Stochastic Hamiltonian-Jacobi-Bellman equation and stochastic Hamiltonian systems
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.
stochastic Hamilton-Jacobi-Bellman equation
(nonlinear) stochastic partial differential equations of first order
scientific article; zbMATH DE number 1431817