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

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
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