Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations

Abstract

Chapter 6. Controllable fast variables 6.1. Bounded-time controllability and ergodicity 6.2. Stabilization and a formula for the effective initial data 6.3. An explicit formula for the effective Hamiltonian and the limit differential game 6.4. Uniform convergence 6.5. The reduction order formula for the effective control problem Chapter 7. Nonresonant fast variables 7.1. Ergodicity 7.2. Stabilization 7.3. Uniform convergence Chapter 8. A counterexample to uniform convergence Chapter 9. Applications to homogenization 9.1. Periodic homogenization of 1st order H-J equations v vi CONTENTS 9.2. Periodic homogenization of 2nd order equations Bibliography 73

Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility

“…This situation is modeled as a 0-sum differential game: its value function is characterized by a Hamilton-Jacobi-Isaacs PDE that can be analyzed in the framework of viscosity solutions [21,3]. In [1,2,3] the disturbanceũ t and/or the controls u t may also affect the fast variables Y t (constrained to a compact set). Then there is no invariant measure and the definition of the effective Hamiltonian and terminal cost is less explicit, but the convergence theorem still holds.…”

“…We consider the diffusion process in R m (25) dY The first result of this section is a Liouville property that replaces the standard strong maximum principle of the periodic case and is the key ingredient for extending some results of [3] to the nonperiodic setting.…”

“…On the fast process Y t we will assume that the matrix τ τ T is positive definite and a condition implying the ergodicity (see (3)). We also take payoff functionals of the form formulas, taking into account the effect of uncertain and changing volatility.…”

“…For results based on probabilistic methods we refer the reader to the books [33,32], the recent papers [38,39,40,12], and the references therein. An approach based on PDEviscosity methods for the HJB equations was developed by Alvarez and one of the authors in [1,2,3]; see also [4] for problems with an arbitrary number of scales. It allows one to identify the appropriate limit PDE governed by the effective Hamiltonian and gives general convergence theorems of the value function of the singularly perturbed system to the solution of the effective PDE, under assumptions that include deterministic control (i.e., σ ≡ 0 and/or τ ≡ 0) as well as differential games, deterministic and stochastic.…”

“…The goal of this paper is extending the methods based on viscosity solutions of [1,2,3] to singular perturbation problems of the form (1), including several models of mathematical finance. The main new difficulty is that the fast variables Y t are unbounded.…”