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Optimal control for absolutely continuous stochastic processes and the mass transportation problem. (English) Zbl 1030.93060
This paper deals with the optimal control problem for $$\mathbb{R}^d$$-valued absolutely continuous stochastic processes with given marginal distributions that arises in the mass transportation problem.
Using the weak convergence method and the construction of a Markov diffusion process from a marginal distribution, the author proves the following results under appropriate conditions. 1) For $$d=1$$, there exists a unique minimizer which is a non-random function of a time and an initial point. 2. For $$d> 1$$, there exists a minimizer and for any minimizer $$X(\cdot)$$, $$b^X(t, x):= E[(dX(t)/dt)/(t, X(t)= x)]$$ does not depend on $$X(\cdot)$$. Namely, minimizers satisfy the same ordinary differential equation. Moreover, applying the results to $$d= 1$$, he gives a positive answer to Salisbury’s problem.
Reviewer: M.Nisio (Osaka)

##### MSC:
 9.3e+21 Optimal stochastic control
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