Stochastic Hamiltonian-Jacobi-Bellman equation and stochastic Hamiltonian systems
type: Item
instance of: scholarly article
title: Stochastic Hamiltonian-Jacobi-Bellman equation and stochastic Hamiltonian systems
zbMATH Open document ID: 0967.60067
DOI: 10.1007/BF02269422
author: Vassili Kolokoltsov
published in: Journal of Dynamical and Control Systems
publication date: 2000-04-13T00:00:00Z
review text: This paper deals with the stochastic Hamilton-Jacobi-Bellman equation \[ d S + H\biggl(x,\frac{\partial S}{\partial x}\biggr) dt + c(x) dW = 0 \] where \(H=H(x,p)\) is a deterministic Hamiltonian function which is convex with respect to \(p\), and \(x,p \in R^n \) satisfy ordinary stochastic differential equations \[ d x = \frac{\partial H}{\partial p} dt, \qquad d p = - \frac{\partial H}{\partial x} dt - c' (x) d W(t) \] where \(W\) is a standard \(m\)-dimensional Wiener process (note that Itô and Stratonovich integrals coincide here). The Cauchy problem and the long-time behavior is studied for the related stochastic systems. The presented theory is applied to quantization of stochastic Hamiltonian systems, to scattering for a stochastic version of the Schrödinger equation, to their quantum versions on manifolds and to a control problem for stochastic differential equations with control variable in its drift part. The author also claims that a generalization of the theory to nonautonomous systems of that mentioned type is simple. At the end an open problem of stochastic control theory is formulated, calling for a proof of the Pontryagin maximum principle for the related control system where the control variable also appears in the diffusion part.
reviewed by: Henri Schurz
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 35F20
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 49L20
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 81U05
zbMATH DE Number: 1431817
zbMATH Keywords: stochastic Hamilton-Jacobi-Bellman equation
zbMATH Keywords: stochastic Hamiltonian systems
zbMATH Keywords: stochastic Schrödinger equation
zbMATH Keywords: stochastic scattering
zbMATH Keywords: (nonlinear) stochastic partial differential equations of first order
zbMATH Keywords: stochastic control theory
zbMATH Keywords: stochastic maximum
MaRDI profile type: Publication
cites work: Hamiltonian systems with a stochastic force:nonlinear versus linear, and a girsanov formula
cites work: Transience of stochastically perturbed classical hamiltonian systems and random wave operators
cites work: The log log law for multidimensional stochastic integrals and diffusion processes
cites work: Stochastic Problems in Physics and Astronomy
cites work: Asymptotic completeness for quantum mechanical potential scattering. I: Short range potentials
cites work: Quantum Ito's formula and stochastic evolutions
cites work: Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate
cites work: Idempotent analysis as a tool of control theory and optimal synthesis. I
cites work: Q5552132
cites work: Q5670045
cites work: On the gap between deterministic and stochastic ordinary differential equations
cites work: Q4226056
cites work: Riemann-Stieltjes approximations of stochastic integrals
description: scientific article; zbMATH DE number 1431817