A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics
Publikationen: Beitrag in Fachzeitschrift › Artikel › Forschung › (peer-reviewed)
Standard
in: Risks, Jahrgang 8, Nr. 4, 136, 09.12.2020, S. 1-18.
Publikationen: Beitrag in Fachzeitschrift › Artikel › Forschung › (peer-reviewed)
Harvard
APA
Vancouver
Author
Bibtex - Download
}
RIS (suitable for import to EndNote) - Download
TY - JOUR
T1 - A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics
AU - Kremsner, Stefan
AU - Steinicke, Alexander
AU - Szölgyenyi, Michaela
PY - 2020/12/9
Y1 - 2020/12/9
N2 - 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.
AB - 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.
KW - backward stochastic differential equations
KW - semilinear elliptic partial differential equations
KW - stochastic optimal control
KW - unbounded random terminal time
KW - machine learning
KW - deep neural networks
UR - http://www.scopus.com/inward/record.url?scp=85097538081&partnerID=8YFLogxK
U2 - 10.3390/risks8040136
DO - 10.3390/risks8040136
M3 - Article
VL - 8
SP - 1
EP - 18
JO - Risks
JF - Risks
SN - 2227-9091
IS - 4
M1 - 136
ER -