scholarly journals Full Stability for a Class of Control Problems of Semilinear Elliptic Partial Differential Equations

2019 ◽  
Vol 57 (4) ◽  
pp. 3021-3045 ◽  
Author(s):  
Nguyen Thanh Qui ◽  
Daniel Wachsmuth
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Partial Differential Equations ◽  
Control Problems ◽  
Partial Differential ◽  
Semilinear Elliptic

scholarly journals A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics

Risks ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 136
Author(s):  
Stefan Kremsner ◽  
Alexander Steinicke ◽  
Michaela Szölgyenyi
Keyword(s):  
Neural Network ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Deep Neural Network ◽  
Risk Measure ◽  
Elliptic Partial Differential Equations ◽  
High Dimensional ◽  
Control Problems ◽  
Partial Differential ◽  
Neural Network Algorithm

In insurance mathematics, optimal control problems over an infinite time horizon arise when computing risk measures. An example of such a risk measure is the expected discounted future dividend payments. In models which take multiple economic factors into account, this problem is high-dimensional. The solutions to such control problems correspond to solutions of deterministic semilinear (degenerate) elliptic partial differential equations. In the present paper we propose a novel deep neural network algorithm for solving such partial differential equations in high dimensions in order to be able to compute the proposed risk measure in a complex high-dimensional economic environment. The method is based on the correspondence of elliptic partial differential equations to backward stochastic differential equations with unbounded random terminal time. In particular, backward stochastic differential equations—which can be identified with solutions of elliptic partial differential equations—are approximated by means of deep neural networks.

scholarly journals An Application of the Browder-Minty Theorem in a Problem of Partial Differential Equations

2019 ◽  
Vol 6 (1) ◽  
pp. 1-12
Author(s):  
Westher Manricky Bernardes Fortunato ◽  
Dassael Fabricio dos Reis Santos
Keyword(s):  
Weak Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Main Tool ◽  
Elliptic Partial Differential Equations ◽  
Hölder Inequality ◽  
Partial Differential ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic ◽  
Continuous Operators

In this work, we will show existence of weak solution for a semilinear elliptic problem using as main tool the Browder-Minty Theorem. First, we will make a brief introduction about basic theory of the Sobolev Spaces to support our study and provide sufficient tools for the development of our work. Then we will take a quick approach on the Browder-Minty Theorem and use this result to show the existence of at least one weak solution to an elliptic Partial Differential Equations (PDE) problem whose nonlinearity, denoted by f, is a known function. For this, in addition to the already mentioned results, we will also use as study tools: Embedding Sobolev Theorems, Linear Continuous Operators Theory, Poincaré Inequality and Hölder Inequality.

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