Some estimates for finite difference approximations. (English) Zbl 0684.93088

Menaldi, Jose-Luis

SIAM J. Control Optimization 27, No. 3, 579-607 (1989).

The author deals with the approximation of optimal control problems for diffusion processes by means of finite difference methods. A typical problem in stochastic control theory is considered. In a complete filtered probability space (\(\Omega\),P,\({\mathcal F},{\mathcal F}(t),t\geq 0)\) suppose we have two progressively measurable processes (y(t),\(\lambda\) (t),t\(\geq 0)\) satisfying the following stochastic differential equation in the Itô sense: \[ dy(t)=g(y(t),\lambda (t))dt+\sigma (y(t),\lambda (t))dw(t),\quad t\geq 0,\quad y(0)=x, \] for given x, g, \(\sigma\), and some n-dimensional Wiener process (w(t),t\(\geq 0)\). The processes (y(t),t\(\geq 0)\) and (\(\lambda\) (t),t\(\geq 0)\) represent the state in \({\mathcal R}^ d\) and the control in \(\Lambda\) (a compact metric space) of the dynamic system, respectively. The cost functional is given by: \[ J(x,\lambda)=E\{\int^{\tau}_{0}f(y(t),\lambda (t))e^{-\alpha t}dt\}, \] where f is a given function, \(\alpha >0\), and \(\tau\) is the first exit time of a domain D in \({\mathcal R}^ d\) for the process (y(t),t\(\geq 0).\)
The author introduces a finite difference operator, satisfying the discrete maximum principle, by which the associated Hamilton-Jacobi- Bellman equation can be given a probabilistic interpretation.
First the one-dimensional case is investigated. Then the general problem is considered.

Reviewer: M.Tibaldi

MSC:

93E20	Optimal stochastic control
93E25	Computational methods in stochastic control (MSC2010)
49M25	Discrete approximations in optimal control
65K10	Numerical optimization and variational techniques
65G99	Error analysis and interval analysis

Keywords:

diffusion processes; finite difference methods; complete filtered probability space; n-dimensional Wiener process; discrete maximum principle; Hamilton-Jacobi-Bellman equation