Archives

26
Fév

Some machine learning schemes for high-dimensional nonlinear PDEs - C. HURE, H. PHAM, X. WARIN

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 [...]

18
Jan

Deep neural networks algorithms for stochastic control problems on nite horizon, Part 2: numerical applications - A. Bachouch, C. Huré, N. Langrené, H. Pham

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 di erent 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 Di erential equations Read more [...]

18
Jan

Deep neural networks algorithms for stochastic control problems on finite horizon, part I : convergence analysis - C. Hure, H. Pham, A. Bachouch and N. Langrené

This paper develops algorithms for high-dimensional stochastic control problems based on deep learning and dynamic programming (DP). Di erently 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 [...]

18
Jan

Optimal electricity demand response contracting with responsiveness incentives - R. Aïd, D. Possamaï, and N. Touzi

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 [...]

17
Jan

Day-ahead probabilistic forecast of solar irradiance: a Stochastic Differential Equation approach - J. Badosa, E. Gobet, M. Grangereau and D. Kim

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 [...]

11
Déc

Regression Monte Carlo for microgrid management - C. Alasseur, A. Balata, S. Ben Aziza, A. Maheshwari, P. Tankov, and X. Warin

We study an islanded microgrid system designed to supply a small village with the power produced by photovoltaic panels, wind turbines and a diesel generator. A battery storage system device is used to shift power from times of high renewable production to times of high demand. We build on the mathematical model introduced in [14] and optimize the diesel con-sumption under a “no-blackout” constraint. We introduce a methodology to solve microgrid man-agement problem using different variants of Read more [...]

11
Déc

Variance optimal hedging with application to Electricity markets - Xavier Warin

In this article, we use the mean variance hedging criterion to value contracts in incomplete markets. Although the problem is well studied in a continuous and even discrete framework, very few works incorporating illiquidity constraints have been achieved and no algorithm is available in the literature to solve this problem. We first show that the valuation problem incorporating illiquidity constraints with a mean variance criterion admits a unique solution. Then we develop two Least Squares Read more [...]

23
Juil

Estimating fast mean-reverting jumps in electricity Market models - Thomas Deschatre, Olivier Féron, and Marc Hoffmann

Based on empirical evidence of fast mean-reverting spikes, we model electricity price processes as the sum of a continuous Itö semimartingale and a a mean-reverting compound Poisson process. In a first part, we investigate the estimation of the two parameters of the Poisson process from discrete observations and establish asymptotic efficiency in various asymptotic settings. In a second part, we discuss the use of our inference results for correcting the value of forward contracts on electricity Read more [...]

18
Juin

Fast and stable multivariate kernel density estimation by fast sum updating - N. Langrené, X. Warin

Kernel density estimation and kernel regression are powerful but computationally expensive techniques: a direct evaluation of kernel density estimates at M evaluation points given N input sample points requires a quadratic O(MN) operations, which is prohibitive for large scale problems. For this reason, approximate methods such as binning with Fast Fourier Transform or the Fast Gauss Transform have been proposed to speed up kernel density estimation. Among these fast methods, the Fast Sum Updating Read more [...]

18
Juin

Monte Carlo for high-dimensional degenerated Semi Linear and Full Non Linear PDEs - Xavier Warin

We extend a recently developed method to solve semi-linear PDEs to the case of a degenerated diffusion. Being a pure Monte Carlo method it does not su er from the so called curse of dimensionality and it can be used to solve problems that were out of reach so far. We give some results of convergence and show numerically that it is effective. Besides we numerically show that the new scheme developed can be used to solve some full non linear PDEs. At last we provide an effective algorithm to implement Read more [...]

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