Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance and Finance

Frey, Rüdiger; Köck, Verena

doi:10.3390/computation10110201

Cited by 4 publications

(3 citation statements)

References 30 publications

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“…Q1.2: Regarding the pricing of non-life products, the problems focus on model simplification through feature selection, data cleaning, and the extraction of outliers, along with techniques to improve prediction capacity, such as RNN and SHAP [25,26], isotonic recalibration [27], tree-based ensemble [28], Hierarchical Risk-factors Adaptive Top-down (PHiRAT) [29], logistic regression, decision tree, random forest, XGBoost, feed-forward network [30], transaction models for in life IBNR, inconclusive [31], integration of graphic themes [32], and extreme event estimation [33]. Moreover, currently, more efficient prediction models have been developed with techniques such as extreme gradient boosting or XGBoost [34], Bayesian CART models [35], boosting [36], and deep neural networks [36][37][38], among others.…”

Section: Results and Findingsmentioning

confidence: 99%

Potential Applications of Explainable Artificial Intelligence to Actuarial Problems

Lozano-Murcia,

Romero,

Serrano-Guerrero

et al. 2024

Mathematics

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Explainable artificial intelligence (XAI) is a group of techniques and evaluations that allows users to understand artificial intelligence knowledge and increase the reliability of the results produced using artificial intelligence. XAI can assist actuaries in achieving better estimations and decisions. This study reviews the current literature to summarize XAI in common actuarial problems. We proposed a research process based on understanding the type of AI used in actuarial practice in the financial industry and insurance pricing and then researched XAI implementation. This study systematically reviews the literature on the need for implementation options and the current use of explanatory artificial intelligence (XAI) techniques for actuarial problems. The study begins with a contextual introduction outlining the use of artificial intelligence techniques and their potential limitations, followed by the definition of the search equations used in the research process, the analysis of the results, and the identification of the main potential fields for exploitation in actuarial problems, as well as pointers for potential future work in this area.

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Section: Results and Findingsmentioning

confidence: 99%

Potential Applications of Explainable Artificial Intelligence to Actuarial Problems

Lozano-Murcia,

Romero,

Serrano-Guerrero

et al. 2024

Mathematics

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“…Actually, In addition to FEM, Frey et al (2022) and Sacchetti et al (2022) have also discussed the feasibility of using NN to solve PDE, and attempted to use NN to solve PDEs with different boundary conditions and types. Their conclusions indicate that using NN to solve PDEs can achieve more accurate results than FEM, especially when it comes to analyze data with time properties or high-dimensional data, but the network structure of NN is often relatively complex [43][44].…”

Section: Solving Pdes Based On Pinnmentioning

confidence: 99%

Compressible Non-Newtonian Fluid Based Road Traffic Flow Equation Solved by Physical-Informed Rational Neural Network

Yang,

Li,

Nai

et al. 2024

IEEE Access

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The study of road traffic flow theory utilizes physics and applied mathematics to analyze relevant parameters and their relationships quanlitatively and quantitatively, in order to explore their dynamic changes. The fluid dynamics model used for traffic flow analysis is highly favored by scholars due to its solid mathematical foundation and good simulation results. However, existing models have two main shortcomings: firstly, existing research is mostly limited to non-viscoelastic fluid equation or incompressible non-Newtonian fluid equation, making it difficult to accurately describe the viscosity state and micro cluster properties of the actual traffic flow; secondly, the existing non-Newtonian fluid partial differential equations (PDEs) rely heavily on the finite element method (FEM) for solving, requiring higher computational cost, larger storage space, and more constraint conditions. Thus, in this paper, a traffic flow equation based on compressible non-Newtonian fluid has been constructed, and it has been solved by using physical-informed rational neural network (PIRNN) and noise heavy-ball acceleration gradient descent (NHAGD) to ensure learning and training speed and accuracy. Numerical results indicate that the proposed method can truly reflect the gradual change process in the viscosity of traffic flow, and has better solving performance than traditional FEM and physical-informed neural network (PINN) with activation functions; under the same conditions, the prediction error of the proposed method is also smaller than that of traditional traffic flow models.

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“…The literature also contains approaches that are more closely related to the machine learning-based algorithm presented here. Frey and Köck [82,83] propose an approximation method for non-local semilinear parabolic PDEs with Dirichlet boundary conditions based on and extending the deep splitting method in [65] and carry out numerical simulations for example PDEs in up to 4 dimensions. Castro [84] proposes a numerical scheme for approximately solving non-local nonlinear PDEs based on [64] and proves convergence results for this scheme.…”

Section: Introductionmentioning

confidence: 99%

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

Boussange,

Becker,

Jentzen

et al. 2023

Partial Differ. Equ. Appl.

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Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for certain non-local phenomena such as, e.g., interactions at a distance. Non-local nonlinear PDE models can accurately capture these phenomena, but traditional numerical approximation methods are infeasible when the considered non-local PDE is high-dimensional. In this article we propose two numerical methods based on machine learning and on Picard iterations, respectively, to approximately solve non-local nonlinear PDEs. The proposed machine learning-based method is an extended variant of a deep learning-based splitting-up type approximation method previously introduced in the literature and utilizes neural networks to provide approximate solutions on a subset of the spatial domain of the solution. The Picard iterations-based method is an extended variant of the so-called full history recursive multilevel Picard approximation scheme previously introduced in the literature and provides an approximate solution for a single point of the domain. Both methods are mesh-free and allow non-local nonlinear PDEs with Neumann boundary conditions to be solved in high dimensions. In the two methods, the numerical difficulties arising due to the dimensionality of the PDEs are avoided by (i) using the correspondence between the expected trajectory of reflected stochastic processes and the solution of PDEs (given by the Feynman–Kac formula) and by (ii) using a plain vanilla Monte Carlo integration to handle the non-local term. We evaluate the performance of the two methods on five different PDEs arising in physics and biology. In all cases, the methods yield good results in up to 10 dimensions with short run times. Our work extends recently developed methods to overcome the curse of dimensionality in solving PDEs.

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Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance and Finance

Cited by 4 publications

References 30 publications

Potential Applications of Explainable Artificial Intelligence to Actuarial Problems

Potential Applications of Explainable Artificial Intelligence to Actuarial Problems

Compressible Non-Newtonian Fluid Based Road Traffic Flow Equation Solved by Physical-Informed Rational Neural Network

Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions

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