogorov operator are also studied.

The optimal stopping problem arising in the pricing of American options can be tackled by the so called dual martingale approach. In this approach, a dual problem is formulated over the space of adapted martingales. A feasible solution of the dual problem yields an upper bound for the solution of the original primal problem. In practice, the optimization is performed over a finite-dimensional subspace of martingales. A sample of paths of the underlying stochastic process is produced by a Monte-Carlo simulation, and the expectation is replaced by the empirical mean. As a rule the resulting optimization problem, which can be written as a linear program, yields a martingale such that the variance of the obtained estimator can be large. In order to decrease this variance, a penalizing term can be added to the objective function of the pathwise optimization problem. In this paper, we provide a rigorous analysis of the optimization problems obtained by adding different penalty functions. In particular, a convergence analysis implies that it is better to minimize the empirical maximum instead of the empirical mean. Numerical simulations confirm the variance reduction effect of the new approach.

LetX be a Markov process taking values in a complete, separable metric spaceE and characterized via a martingale problem for an operatorA. We develop a criterion for invariant measures when rangeA is a subset of continuous functions onE. Using this, uniqueness in the class of all positive finite measures of solutions to a (perturbed) measure-valued evolution equation is proved when the test functions are taken from the domain ofA. As a consequence, it is shown that in the characterization of the optimal filter (in the white-noise theory of filtering) as the unique solution to an analogue of Zakai (as well as Fujisaki-Kallianpur-Kunita) equation, it suffices to take domainA as the class of test functions where the signal process is the solution to the martingale problem forA.

We show that the rate of convergence of solutions of finite-difference approximations for uniformly elliptic Bellman’s equations is of order at least h 2/3, where h is the mesh size. The equations are considered in smooth bounded domains.