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The objective of this paper is to study the optimal consumption and portfolio choice problem of risk-controlled investors who strive to maximize total expected discounted utility of both consumption and terminal wealth. Risk is measured by the variance of terminal wealth, which introduces a nonlinear function of the expected value into the control problem. The control problem presented is no longer a standard stochastic control problem but rather, a mean field-type control problem. The optimal portfolio and consumption rules are obtained explicitly. Numerical results shed light on the importance of controlling variance risk. The optimal investment policy is nonmyopic, and consumption is not sacrificed. 

In our present article, we follow our way of developing mean field type control theory in our earlier works [Bensoussanet 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 Bandiniet al.[Stochastic Process. Appl.129(2019) 674–711], which is fundamentally different from our present proposed framework.

 

This paper considers a class of linear-quadratic-Gaussian (LQG) mean-field games (MFGs) with partial observation structure for individual agents. Unlike other literature, there are some special features in our formulation. First, the individual state is driven by some common-noise due to the external factor and the state-average thus becomes a random process instead of a deterministic quantity. Second, the sensor function of individual observation depends on state-average thus the agents are coupled in triple manner: not only in their states and cost functionals, but also through their observation mechanism. The decentralized strategies for individual agents are derived by the Kalman filtering and separation principle. The consistency condition is obtained which is equivalent to the wellposedness of some forward-backward stochastic differential equation (FBSDE) driven by common noise. Finally, the related ϵ-Nash equilibrium property is verified.