Browsing by Author "Yam, S. C. P."
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Item Mean Field Stackelberg Games: Aggregation of Delayed InstructionsBensoussan, Alain; Chau, M. H. M.; Yam, S. C. P.; 0000000113231180 (Bensoussan, A)In this paper, we consider an N-player interacting strategic game in the presence of a (endogenous) dominating player, who gives direct influence on individual agents, through its impact on their control in the sense of Stackelberg game, and then on the whole community. Each individual agent is subject to a delay effect on collecting information, specifically at a delay time, from the dominating player. The size of his delay is completely known by the agent, while to others, including the dominating player, his delay plays as a hidden random variable coming from a common fixed distribution. By invoking a noncanonical fixed point property, we show that for a general class of finite N-player games, each of them converges to the mean field counterpart which may possess an optimal solution that can serve as an epsilon-Nash equilibrium for the corresponding finite N-player game. Second, we provide, with explicit solutions, a comprehensive study on the corresponding linear quadratic mean field games of small agents with delay from a dominating player. Given the information flow obtained from both the dominating player and the whole community via the mean field term, the filtration to which the control of the representative agent adapted is non-Brownian. Therefore, we propose to utilize backward stochastic dynamics (instead of the common approach through backward stochastic differential equations) for the construction of adjoint process for the resolution of his optimal control. A simple sufficient condition for the unique existence of mean field equilibrium is provided by tackling a class of nonsymmetric Riccati equations. Finally, via a study of a class of forward-backward stochastic functional differential equations, the optimal control of the dominating player is granted given the unique existence of the mentioned mean field equilibrium for small players.Item Time-Consistent Portfolio Selection Under Short-Selling Prohibition: From Discrete to Continuous SettingBensoussan, Alain; Wong, K. C.; Yam, S. C. P.; Yung, S. P.; 0000 0001 1323 1180 (Bensoussan, A); 2002119562 (Bensoussan, A); 0000-0003-0743-498X (Bensoussan, A)In this paper, we study the time consistent strategies in the mean-variance portfolio selection with short-selling prohibition in both discrete and continuous time settings. Recently, [T. Björk, A. Murgoci, and X. Y. Zhou, Math. Finance, 24 (2014), pp. 1-24] considered the problem with state dependent risk aversion in the sense that the risk aversion is inversely proportional to the current wealth, and they showed that the time consistent control is linear in wealth. Considering the counterpart of their continuous time equilibrium control in the discrete time framework, the corresponding "optimal" wealth process can take negative values; and this negativity in wealth will lead the investor to a risk seeker which results in an unbounded value function that is economically unsound; even more, the limiting of the discrete solutions has shown to be their obtained continuous solution in [T. Björk, A. Murgoci, and X. Y. Zhou, Math. Finance, 24 (2014), pp. 1-24]. To deal this limitation, we eliminate the chance of getting nonpositive wealth by prohibiting short-selling. Using backward induction, the equilibrium control in discrete time setting is shown to be linear in wealth. An application of the extended Hamilton-Jacobi-Bellman equation (see [T. Björk and A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, working paper, Stockholm School of Economics, Stockholm, Sweden, 2010]) makes us also conclude that the continuous time equilibrium control is also linear in wealth with investment to wealth ratio satisfying an integral equation uniquely. We also show that the discrete time equilibrium controls converge to that in continuous time setting. Finally, in numerical studies, we illustrate that the constrained strategy in continuous setting can outperform the unconstrained one in some situations as depicted in Figure 8.