Emmanuel Gobet

Decentralized Finance (DeFi) lending and borrowing protocols enable investors to take leveraged long and short positions on digital assets without centralized intermediaries, but expose them to a distinctive form of risk: on-chain liquidation triggered by debt and collateral value fluctuations. In this work, we provide a detailed formalization of Aave's lending, borrowing, and liquidation mechanisms, grounded in the protocol's open-source implementation. In doing so, we propose a mathematical modeling of the risk of liquidation, including some stochastic approximations with the purpose of efficient analysis, with different applications. Among them, portfolio optimization problem.

We propose a non‐asymptotic convergence analysis of a two‐step approach to learn a conditional value‐at‐risk (VaR) and a conditional expected shortfall (ES) using Rademacher bounds, in a non‐parametric setup allowing for heavy‐tails on the financial loss. Our approach for the VaR is extended to the problem of learning at once multiple VaRs corresponding to different quantile levels. This results in efficient learning schemes based on neural network quantile and least‐squares regressions. An a posteriori Monte Carlo procedure is introduced to estimate distances to the ground‐truth VaR and ES. This is illustrated by numerical experiments in a Student‐ toy model and a financial case study where the objective is to learn a dynamic initial margin.

We study price aggregation methodologies applied to crypto‐currency prices with quotations fragmented on different platforms. An intrinsic difficulty is that the price returns and volumes are heavy‐tailed, with many outliers, making averaging and aggregation challenging. While conventional methods rely on volume‐weighted average prices (called VWAPs), or volume‐weighted median prices (called VWMs), we develop a new robust weighted median (RWM) estimator that is robust to price and volume outliers. Our study is based on new probabilistic concentration inequalities for weighted means and weighted quantiles under different tail assumptions (heavy tails, sub‐gamma tails, sub‐Gaussian tails). This justifies that fluctuations of VWAP and VWM are statistically important given the heavy‐tailed properties of volumes and/or prices. We show that our RWM estimator overcomes this problem and also satisfies all the desirable properties of a price aggregator. We illustrate the behavior of RWM on synthetic data (within a parametric model close to real data): Our estimator achieves a statistical accuracy twice as good as its competitors, and also allows to recover realized volatilities in a very accurate way. Tests on real data are also performed and confirm the good behavior of the estimator on various use cases.

Our study focuses on climate models extensively employed in climate science and climate-economy research, which project temperature outcomes from carbon emission trajectories. Addressing the need for rapid evaluation in Integrated Assessment Models (IAMs) -- critical tools for carbon emission mitigation policy analysis -- we design a neural network (NN) meta-model as an efficient surrogate for mapping, in an infinite horizon setting, emission trajectories into temperature trajectories (usually modeled as coupled systems of differential equations). Our approach combines a projection on Generalized Dirichlet polynomials, whose coefficients are inputs of the NN and a suitable time change for handling infinite horizon: we prove that the quantity of interest is, under some assumptions, a smooth function of the inputs and therefore, is prone to accurate NN approximation. After a training with augmented Shared Socioeconomic Pathway scenarios, the NN achieves high-fidelity approximations of the original climate model. Additionally, we establish theoretical accuracy guarantees for both the encoding and neural network approximation. Our numerical experiments demonstrate the framework's accuracy, and computational efficiency is improved by a factor of 100 in comparison to traditional ODE solvers. As an application to Actuarial Sciences, we illustrate the use of the metamodel to quantify the distribution of future scorching days.