Volume 27, 2021
Special issue in honor of Enrique Zuazua's 60th birthday
|Number of page(s)||27|
|Published online||24 August 2021|
Mean field approach to stochastic control with partial information
International Center for Decision and Risk Analysis, Jindal School of Management, University of Texas at Dallas.
2 School of Data Science, The City University of Hong Kong.
3 Department of Statistics, The Chinese University of Hong Kong.
* Research supported by grant from the National Science Foundation 1905449.
** Phillip Yam acknowledges the financial supports from HKGRF-14300717 with the project title “New kinds of Forward-backward Stochastic Systems with Applications”, HKGRF-14300319 with the project title “Shape-constrained Inference: Testing for Monotonicity”, and HKGRF-14301321 with the project title “General Theory for Infinite Dimensional Stochastic Control: Mean Field and Some Classical Problems”. He also thanks Columbia University for the kind invitation to be a visiting faculty member in the Department of Statistics during his sabbatical leave. He also recalled the unforgettable moments and the happiness shared with his beloved father and used this work in memory of his father's brave battle against liver cancer.
*** Corresponding author: firstname.lastname@example.org
Accepted: 4 August 2021
In our present article, we follow our way of developing mean field type control theory in our earlier works [Bensoussan et al., Mean Field Games and Mean Field Type Control Theory. Springer, New York (2013)], by first introducing the Bellman and then master equations, the system of Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations, and then tackling them by looking for the semi-explicit solution for the linear quadratic case, especially with an arbitrary initial distribution; such a problem, being left open for long, has not been specifically dealt with in the earlier literature, such as Bensoussan [Stochastic Control of Partially Observable Systems. Cambridge University Press, (1992)] and Nisio [Stochastic control theory: Dynamic programming principle. Springer (2014)], which only tackled the linear quadratic setting with Gaussian initial distributions. Thanks to the effective mean-field theory, we propose a solution to this long standing problem of the general non-Gaussian case. Besides, our problem considered here can be reduced to the model in Bandini et al. [Stochastic Process. Appl. 129 (2019) 674–711], which is fundamentally different from our present proposed framework.
Mathematics Subject Classification: 49N30 / 49N70 / 49N90 / 60H15 / 60H30 / 91A16
Key words: Duncan-Mortensen-Zakai equations / mean field type control problem / Bellman and Master equations / filtering formulae with non-Gaussian initial conditions / linear dynamics and quadratic payoff / settings with Gaussian or non-Gaussian initial distributions / Riccati equations
© The authors. Published by EDP Sciences, SMAI 2021
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