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On the relation between deterministic and stochastic optimal control. (English) Zbl 0794.93106

Da Prato, G. (ed.) et al., Stochastic partial differential equations and applications. Proceedings of the third meeting on stochastic partial differential equations and applications held at Villa Madruzzo, Trento, Italy, January 1990. Harlow: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 268, 124-157 (1992).
The authors establish the assumptions to guarantee that if the feedback \(u^*(t,x)\) is optimal for the nonlinear stochastic non-anticipative control problem with control values in a compact set: \(v^*(t,x)=\inf E\theta(x_ T)\) subject to \(dx_ t= f(x_ t,u_ t)dt+ g(x_ t)dw_ t\), \(x_ 0\in R^ d\), then \(u^*(t,\xi_ t(\eta))\) is also optimal for the anticipative control problem \((P^ \omega)\) for almost all \(\omega\in\Omega\): \[ W(t,\eta)=\inf\left[\int^ T_ 0 \lambda'(t,\eta_ t,\omega)u_ t dt+ \theta\circ \xi_ T(\eta_ T)\right] \] subject to \(\dot\eta_ t= (\partial\xi_ t/\partial x)^{- 1}(\eta_ t) f(\xi+t(\eta_ t), u_ t)-1/2\sum^ d_{i=1} \partial g_ i/\partial x\cdot g_ i(\xi_ t(\eta_ t))\), \(\eta_ 0= x_ 0\), where \(\xi_ t(\eta)\) is the solution of \(d\xi_ t= g(\xi_ t)\circ dw_ t\), \(\xi_ 0= \eta\). Here these problems \((P^ \omega)\) arise by allowing a wider class of possibly anticipate controls and by imposing nonanticipativity as an equality constraint and then are parametrized by the sample functions of the Wiener process \(w_ t\) and of the newly introduced Lagrange multiplier \(\lambda(t,\eta_ t,\omega)\). Moreover, \(\text{EW}(t,\xi^{-1}_ t(x))= v^*(t,x)\). This framework is also generalized to the stochastic control problem with integral cost and the LQG problem is considered as an example. One half of the paper is devoted to the proof of the mentioned Lagrange multiplier theorem for anticipative control and the whole paper is a resource on the techniques for representing the value function of \((P^ \omega)\), constructing its equation and recovering the optimal cost of the original stochastic problem.
For the entire collection see [Zbl 0780.00023].
Reviewer: T.N.Pham (Hanoi)

MSC:

93E20 Optimal stochastic control
49N99 Miscellaneous topics in calculus of variations and optimal control
34F05 Ordinary differential equations and systems with randomness
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