{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T17:20:56Z","timestamp":1778088056193,"version":"3.51.4"},"reference-count":33,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T00:00:00Z","timestamp":1668038400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this paper we study deep neural network algorithms for solving linear and semilinear parabolic partial integro-differential equations with boundary conditions in high dimension. Our method can be considered as an extension of the deep splitting method for PDEs to equations with non-local terms. To show the viability of our approach, we discuss several case studies from insurance and finance.<\/jats:p>","DOI":"10.3390\/computation10110201","type":"journal-article","created":{"date-parts":[[2022,11,10]],"date-time":"2022-11-10T19:17:34Z","timestamp":1668107854000},"page":"201","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance and Finance"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8402-4653","authenticated-orcid":false,"given":"R\u00fcdiger","family":"Frey","sequence":"first","affiliation":[{"name":"Institute for Statistics and Mathematics, Vienna University of Economics and Business, Welthandelsplatz 1, 1020 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Verena","family":"K\u00f6ck","sequence":"additional","affiliation":[{"name":"Institute for Statistics and Mathematics, Vienna University of Economics and Business, Welthandelsplatz 1, 1020 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1596","DOI":"10.1137\/S0036142903436186","article-title":"A finite difference scheme for option pricing in jump diffusion and exponential L\u00e9vy models","volume":"43","author":"Cont","year":"2005","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1023\/A:1011354913068","article-title":"Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing","volume":"4","author":"Andersen","year":"2000","journal-title":"Rev. Deriv. Res."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1051\/m2an:2004003","article-title":"Fast deterministic pricing of options on L\u00e9vy driven assets","volume":"38","author":"Matache","year":"2004","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2598","DOI":"10.1137\/090777529","article-title":"A second-order finite difference method for option pricing under jump-diffusion models","volume":"49","author":"Kwon","year":"2011","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s10092-007-0128-x","article-title":"Implicit\u2013explicit numerical schemes for jump\u2013diffusion processes","volume":"44","author":"Briani","year":"2007","journal-title":"Calcolo"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1287\/opre.1070.0496","article-title":"Multilevel Monte Carlo path simulation","volume":"56","author":"Giles","year":"2008","journal-title":"Oper. Res."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering, Springer.","DOI":"10.1007\/978-0-387-21617-1"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"43","DOI":"10.3905\/jod.2002.319189","article-title":"Using Brownian bridge for fast simulation of jump-diffusion processes and barrier options","volume":"10","author":"Metwally","year":"2002","journal-title":"J. Deriv."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"8505","DOI":"10.1073\/pnas.1718942115","article-title":"Solving high-dimensional partial differential equations using deep learning","volume":"115","author":"Han","year":"2018","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/s40304-017-0117-6","article-title":"Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations","volume":"5","author":"Han","year":"2017","journal-title":"Commun. Math. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1186\/s41546-020-00047-w","article-title":"Convergence of the deep BSDE method for coupled FBSDEs","volume":"5","author":"Han","year":"2020","journal-title":"Probab. Uncertain. Quant. Risk"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Kremsner, S., Steinicke, A., and Sz\u00f6lgyenyi, M. (2020). A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics. Risks, 8.","DOI":"10.3390\/risks8040136"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1547","DOI":"10.1090\/mcom\/3514","article-title":"Deep backward schemes for high-dimensional nonlinear PDEs","volume":"89","author":"Pham","year":"2020","journal-title":"Math. Comput."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"A3135","DOI":"10.1137\/19M1297919","article-title":"Deep splitting method for parabolic PDEs","volume":"43","author":"Beck","year":"2021","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s10915-021-01590-0","article-title":"Solving the Kolmogorov PDE by means of deep learning","volume":"88","author":"Beck","year":"2021","journal-title":"J. Sci. Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1007\/s42985-020-00062-8","article-title":"Neural networks-based backward scheme for fully nonlinear PDEs","volume":"2","author":"Pham","year":"2021","journal-title":"SN Partial. Differ. Equ. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"A28","DOI":"10.1137\/20M1355355","article-title":"Approximation error analysis of some deep backward schemes for nonlinear PDEs","volume":"44","author":"Germain","year":"2022","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Castro, J. (2021). Deep Learning Schemes For Parabolic Nonlocal Integro-Differential Equations. arXiv.","DOI":"10.1007\/s42985-022-00213-z"},{"key":"ref_19","unstructured":"Al-Aradi, A., Correia, A., Naiff, D.d.F., Jardim, G., and Saporito, Y. (2019). Applications of the Deep Galerkin Method to Solving Partial Integro-Differential and Hamilton-Jacobi-Bellman Equations. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1339","DOI":"10.1016\/j.jcp.2018.08.029","article-title":"DGM: A deep learning algorithm for solving partial differential equations","volume":"375","author":"Sirignano","year":"2018","journal-title":"J. Comput. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Boussange, V., Becker, S., Jentzen, A., Kuckuck, B., and Pellissier, L. (2022). Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions. arXiv.","DOI":"10.1007\/s42985-023-00244-0"},{"key":"ref_22","unstructured":"Frey, R., and K\u00f6ck, V. (2022). Convergence Analysis of the Deep Splitting Scheme: The Case of Partial Integro-Differential Equations and the associated FBSDEs with Jumps. arXiv."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Gihman, I., and Skohorod, A. (1980). The Theory of Stochastic Processes, Springer.","DOI":"10.1007\/978-1-4612-6202-2"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1415","DOI":"10.1109\/18.59936","article-title":"On the unnormalized solution of the filtering problem with counting process observations","volume":"36","author":"Kliemann","year":"1990","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Ethier, S., and Kurtz, T.G. (1986). Markov Processes: Characterization and Convergence, Wiley.","DOI":"10.1002\/9780470316658"},{"key":"ref_26","first-page":"1","article-title":"Optimal stopping of controlled jump diffusion processes: A viscosity solution approach","volume":"8","author":"Pham","year":"1998","journal-title":"J. Math. Syst. Estim. Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1016\/j.insmatheco.2021.09.003","article-title":"Classical Solutions of the Backward PIDE for Markov Modulated Marked Point Processes and Applications to CAT Bonds","volume":"101","author":"Colaneri","year":"2021","journal-title":"Insur. Math. Econ."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1137\/19M1283045","article-title":"Value adjustments and dynamic hedging of reinsurance counterparty risk","volume":"11","author":"Ceci","year":"2020","journal-title":"SIAM J. Financ. Math."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Frey, R., and K\u00f6ck, V. (2021). Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance Mathematics. arXiv.","DOI":"10.1007\/978-3-030-99638-3_44"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1016\/j.insmatheco.2009.05.012","article-title":"Approximate basket options valuation for a jump-diffusion model","volume":"45","author":"Xu","year":"2009","journal-title":"Insur. Math. Econ."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s11579-017-0206-z","article-title":"Mean Field Game of Controls and an Application to Trade Crowding","volume":"12","author":"Cardaliaguet","year":"2018","journal-title":"Math. Financ. Econ."},{"key":"ref_32","unstructured":"Cartea, \u00c1., Jaimungal, S., and Penalva, J. (2015). Algorithmic and High-Frequency Trading, Cambridge University Press."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"\u00d8ksendal, B.K., and Sulem, A. (2007). Applied Stochastic Control of Jump Diffusions, Springer.","DOI":"10.1007\/978-3-540-69826-5"}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/11\/201\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:14:08Z","timestamp":1760145248000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/11\/201"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,10]]},"references-count":33,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2022,11]]}},"alternative-id":["computation10110201"],"URL":"https:\/\/doi.org\/10.3390\/computation10110201","relation":{},"ISSN":["2079-3197"],"issn-type":[{"value":"2079-3197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,10]]}}}