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**A stochastic maximum principle for optimal control of diffusions.**
*(English)*
Zbl 0616.93076

Pitman Research Notes in Mathematics Series, 151. Harlow, Essex, England: Longman Scientific & Technical. Copubl. in the United States with John Wiley & Sons, Inc., New York. V, 109 p. Ł 12.00 (1986).

The author derives a maximum principle of the Pontryagin type for the following stochastic optimal control problem: minimize \(J_ 0(u)\) in some class of controls \(u_ t=u(t,x)\) of feedback form with values in a given set of an Euclidean space subject to the constraints \(J_ i(u)=0\), \(i=-m_ 1,...,-1\), \(J_ i(u)\leq 0\), \(i=1,...,m_ 2\). Here \(J_ i(u)=E\{\int^{T}_{a}\ell_ i(t,x_ t,u_ t)dt+c_ i(x_ T)\}\), \(i=-m_ 1,...,0,...,m_ 2\), \(x_ t\) is a weak solution of the stochastic differential equation (SDE) \(dx_ t=f(t,x_ t,u_ t)dt+\sigma (t,x_ t)dw_ t\), \(\{w_ t\}\) is a Brownian motion and f, \(\sigma\), \(\ell_{-m_ 1},..., \ell_{m_ 2}\), \(c_{-m_ 1},..., c_{m_ 2}\) are given functions.

The notes consist of 13 sections. Auxiliary results such as basic terminology, the Girsanov theorem, some facts on weak solutions of SDE and the Lagrange multiplier theorem are given in sections 0,1,2,4. The problem is formulated in section 3. A cone of variations is constructed in section 5. To this end perturbations of the optimal control over short time intervals are used. The general necessary conditions are derived in section 6 using the adjoint process defined by the martingale representation theorem. The case of complete information is studied in sections 7-11. The necessary conditions in this case are given in section 8 with explicitly identified adjoint process, and some examples are solved in section 9. In sections 10, 11 it is shown that necessary conditions are sufficient for some linear convex control problems.

The notes are based, in general, on the author’s results. Some comments on other approaches are given in section 12.

The notes consist of 13 sections. Auxiliary results such as basic terminology, the Girsanov theorem, some facts on weak solutions of SDE and the Lagrange multiplier theorem are given in sections 0,1,2,4. The problem is formulated in section 3. A cone of variations is constructed in section 5. To this end perturbations of the optimal control over short time intervals are used. The general necessary conditions are derived in section 6 using the adjoint process defined by the martingale representation theorem. The case of complete information is studied in sections 7-11. The necessary conditions in this case are given in section 8 with explicitly identified adjoint process, and some examples are solved in section 9. In sections 10, 11 it is shown that necessary conditions are sufficient for some linear convex control problems.

The notes are based, in general, on the author’s results. Some comments on other approaches are given in section 12.

Reviewer: H.Pragarauskas

### MSC:

93E20 | Optimal stochastic control |

49K45 | Optimality conditions for problems involving randomness |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60J60 | Diffusion processes |