**Abstract:**We propose a mean field game model to study the question of how centralization of reward and computational power occur in the Bitcoin-like cryptocurrencies. Miners compete against each other for mining rewards by increasing their computational power. This leads to a novel mean field game of jump intensity control, which we solve explicitly for miners maximizing exponential utility, and handle numerically in the case of miners with power utilities. We show that the heterogeneity of their initial wealth distribution leads to greater imbalance of the reward distribution, and increased wealth heterogeneity over time, or a “rich get richer” effect. This concentration phenomenon is aggravated by a higher bitcoin price, and reduced by competition. Additionally, an advanced miner with cost advantages such as access to cheaper electricity, contributes a significant amount of computational power in equilibrium. Hence, cost efficiency can also result in the type of centralization seen among miners of cryptocurrencies.

**Abstract:**The dynamics of many models in Biology and Ecology such as: epidemic models, tumor-immune models, chemostat models, prey-predator models, competitive models, and among others can be mathematically described. The earliest and simplest mathematical models are given by ordinary differential equations (ODE). Long-standing and important questions in mathematical biology are that: How is the long-time behavior of the system? Does one group of populations come extinct or persistent? Under which condition, the disease will be controlled in the epidemic systems?. In longtime, which species dominates the others? and among others. This talk focuses on modeling these above biological and ecological systems and answering such problems when the random factors (leading to stochastic system), past-dependence (leading to delay system), spatial inhomogeneity (leading to reaction-diffusion models) are taken into consideration, which are described under stochastic differential equations (SDEs), stochastic functional differential equations (SFDEs) and stochastic partial differential equations (SPDEs) framework. .

**Abstract:**We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with random weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. A law of large numbers result is established as the system size increases and the underlying graphons converge. Under suitable additional assumptions, we show the exponential ergodicity for the system, establish the uniform in time law of large numbers, and introduce the uniform in time Euler approximation. The precise rate of convergence of the Euler approximation is provided. Based on joint works with Erhan Bayraktar and Suman Chakraborty.

We are pleased to announce free online events for students:

*BIG Math Network Industry Connection Series*

*Interactive “office hours” with mathematical scientists working in industry. *

The first event is coming up on February 17, 2021. Both undergraduate and graduate students are encouraged to participate.

]]>**Time**: 02/18/2021, 2:00pm -2:50pm- Zoom ID: 992 7853 8762
**Abstract**:We consider the optimal investment problem with both probability distortion/weighting (e.g. inverse S-shaped probability weighting) and general non-concave utility functions (e.g. S-shape utility). The regular method to solve this type of problems is to apply the concavification technique and then refer to Lagrange duality to generate the optimal solutions. Existing literature have shown the equivalent relationships (strong duality) between the concavified problem and the original one by either assuming the appearance of probability weighting or the non-concavity of utility functions, but not both. In contrast, we have shown that this strong duality relationship would not be maintained unconditionally if we try to combine both factors together. A step-wise relaxation is proposed to handle miscellaneous general non-concave utility and probability distortion functions. The necessary and sufficient conditions on eliminating the duality gap for the Lagrange method based on the step-wise relaxation have been provided under this circumstance. We have applied this solution method to a couple of typical examples in behavioral finance including the CPT model, VAR-RM model with distortions, Yarri’s dual model and the goal reaching model. The closed form solutions on the optimal trading strategies are obtained for a special example of the CPT model where a ”distorted” Merton line has been shown exactly.

**Time:**12/10/2020 2-2:50pm

**Zoom:**992 7853 8762

**Abstract:***In this talk, we propose a definition for the price of an American option in an incomplete market using the methodology of indifference pricing. In particular, we price options using dynamic monetary convex risk measures given by backward stochastic differential equations, with the driver of the BSDE (R,Z) assumed to be of quadratic growth in the variable z. The proposed risk-indifference price of the American option has the form of a sum of a BSDE term and a continuously reflected BSDE term. To compute the price numerically, we show that the continuously reflected BSDE term can be approximated by a BSDE which is reflected at discretely many time points.*

**Time**: 11/05/2020, 2:00pm -2:50pm

**Zoom**: 992 7853 8762

**Abstract**: We state Itô’s formula along a flow of probability measures associated with general semimartingales. This extends recent existing results for flow of measures on Itô processes. Our approach is to first prove Itô’s formula for cylindrical polynomials and then use function approximation for the general case. Some applications to McKean-Vlasov controls of jump-diffusion processes and McKean-Vlasov singular controls are developed. This is a joint work with Xin Guo, Huyên PHAM.

**Time:** 11/19/2020, 2:00 pm -2:50pm

**Zoom:** 992 7853 8762

**Abstract**: A monopolist platform (the principal) shares profits with a population of affiliates (the agents), heterogeneous in skill, by offering them a common nonlinear contract contingent on individual output. The principal cannot discriminate across individual skill, but knows its distribution and aims at maximizing profits. This paper identifies the optimal contract, its implied profits, and agents’ effort as the unique solution to an equation depending on skill distribution and agents’ costs of effort. If skill is Pareto-distributed and agents’ costs include linear and power components, closed-form solutions highlight two regimes: If linear costs are low, the principal’s share of revenues is insensitive to the distribution of skills, and decreases as agents’ costs increase. If linear costs are high, the principal’s share is insensitive to the agents’ costs, while it decreases as skill diversity increases.

**Time:** 10/8/2020, 9am-9:50am

**Abstract:** We develop a novel algorithm for the analysis of drawdown in general one-dimensional Markovian models. We compute the Laplace transform of the first passage time of the drawdown process based on continuous time Markov chain (CTMC) approximation and numerically invert the Laplace transform to obtain first passage probabilities of drawdown and the distribution of maximum drawdown. Our method has important applications in financial markets where drawdown risk is a major concern and we apply it to price and hedge several types of drawdown derivatives. We prove convergence of our algorithm for general Markovian models and provide sharp estimates of the convergence rates for a general class of jump-diffusion models. Various numerical experiments document the computational efficiency of our method and its advantages over popular alternatives. In an empirical study on bitcoin, we show that some drawdown derivatives can adequately safeguard extreme price risk and that a dynamic hedging strategy can effectively hedge the risk of selling drawdown derivatives. Extensions of our method to analyze drawdown problems in models with time dependence or stochastic volatility or regime switching as well as to quantify portfolio-level drawdown risk are also provided.

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