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We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, through a sequence of learning problems obtained from the minimization of suitable quadratic loss functions and training simulations. This methodology extends to the fully non- linear case the approach recently proposed in (Huré, Pham, Warin, 2019) for semi-linear Read more [...]

We statistically analyse a multivariate HJM diffusion model with stochastic volatility. The volatility process of the first factor is left totally unspecified while the volatility of the second factor is the product of an unknown process and an exponential function of time to maturity. This exponential term includes some real parameter measuring the rate of increase of the second factor as time goes to maturity. From historical data, we efficiently estimate the time to maturity parameter in the sense Read more [...]

The assessment of fuel poverty in mainland France is based mainly on data provided by the French national housing survey (ENL). However, the last two surveys date from 2006 and 2014. To understand the change in the number of fuel poverty households, we have developed a micro simulation tool that takes into account the three predominant factors in the notion of fuel poverty, that is, household resources, energy prices and dwelling quality. Our tool includes three multiple linear models for estimating Read more [...]

Twenty percent of French non-fuel poor households will fall into fuel poverty. The existence of energy insurance can reduce this percentage. This article focuses on non-fuel poor households that can buy insurance that provides a basic level of energy for one year after a significant loss of income. A model of household willingness to pay for energy insurance is proposed. Several simulations are performed with French data. Given the values of the utility function parameters and the energy prices, Read more [...]

We propose new machine learning schemes for solving high dimensional nonlinear partial differential equations (PDEs). Relying on the classical backward stochastic differential equation (BSDE) representation of PDEs, our algorithms estimate simultaneously the solution and its gradient by deep neural networks. These approximations are performed at each time step from the minimization of loss functions dened recursively by backward induction. The methodology is extended to variational inequalities Read more [...]

Systemic risk is a multifaceted concept that is of crucial importance for regulators. In order to ensure financial stability, they need to properly assess this risk, preventing financial shocks from affecting the real economy. In this study, we evaluate the extent to which the financialization of commodity markets contributes to systemic risk. We consider a system consisting of both commodity futures and financial markets in a sparse Vector AutoRegression (VAR) framework. It allows to distinguish Read more [...]

This paper presents several numerical applications of deep learning-based algorithms that have been analyzed in [11]. Numerical and comparative tests using TensorFlow illustrate the performance of our dierent algorithms, namely control learning by performance iteration (algorithms NNcontPI and ClassifPI), control learning by hybrid iteration (algorithms Hybrid-Now and Hybrid-LaterQ), on the 100-dimensional nonlinear PDEs examples from [6] and on quadratic Backward Stochastic Dierential equations Read more [...]

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming (DP). Dierently from the classical approximate DP approach, we rst approximate the optimal policy by means of neural networks in the spirit of deep reinforcement learning, and then the value function by Monte Carlo regression. This is achieved in the DP recursion by performance or hybrid iteration, and regress now or later/quantization methods from numerical probabilities. Read more [...]

Despite the success of demand response programs in retail electricity markets in reducing average consumption, the literature shows failure to reduce the variance of consumers’ responses. This paper aims at designing demand response contracts which allow to act on both the average consumption and its variance. The interaction between the producer and the consumer is modeled as a Principal-Agent problem, thus accounting for the moral hazard underlying demand response programs. The producer, Read more [...]

In this work, we derive a probabilistic forecast of the solar irradiance during a day at a given location, using a stochastic differential equation (SDE for short) model. We propose a procedure that transforms a deterministic forecast into a probabilistic forecast: the input parameters of the SDE model are the AROME numerical weather predictions computed at day D-1 for the day D. The model also accounts for the maximal irradiance from the clear sky model. The SDE model is mean-reverting towards Read more [...]