## Electronic Communications in Probability

### Optimal Control for Absolutely Continuous Stochastic Processes and the Mass Transportation Problem

Toshio Mikami

#### Abstract

We study the optimal control problem for $\mathbb{R}^d$-valued absolutely continuous stochastic processes with given marginal distributions at every time. When $d=1$, we show the existence and the uniqueness of a minimizer which is a function of a time and an initial point. When $d \gt 1$,  we show that a minimizer exists and that  minimizers satisfy the same ordinary differential equation.

#### Article information

Source
Electron. Commun. Probab., Volume 7 (2002), paper no. 20, 199-213.

Dates
Accepted: 29 October 2002
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.ecp/1463434787

Digital Object Identifier
doi:10.1214/ECP.v7-1061

Mathematical Reviews number (MathSciNet)
MR1937905

Zentralblatt MATH identifier
1030.93060

Subjects
Primary: 93E20: Optimal stochastic control

Rights

#### Citation

Mikami, Toshio. Optimal Control for Absolutely Continuous Stochastic Processes and the Mass Transportation Problem. Electron. Commun. Probab. 7 (2002), paper no. 20, 199--213. doi:10.1214/ECP.v7-1061. https://projecteuclid.org/euclid.ecp/1463434787

#### References

• Breiman, L. (1992), Probability, SIAM, Philadelphia.
• Brenier, Y. and Benamou, J. D. (1999), A numerical method for the optimal mass transport problem and related problems, Contemporary Mathematics 226, AMS, Providence, 1-11.
• Brenier, Y. and Benamou, J. D. (2000), A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math. 84, 375-393.
• Carlen, E. A. (1984), Conservative diffusions, Commun. Math. Phys. 94, 293-315.
• Carlen, E. A. (1986), Existence and sample path properties of the diffusions in Nelson's stochastic machanics, Lecture Notes in Mathematics1158, Springer, Berlin Heidelberg New York, 25-51.
• Çinlar, E. and Jacod, J. (1981), Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures, Seminar on Stochastic Processes 1981, Birkhauser, Boston Basel Berlin, 159-242.
• Dall'Aglio, G. (1991), Frèchet classes: the beginning, Mathematics and its applications 67, Kluwer Academic Publishers, Dordrecht Boston London,1-12.
• Doob, J. L. (1990), Stochastic processes, John Wiley & Sons, Inc., New York.
• Evans, L. C. (1998), Partial differential equations, AMS, Providence.
• Evans, L. C. and Gangbo, W. (1999), Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc. 137, No. 653.
• Fleming, W. H. and Soner, H. M. (1993), Controlled Markov Processes and Viscosity Solutions, Springer, Berlin Heidelberg New York.
• Gangbo, W. and McCann, R. J. (1996), The geometry of optimal transportation, Acta Math. 177, 113-161.
• Ikeda, N. and Watanabe, S. (1981), Stochastic differential equations and diffusion processes, North-Holland/Kodansha, Amsterdam New York Oxford Tokyo.
• Jordan, R, Kinderlehrer, D. and Otto, F. (1998), The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal. 29, 1-17.
• Mikami, T. (1990), Variational processes from the weak forward equation, Commun. Math. Phys. 135, 19-40.
• Mikami, T. (2000), Dynamical systems in the variational formulation of the Fokker-Planck equation by the Wasserstein metric, Appl. Math. Optim. 42, 203-227.
• Nelsen, R. B. (1999), An Introduction to Copulas, Lecture Notes in Statistics 139, Springer, Berlin Heidelberg New York.
• Otto, F. (2001), The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26, 101-174.
• Quastel, J. and Varadhan, S.R.S. (1997), Diffusion semigroups and diffusion processes corresponding to degenerate divergence form operators, Comm. Pure Appl. Math. 50, 667-706.
• Rachev, S. T. and Rüschendorf, L. (1998), Mass transportation problems, Vol. I: Theory, Springer, Berlin Heidelberg New York.
• Salisbury, T. S. (1986), An increasing diffusion, Seminar on Stochastic Processes 1984, Birkhäuser, Boston Basel Berlin, 173-194.