Stochastic Graphon Mean Field Games with Jumps and Approximate Nash Equilibria

We study continuous stochastic games with inhomogeneous mean field interactions on large networks and explore their graphon limits. We consider a model with a continuum of players, where each player's dynamics involve not only mean field interactions but also individual jumps induced by a Poisson random measure. We examine the case of controlled dynamics, with control terms present in the drift, diffusion, and jump components. We introduce the graphon game model based on a graphon controlled stochastic differential equation (SDE) system with jumps, which can be regarded as the limiting case of a finite game's dynamic system as the number of players goes to infinity. Under some general assumptions, we establish the existence and uniqueness of Markovian graphon equilibria. We then provide convergence results on the state trajectories and their laws, transitioning from finite game systems to graphon systems. We also study approximate equilibria for finite games on large networks, using the graphon equilibrium as a benchmark. The rates of convergence are analyzed under various underlying graphon models and regularity assumptions.Comment: 37 page

Ruin Probabilities for Risk Processes in Stochastic Networks

We study multidimensional Cram\'er-Lundberg risk processes where agents, located on a large sparse network, receive losses form their neighbors. To reduce the dimensionality of the problem, we introduce classification of agents according to an arbitrary countable set of types. The ruin of any agent triggers losses for all of its neighbours. We consider the case when the loss arrival process induced by the ensemble of ruined agents follows a Poisson process with general intensity function that scales with the network size. When the size of the network goes to infinity, we provide explicit ruin probabilities at the end of the loss propagation process for agents of any type. These limiting probabilities depend, in addition to the agents' types and the network structure, on the loss distribution and the loss arrival process. For a more complex risk processes on open networks, when in addition to the internal networked risk processes the agents receive losses from external users, we provide bounds on ruin probabilities.Comment: 31 page

Limit Theorems for Default Contagion and Systemic Risk

We consider a general tractable model for default contagion and systemic risk in a heterogeneous financial network, subject to an exogenous macroeconomic shock. We show that, under some regularity assumptions, the default cascade model could be transferred to a death process problem represented by balls-and-bins model. We also reduce the dimension of the problem by classifying banks according to di↵erent types, in an appropriate type space. These types may be calibrated to real-world data by using machine learning techniques. We then state various limit theorems regarding the final size of default cascade over di↵erent types. In particular, under suitable assumptions on the degree and threshold distributions, we show that the final size of default cascade has asymptotically Gaussian fluctuations. We next state limit theorems for di↵erent system-wide wealth aggregation functions and show how the systemic risk measure, in a given stress test scenario, could be related to the structure and heterogeneity of financial networks. We finally show how these results could be used by a social planner to optimally target interventions during a financial crisis, with a budget constraint and under partial information of the financial network

DAG-FL: Direct Acyclic Graph-based Blockchain Empowers On-Device Federated Learning

Due to the distributed characteristics of Federated Learning (FL), the vulnerability of global model and coordination of devices are the main obstacle. As a promising solution of decentralization, scalability and security, leveraging blockchain in FL has attracted much attention in recent years. However, the traditional consensus mechanisms designed for blockchain like Proof of Work (PoW) would cause extreme resource consumption, which reduces the efficiency of FL greatly, especially when the participating devices are wireless and resource-limited. In order to address device asynchrony and anomaly detection in FL while avoiding the extra resource consumption caused by blockchain, this paper introduces a framework for empowering FL using Direct Acyclic Graph (DAG)-based blockchain systematically (DAG-FL). Accordingly, DAG-FL is first introduced from a three-layer architecture in details, and then two algorithms DAG-FL Controlling and DAG-FL Updating are designed running on different nodes to elaborate the operation of DAG-FL consensus mechanism. The extensive simulations show that DAG-FL can achieve the better performance in terms of training efficiency and model accuracy compared with the typical existing on-device federated learning systems as the benchmarks

Expression of mTOR conduction pathway in human osteosarcoma MG-63 cells and their stem cells, and the inhibitory effect of different doses of rapamycin

Purpose: To investigate the expressions of rapamycin target protein (mTOR) conduction pathway in human osteosarcoma MG-63 cells and their stem cells, and to examine the inhibitory effect of different doses of rapamycin.Methods: mTOR mRNA in osteosarcoma stem-like cells and human osteosarcoma MG-63 cells were determined by quantitative reverse transcription polymerase chain reaction (qRT-PCR). The cells were treated with different doses of rapamycin and divided into low dose group (0.5 mg), medium dose group (1.0 mg), high dose group (2.0 mg) and blank (control) group. Apoptosis and cell cycle of MG-63 cells were determined by flow cytometry, while proliferation of MG-63 cells up was assessed by CCK-8 kit.Results: mTOR in human osteosarcoma MG-63 cells was significantly lower than that in osteosarcoma stem-like cells. Compared with the control group, mRNA expression levels of mTOR in MG-63 cells and osteosarcoma stem-like cells were significantly decreased after treatment with different concentrations of rapamycin (p < 0.05). MG-63 cells treated with various doses of rapamycin exhibited a significant decrease in their proliferation, compared with control group, while only the high rapamycin concentration group exhibited a significant decrease in osteosarcoma stem-like cell proliferation (p < 0.05). Treatment with rapamycin in MG-63 cells and osteosarcoma stem-like cells resulted in a significant increase in apoptosis, prolonged G0/G1 phase and shortened S phase (p < 0.05).Conclusion: Rapamycin inhibits the expression of mTOR mRNA in osteosarcoma stem-like and MG-63 cells. It also inhibits the proliferation and cell cycle formation of osteosarcoma stem-like cells and MG-63 cells via mTOR signal pathway. These findings may provide a new target for the treatment of osteosarcoma

Regularization of point vortices for the Euler equation in dimension two

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem [ -\ep^2 \Delta u=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep})_+^p, \quad & x\in\Omega, u=0, \quad & x\in\partial\Omega, ] where $p>1$, $\Omega\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $\Omega$ is simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1,...,x_m)$ with the same strength $\kappa>0$, there is a stationary classical solution approximating stationary $m$ points vortex solution of incompressible Euler equations with vorticity $m\kappa$. Existence and asymptotic behavior of single point non-vanishing vortex solutions were studied by D. Smets and J. Van Schaftingen (2010).Comment: 32page

Risque systémique, réseaux financiers complexes et systèmes interactifs de type graphon champ moyen

This thesis is divided in two parts. The first part considers the issues of stability and systemic risk in large complex financial networks, including the study of default contagion, fire sales and risk processes on networks. We first prove limit theorems (law of large numbers and central limit theorem types) for the contagion dynamics. We show how to quantify the systemic risk for a financial network under partial information facing an outside shock. Then we present a general tractable framework for understanding the joint impact of fire sales and default cascades on systemic risk in complex financial networks. We finally study risk processes on large financial systems, when agents, located on a large network, receive losses from their neighbors. The second part of the thesis focuses on graphon mean field interacting systems with jumps and graphon mean field games. Here, the financial network is seen as a large interacting system, with a graphon mean field structure depending on the underlying graph structure of the network. We first conduct a comprehensive study of graphon mean field backward stochastic differential equations (BSDEs) with jumps and associated global dynamic risk measures. We then study continuous stochastic games with heterogeneous mean field interactions on large networks and investigate their graphon limits. We provide approximate Nash equilibria for finite games with heterogeneous interactions, using their graphon equilibria as benchmarks.Cette thèse est divisée en deux parties. La première partie étudie la stabilité et le risque systémique de réseaux financiers complexes, soumis à des processus de contagion de défauts, et de ventes forcées. Nous prouvons des théorèmes limites de type loi des grands nombres et limite centrale sur la dynamique de contagion. Nous montrons comment quantifier le risque systémique d'un réseau financier en présence d'une perturbation externe et sous information partielle. Nous étudions ensuite les processus de risque multidimensionnels de Cramér-Lundberg où les agents, situés sur un grand réseau, subissent des pertes de la part de leurs voisins. Nous présentons enfin un cadre général abordable pour comprendre l'impact conjoint de liquidations et de cascades de défauts sur le risque systémique dans les réseaux financiers complexes. La deuxième partie de la thèse est consacrée à l'étude et le contrôle de systèmes interactifs de type graphon champ moyen. Le réseau financier est ici considéré comme un grand système interactif, ce qui établit un lien avec la théorie des jeux à champ moyen. La structure en champ moyen repose sur la structure de graphe sous-jacente du réseau, appelée champ moyen graphon. Nous commençons par une étude systématique des équations différentielles stochastiques rétrogrades (EDSR) avec sauts de type graphon champ moyen et ses mesures de risque dynamiques associées. Nous étudions ensuite des jeux stochastiques continus avec interactions non homogènes de type champ moyen sur de vastes réseaux et explorons leurs limites graphon champ moyen. Nous proposons des équilibres de Nash approximés pour les jeux finis sur les réseaux, utilisant les équilibres en champ moyen graphon associés comme référence