LQG Risk-Sensitive Mean Field Games with a Major Agent: A Variational Approach

Risk sensitivity plays an important role in the study of finance and economics as risk-neutral models cannot capture and justify all economic behaviors observed in reality. Risk-sensitive mean field game theory was developed recently for systems where there exists a large number of indistinguishable, asymptotically negligible and heterogeneous risk-sensitive players, who are coupled via the empirical distribution of state across population. In this work, we extend the theory of Linear Quadratic Gaussian risk-sensitive mean-field games to the setup where there exists one major agent as well as a large number of minor agents. The major agent has a significant impact on each minor agent and its impact does not collapse with the increase in the number of minor agents. Each agent is subject to linear dynamics with an exponential-of-integral quadratic cost functional. Moreover, all agents interact via the average state of minor agents (so-called empirical mean field) and the major agent's state. We develop a variational analysis approach to derive the best response strategies of agents in the limiting case where the number of agents goes to infinity. We establish that the set of obtained best-response strategies yields a Nash equilibrium in the limiting case and an $\varepsilon$-Nash equilibrium in the finite player case. We conclude the paper with an illustrative example

On Talagrand's functional and generic chaining

In the study of the supremum of stochastic processes, Talagrand's chaining functionals and his generic chaining method are heavily related to the distribution of stochastic processes. In the present paper, we construct Talagrand's type functionals in the general distribution case and obtain the upper bound for the suprema of all $p$-th moments of the stochastic process using the generic chaining method. As applications, we obtained the Johnson-Lindenstrauss lemma, the upper bound for the supremum of all $p$-th moment of order 2 Gaussian chaos, and convex signal recovery in our setting

Potential pollen evidence for the 1933 M 7.5 Diexi earthquake and implications for post-seismic landscape recovery

The relationships between strong earthquakes, landslides, and vegetation destruction and the process of post-seismic recovery in tectonically active alpine valley areas have not been adequately documented. Here we show detailed pollen study results from a swamp located near the epicenter of the 1933 M 7.5 Diexi earthquake in eastern Qinghai-Tibetan Plateau (QTP) to reveal the impact of earthquake on vegetation, and the post-seismic recovery process. Based on(210)Pb-Cs-137 age model, the seismic event layer is well constrained. The earthquake event corresponds stratigraphically to a zone with the lowest pollen concentrations, the lowest pollen diversity, and a high frequency of non-arboreal pollen. Elaeagnaceae scrubs rapidly developed in post-seismic landscape recovery processes, which is important for reducing soil erosion and landslide activities. Natural ecological recovery is slow due to increasing human activities and historical climatic fluctuations

When one stock share is a biological individual: a stylized simulation of the population dynamics in an order-driven market

The demand-supply relationship plays an important role in an order-driven stock market. In this thesis, we propose a stylized model by defining demand (supply) over a stock at a certain time as how many shares are on the bid (ask) side, which includes all buy (sell) limit orders and buy (sell) market orders. We treat two types of shares as two different species with an interaction effect and construct generalized Lotka-Volterra equations based on some properties or assumptions of an order-driven market. Also, we apply the model to simulate how the population of the two types of shares evolves over time under the condition that there is no signal information influencing the decisions of investors. The model suggests that the population of bid and ask shares moves either to a fixed point in the phase space or exhibits periodical dynamics. Also, our model explains, though not perfectly, why it is that stock prices sometimes behave chaotically