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We develop multistep machine learning schemes for solving nonlinear partial differential equations (PDEs) in high dimension. The method is based on probabilistic representation of PDEs by backward stochastic differential equations (BSDEs) and its iterated time discretization. In the case of semilinear PDEs, our algorithm estimates simultaneously by backward induction the solution and its gradient by neural networks through sequential minimizations of suitable quadratic loss functions that are performed Read more [...]

This paper revisits the problem of computing empirical cumulative distribution functions (ECDF) efficiently on large, multivariate datasets. Computing an ECDF at one evaluation point requires O(N) operations on a dataset composed of N data points. Therefore, a direct evaluation of ECDFs at N evaluation points requires a quadratic O(N^2) operations, which is prohibitive for large-scale problems. Two fast and exact methods are proposed and compared. The first one is based on fast summation in lexicographical Read more [...]

The large reductions in electricity demand caused by the COVID-19 crisis have disrupted electricity systems worldwide. This article draws insights from New York into the consequences of the pandemic for electricity markets. It disentangles the effects of the demand reductions, increased forecast errors, and fuel price drops on the day-ahead and real-time markets. From March 16 to May 31, New York has experienced a 6.5% demand reduction, prices have dropped, and producers have lost $87 million (-18%). Read more [...]

In this paper we study the calibration of specific multi-factorial Heath-Jarrow-Morton models to electricity market prices, with a focus on the estimation of the optimal number of factors. We describe a common statistical procedure based on likelihood maximisation and Akaike / Bayesian information criteria, in the case of calibration on futures prices, as well as on both spot and futures prices. We perform a detailed analysis on 6 European markets: Belgium, France, Germany, Italy, Switzerland and Read more [...]

We propose to study electricity capacity remuneration mechanism design through a Principal-Agent approach. The Principal represents the aggregation of electricity consumers (or a representative entity), subject to the physical risk of shortage, and the Agent represents the electricity capacity owners, who invest in capacity and produce electricity to satisfy consumers’ demand, and are subject to financial risks. Following the methodology of Cvitanic et al. (2017), we propose an optimal contract, Read more [...]

We propose several algorithms to solve McKean-Vlasov Forward Backward Stochastic Differential Equations. Our schemes rely on the approximating power of neural networks to estimate the solution or its gradient through minimization problems. As a consequence, we obtain methods able to tackle both mean field games and mean field control problems in high dimension. We analyze the numerical behavior of our algorithms on several examples including non linear quadratic models.

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 [...]