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My research interests include optimal control, optimal transport, calculus of variations, and partial differential equations. I enjoy learning how these branches of mathematics merge when studying applied mathematics.
Currently, I study the theory and applications of mean-field games. This novel area borrows ideas from mean-field theory to approximate the Nash equilibrium of finite-population symmetric games. It has found applications in several fields and has motivated deep mathematical investigation.
In mean-field games, one is interested in determining two functions: a value function and a density. In the following five (5) papers, I have studied the price-formation mean-field game model (in three functions: a value function, a density, and a price) using different approaches.
We use machine learning to solve the deterministic price-formation mean-field games model numerically. We develop a-posteriori estimates to guarantee the convergence of the training. Our approach relies on the calculus of variations theory and recurrent neural networks.
We use machine learning to solve the deterministic price-formation mean-field games model numerically. We develop a-posteriori estimates to guarantee the convergence of the training. Our approach relies on the calculus of variations theory and recurrent neural networks.
Applied Mathematics & Optimization
Applied Mathematics & Optimization
We consider both the finite players' game with common noise and its mean-field limit. The price arises as a Lagrange multiplier of a market-clearing condition. Our approach relies on the calculus of variations theory.
We consider both the finite players' game with common noise and its mean-field limit. The price arises as a Lagrange multiplier of a market-clearing condition. Our approach relies on the calculus of variations theory.
SIAM Journal on Financial Mathematics
SIAM Journal on Financial Mathematics
We study the deterministic price formation problem using the Aubry-Mather theory. We obtain a variational problem on a space of measures. Our approach relies on the optimal transport (duality) theory.
We study the deterministic price formation problem using the Aubry-Mather theory. We obtain a variational problem on a space of measures. Our approach relies on the optimal transport (duality) theory.
Minimax Theory and its Applications
Minimax Theory and its Applications
We introduce the so-called potential approach for the deterministic price-formation mean-field game model. Using Poincaré Lemma, we obtain a variational problem for a single function (instead of the original three). Our approach relies on the calculus of variations theory.
We introduce the so-called potential approach for the deterministic price-formation mean-field game model. Using Poincaré Lemma, we obtain a variational problem for a single function (instead of the original three). Our approach relies on the calculus of variations theory.
Communications in Mathematical Sciences
Communications in Mathematical Sciences
We propose the stochastic version (common noise) of the price-formation mean-field game model introduced by Gomes & Saúde in the deterministic setting. Our approach relies on the optimal control theory.
We propose the stochastic version (common noise) of the price-formation mean-field game model introduced by Gomes & Saúde in the deterministic setting. Our approach relies on the optimal control theory.
AIMS Press
AIMS Press
(Conference paper)
We merge the potential approach and machine learning methods to solve both the deterministic and the stochastic mean-field game price formation model.
We merge the potential approach and machine learning methods to solve both the deterministic and the stochastic mean-field game price formation model.
Our approach relies on the (stochastic) calculus of variations theory and recurrent neural networks.
Our approach relies on the (stochastic) calculus of variations theory and recurrent neural networks.
61st IEEE Conference on Decision and Control
61st IEEE Conference on Decision and Control
We extend the machine learning-based approach introduced in our previous work to include the price formation problem with common noise, including using a-posteriori estimates to guarantee convergence.
We extend the machine learning-based approach introduced in our previous work to include the price formation problem with common noise, including using a-posteriori estimates to guarantee convergence.
62nd IEEE Conference on Decision and Control
62nd IEEE Conference on Decision and Control