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Computing the Gerber-Shiu function with interest and a constant dividend barrier by physics-informed neural networks (2401.04378v1)

Published 9 Jan 2024 in math.NA, cs.NA, math.PR, and q-fin.RM

Abstract: In this paper, we propose a new efficient method for calculating the Gerber-Shiu discounted penalty function. Generally, the Gerber-Shiu function usually satisfies a class of integro-differential equation. We introduce the physics-informed neural networks (PINN) which embed a differential equation into the loss of the neural network using automatic differentiation. In addition, PINN is more free to set boundary conditions and does not rely on the determination of the initial value. This gives us an idea to calculate more general Gerber-Shiu functions. Numerical examples are provided to illustrate the very good performance of our approximation.

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References (47)
  1. \APACrefYearMonthDay2005. \BBOQ\APACrefatitleOn the distribution of dividend payments in a Sparre Andersen model with generalized Erlang (n) interclaim times On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang (n) interclaim times.\BBCQ \APACjournalVolNumPagesInsurance: Mathematics and Economics372324–334. \PrintBackRefs\CurrentBib
  2. \APACrefYearMonthDay2010. \BBOQ\APACrefatitleA Direct Approach to the Discounted Penalty Function A direct approach to the discounted penalty function.\BBCQ \APACjournalVolNumPagesNorth American Actuarial Journal144420-434. \PrintBackRefs\CurrentBib
  3. \APACrefYearMonthDay1995. \BBOQ\APACrefatitleA limited memory algorithm for bound constrained optimization A limited memory algorithm for bound constrained optimization.\BBCQ \APACjournalVolNumPagesSIAM Journal on scientific computing1651190–1208. \PrintBackRefs\CurrentBib
  4. \APACrefYearMonthDay2002. \BBOQ\APACrefatitleOn the expected discounted penalty function at ruin of a surplus process with interest On the expected discounted penalty function at ruin of a surplus process with interest.\BBCQ \APACjournalVolNumPagesInsurance: Mathematics and Economics303389–404. \PrintBackRefs\CurrentBib
  5. \APACrefYearMonthDay2009. \BBOQ\APACrefatitleAnalysis of the compound Poisson surplus model with liquid reserves, interest and dividends Analysis of the compound poisson surplus model with liquid reserves, interest and dividends.\BBCQ \APACjournalVolNumPagesASTIN Bulletin: The Journal of the IAA391225–247. \PrintBackRefs\CurrentBib
  6. \APACrefYearMonthDay2015. \BBOQ\APACrefatitleFourier-Cosine method for Gerber–Shiu functions Fourier-Cosine method for Gerber–Shiu functions.\BBCQ \APACjournalVolNumPagesInsurance: Mathematics and Economics61170-180. \PrintBackRefs\CurrentBib
  7. \APACrefYearMonthDay2020. \BBOQ\APACrefatitlePhysics-informed neural networks for inverse problems in nano-optics and metamaterials Physics-informed neural networks for inverse problems in nano-optics and metamaterials.\BBCQ \APACjournalVolNumPagesOptics express28811618–11633. \PrintBackRefs\CurrentBib
  8. \APACrefYearMonthDay2010. \BBOQ\APACrefatitleStructural properties of Gerber–-Shiu functions in dependent Sparre Andersen models Structural properties of Gerber–-Shiu functions in dependent Sparre Andersen models.\BBCQ \APACjournalVolNumPagesInsurance: Mathematics and Economics461117-126. \PrintBackRefs\CurrentBib
  9. \APACrefYearMonthDay2019. \BBOQ\APACrefatitlePeriodic threshold-type dividend strategy in the compound Poisson risk model Periodic threshold-type dividend strategy in the compound poisson risk model.\BBCQ \APACjournalVolNumPagesScandinavian Actuarial Journal201911–31. \PrintBackRefs\CurrentBib
  10. \APACrefYearMonthDay2011. \BBOQ\APACrefatitleOn the threshold dividend strategy for a generalized jump–diffusion risk model On the threshold dividend strategy for a generalized jump–diffusion risk model.\BBCQ \APACjournalVolNumPagesInsurance: Mathematics and Economics483326–337. \PrintBackRefs\CurrentBib
  11. \APACrefYearMonthDay2023. \BBOQ\APACrefatitleSystems biology: Identifiability analysis and parameter identification via systems-biology-informed neural networks Systems biology: Identifiability analysis and parameter identification via systems-biology-informed neural networks.\BBCQ \BIn \APACrefbtitleComputational Modeling of Signaling Networks Computational modeling of signaling networks (\BPGS 87–105). \APACaddressPublisherSpringer. \PrintBackRefs\CurrentBib
  12. \APACinsertmetastarde1957impostazione{APACrefauthors}De Finetti, B.  \APACrefYearMonthDay1957. \BBOQ\APACrefatitleSu un’impostazione alternativa della teoria collettiva del rischio Su un’impostazione alternativa della teoria collettiva del rischio.\BBCQ \BIn \APACrefbtitleTransactions of the XVth international congress of Actuaries Transactions of the xvth international congress of actuaries (\BVOL 2, \BPGS 433–443). \PrintBackRefs\CurrentBib
  13. \APACrefYearMonthDay2018. \BBOQ\APACrefatitleAn individual claims history simulation machine An individual claims history simulation machine.\BBCQ \APACjournalVolNumPagesRisks6229. \PrintBackRefs\CurrentBib
  14. \APACrefYearMonthDay2008. \BBOQ\APACrefatitleThe perturbed Sparre Andersen model with a threshold dividend strategy The perturbed Sparre Andersen model with a threshold dividend strategy.\BBCQ \APACjournalVolNumPagesJournal of Computational and Applied Mathematics2201-2394–408. \PrintBackRefs\CurrentBib
  15. \APACrefYearMonthDay2010. \BBOQ\APACrefatitleThe perturbed compound Poisson risk model with constant interest and a threshold dividend strategy The perturbed compound poisson risk model with constant interest and a threshold dividend strategy.\BBCQ \APACjournalVolNumPagesJournal of computational and applied mathematics23392181–2188. \PrintBackRefs\CurrentBib
  16. \APACrefYearMonthDay2006. \BBOQ\APACrefatitleOn The Expected Discounted Penalty function for Lévy Risk Processes On the expected discounted penalty function for Lévy risk processes.\BBCQ \APACjournalVolNumPagesNorth American Actuarial Journal104196-216. \PrintBackRefs\CurrentBib
  17. \APACrefYearMonthDay1998. \BBOQ\APACrefatitleOn the Time Value of Ruin On the time value of ruin.\BBCQ \APACjournalVolNumPagesNorth American Actuarial Journal2148-72. \PrintBackRefs\CurrentBib
  18. \APACinsertmetastarhainaut2018neural{APACrefauthors}Hainaut, D.  \APACrefYearMonthDay2018. \BBOQ\APACrefatitleA neural-network analyzer for mortality forecast A neural-network analyzer for mortality forecast.\BBCQ \APACjournalVolNumPagesASTIN Bulletin: The Journal of the IAA482481–508. \PrintBackRefs\CurrentBib
  19. \APACrefYearMonthDay2021. \BBOQ\APACrefatitlePhysics-informed machine learning Physics-informed machine learning.\BBCQ \APACjournalVolNumPagesNature Reviews Physics36422–440. \PrintBackRefs\CurrentBib
  20. \APACrefYearMonthDay2015. \BBOQ\APACrefatitleAdam: A method for stochastic optimization. conference paper Adam: A method for stochastic optimization. conference paper.\BBCQ \BIn \APACrefbtitle3rd International Conference for Learning Representations 3rd international conference for learning representations (\BVOL 2015). \PrintBackRefs\CurrentBib
  21. \APACinsertmetastarkuo2019deeptriangle{APACrefauthors}Kuo, K.  \APACrefYearMonthDay2019. \BBOQ\APACrefatitleDeeptriangle: A deep learning approach to loss reserving Deeptriangle: A deep learning approach to loss reserving.\BBCQ \APACjournalVolNumPagesRisks7397. \PrintBackRefs\CurrentBib
  22. \APACrefYearMonthDay2014. \BBOQ\APACrefatitleComputing the finite–time expected discounted penalty function for a family of Lévy risk processes Computing the finite–time expected discounted penalty function for a family of Lévy risk processes.\BBCQ \APACjournalVolNumPagesScandinavian Actuarial Journal201411-31. \PrintBackRefs\CurrentBib
  23. \APACrefYearMonthDay2008. \BBOQ\APACrefatitleOn the Gerber–-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution On the Gerber–-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution.\BBCQ \APACjournalVolNumPagesInsurance: Mathematics and Economics422600-608. \PrintBackRefs\CurrentBib
  24. \APACinsertmetastarli2006distribution{APACrefauthors}Li, S.  \APACrefYearMonthDay2006. \BBOQ\APACrefatitleThe distribution of the dividend payments in the compound Poisson risk model perturbed by diffusion The distribution of the dividend payments in the compound Poisson risk model perturbed by diffusion.\BBCQ \APACjournalVolNumPagesScandinavian Actuarial Journal2006273–85. \PrintBackRefs\CurrentBib
  25. \APACrefYearMonthDay2008. \BBOQ\APACrefatitleThe Decompositions of the Discounted Penalty Functions and Dividends-Penalty Identity in a Markov-Modulated Risk Model The decompositions of the discounted penalty functions and dividends-penalty identity in a Markov-modulated risk model.\BBCQ \APACjournalVolNumPagesASTIN Bulletin: The Journal of the IAA38153–71. \PrintBackRefs\CurrentBib
  26. \APACrefYearMonthDay2006. \BBOQ\APACrefatitleThe compound Poisson risk model with a threshold dividend strategy The compound poisson risk model with a threshold dividend strategy.\BBCQ \APACjournalVolNumPagesInsurance: mathematics and Economics38157–80. \PrintBackRefs\CurrentBib
  27. \APACrefYearMonthDay2003. \BBOQ\APACrefatitleThe classical risk model with a constant dividend barrier: analysis of the Gerber–Shiu discounted penalty function The classical risk model with a constant dividend barrier: analysis of the Gerber–Shiu discounted penalty function.\BBCQ \APACjournalVolNumPagesInsurance: Mathematics and Economics333551–566. \PrintBackRefs\CurrentBib
  28. \APACrefYearMonthDay2021. \BBOQ\APACrefatitleDeepXDE: A deep learning library for solving differential equations Deepxde: A deep learning library for solving differential equations.\BBCQ \APACjournalVolNumPagesSIAM review631208–228. \PrintBackRefs\CurrentBib
  29. \APACrefYearMonthDay2008. \BBOQ\APACrefatitleNonparametric estimation of ruin probabilities given a random sample of claims Nonparametric estimation of ruin probabilities given a random sample of claims.\BBCQ \APACjournalVolNumPagesMathematical Methods of Statistics1735–43. \PrintBackRefs\CurrentBib
  30. \APACrefYearMonthDay2020. \BBOQ\APACrefatitleCase study: French motor third-party liability claims Case study: French motor third-party liability claims.\BBCQ {APACrefDOI} \doihttps://papers.ssrn.com/sol3/papers.cfm?abstract_id=3164764 \PrintBackRefs\CurrentBib
  31. \APACrefYearMonthDay2019\BCnt2. \BBOQ\APACrefatitlePhysics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.\BBCQ \APACjournalVolNumPagesJournal of Computational Physics378686-707. \PrintBackRefs\CurrentBib
  32. \APACrefYearMonthDay2019\BCnt1. \BBOQ\APACrefatitlePhysics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.\BBCQ \APACjournalVolNumPagesJournal of Computational physics378686–707. \PrintBackRefs\CurrentBib
  33. \APACrefYearMonthDay2020. \BBOQ\APACrefatitleHidden fluid mechanics: Learning velocity and pressure fields from flow visualizations Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations.\BBCQ \APACjournalVolNumPagesScience36764811026–1030. \PrintBackRefs\CurrentBib
  34. \APACinsertmetastarshimizu2011estimation{APACrefauthors}Shimizu, Y.  \APACrefYearMonthDay2011. \BBOQ\APACrefatitleEstimation of the expected discounted penalty function for Lévy insurance risks Estimation of the expected discounted penalty function for Lévy insurance risks.\BBCQ \APACjournalVolNumPagesMathematical Methods of Statistics202125–149. \PrintBackRefs\CurrentBib
  35. \APACinsertmetastarshimizu2012non{APACrefauthors}Shimizu, Y.  \APACrefYearMonthDay2012. \BBOQ\APACrefatitleNon-parametric estimation of the Gerber–Shiu function for the Wiener–Poisson risk model Non-parametric estimation of the Gerber–Shiu function for the Wiener–Poisson risk model.\BBCQ \APACjournalVolNumPagesScandinavian Actuarial Journal2012156–69. \PrintBackRefs\CurrentBib
  36. \APACrefYearMonthDay2019. \BBOQ\APACrefatitleComputing the Gerber–Shiu function by frame duality projection Computing the Gerber–Shiu function by frame duality projection.\BBCQ \APACjournalVolNumPagesScandinavian Actuarial Journal20194291–307. \PrintBackRefs\CurrentBib
  37. \APACrefYearMonthDay2023. \BBOQ\APACrefatitleEffective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics.\BBCQ \APACjournalVolNumPagesApplied mathematics and mechanics4471039–1068. \PrintBackRefs\CurrentBib
  38. \APACinsertmetastarwuthrich2018neural{APACrefauthors}Wüthrich, M\BPBIV.  \APACrefYearMonthDay2018. \BBOQ\APACrefatitleNeural networks applied to chain–ladder reserving Neural networks applied to chain–ladder reserving.\BBCQ \APACjournalVolNumPagesEuropean Actuarial Journal8407–436. \PrintBackRefs\CurrentBib
  39. \APACrefYearMonthDay2023. \BBOQ\APACrefatitleData analytics for non-life insurance pricing Data analytics for non-life insurance pricing.\BBCQ \APACjournalVolNumPagesSwiss Finance Institute Research Paper16-68. \PrintBackRefs\CurrentBib
  40. \APACrefYearMonthDay2022. \BBOQ\APACrefatitleGradient-enhanced physics-informed neural networks for forward and inverse PDE problems Gradient-enhanced physics-informed neural networks for forward and inverse pde problems.\BBCQ \APACjournalVolNumPagesComputer Methods in Applied Mechanics and Engineering393114823. \PrintBackRefs\CurrentBib
  41. \APACrefYearMonthDay2023. \BBOQ\APACrefatitleComputing Gerber-Shiu function in the classical risk model with interest using collocation method Computing gerber-shiu function in the classical risk model with interest using collocation method.\BBCQ \APACjournalVolNumPagesarXiv preprint arXiv:2312.16004. \PrintBackRefs\CurrentBib
  42. \APACrefYearMonthDay2007. \BBOQ\APACrefatitleThe Gerber–Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier The Gerber–Shiu expected discounted penalty function for risk processes with interest and a constant dividend barrier.\BBCQ \APACjournalVolNumPagesInsurance: Mathematics and Economics401104–112. \PrintBackRefs\CurrentBib
  43. \APACinsertmetastarzhang_2017{APACrefauthors}Zhang, Z.  \APACrefYearMonthDay2017\BCnt1. \BBOQ\APACrefatitleAPPROXIMATING THE DENSITY OF THE TIME TO RUIN VIA FOURIER-COSINE SERIES EXPANSION Approximating the density of the time to ruin via Fourier-Cosine series expansion.\BBCQ \APACjournalVolNumPagesASTIN Bulletin: The Journal of the IAA471169–198. \PrintBackRefs\CurrentBib
  44. \APACinsertmetastarzhang2017estimating{APACrefauthors}Zhang, Z.  \APACrefYearMonthDay2017\BCnt2. \BBOQ\APACrefatitleEstimating the Gerber–Shiu function by Fourier–Sinc series expansion Estimating the Gerber–Shiu function by Fourier–Sinc series expansion.\BBCQ \APACjournalVolNumPagesScandinavian Actuarial Journal201710898–919. \PrintBackRefs\CurrentBib
  45. \APACrefYearMonthDay2017. \BBOQ\APACrefatitleThe compound Poisson risk model under a mixed dividend strategy The compound poisson risk model under a mixed dividend strategy.\BBCQ \APACjournalVolNumPagesApplied Mathematics and Computation3151–12. \PrintBackRefs\CurrentBib
  46. \APACrefYearMonthDay2018. \BBOQ\APACrefatitleA new efficient method for estimating the Gerber-Shiu function in the classical risk model A new efficient method for estimating the Gerber-Shiu function in the classical risk model.\BBCQ \APACjournalVolNumPagesScandinavian Actuarial Journal20185426–449. \PrintBackRefs\CurrentBib
  47. \APACrefYearMonthDay2019. \BBOQ\APACrefatitleEstimating the Gerber–Shiu function in a Lévy risk model by Laguerre series expansion Estimating the Gerber–Shiu function in a Lévy risk model by laguerre series expansion.\BBCQ \APACjournalVolNumPagesJournal of Computational and Applied Mathematics346133–149. \PrintBackRefs\CurrentBib

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