Acknowledgement
Hyungbin Park was supported by Research Resettlement Fund for the new faculty of Seoul National University, South Korea. In addition, Hyungbin Park was supported by the National Research Foundation of Korea (NRF) grants funded by the Ministry of Science and ICT (No. 2017R1A5A1015626, No. 2018R1C1B5085491 and No. 2021R1C1C1011675) and the Ministry of Education (No. 2019R1A6A1A10073437) through the Basic Science Research Program. Financial support from the Institute for Research in Finance and Economics of Seoul National University is gratefully acknowledged.
References
- Y. Achdou and O. Pironneau, Computational methods for option pricing, Frontiers in Applied Mathematics, 30, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. https://doi.org/10.1137/1.9780898717495
- A. Al-Aradi, A. Correia, G. Jardim, D. Naiff, and Y. Saporito, Solving nonlinear and high-dimensional partial differential equations via deep learning, arXiv preprint arXiv:1811.08782.
- A. Al-Aradi, A. Correia, G. Jardim, D. Naiff, and Y. Saporito, Applications of the deep Galerkin method to solving partial integro-differential and Hamilton-Jacobi-Bellman equations, arXiv preprint arXiv:1912.01455.
- F. E. Benth, K. H. Karlsen, and K. Reikvam, Merton's portfolio optimization problem in a Black and Scholes market with non-Gaussian stochastic volatility of Ornstein-Uhlenbeck type, Math. Finance 13 (2003), no. 2, 215-244. https://doi.org/10.1111/1467-9965.00015
- T. Bjork, Arbitrage Theory in Continuous Time, Oxford University Press, Oxford, 2009.
- L. Boccardo, A. Dall'Aglio, T. Gallouet, and L. Orsina, Nonlinear parabolic equations with measure data, J. Funct. Anal. 147 (1997), no. 1, 237-258. https://doi.org/10.1006/jfan.1996.3040
- L. Boccardo, M. M. Porzio, and A. Primo, Summability and existence results for nonlinear parabolic equations, Nonlinear Anal. 71 (2009), no. 3-4, 978-990. https://doi.org/10.1016/j.na.2008.11.066
- R. L. Burden, J. D. Faires, and A. C. Reynolds, Numerical Analysis, Prindle, Weber & Schmidt, Boston, MA, 1978.
- G. Callegaro, M. Gaigi, S. Scotti, and C. Sgarra, Optimal investment in markets with over and under-reaction to information, Math. Financ. Econ. 11 (2017), no. 3, 299-322. https://doi.org/10.1007/s11579-016-0182-8
- R. Crisostomo, An analysis of the Heston stochastic volatility model: Implementation and calibration using Matlab, CNMV Working Paper (2014), no. 58.
- A. Danilova, M. Monoyios, and A. Ng, Optimal investment with inside information and parameter uncertainty, Math. Financ. Econ. 3 (2010), no. 1, 13-38. https://doi.org/10.1007/s11579-010-0025-y
- L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, third edition, Birkhauser/Springer, New York, 2012. https://doi.org/10.1007/978-0-8176-8265-1
- R. Di Nardo, F. Feo, and O. Guibe, Existence result for nonlinear parabolic equations with lower order terms, Anal. Appl. (Singap.) 9 (2011), no. 2, 161-186. https://doi.org/10.1142/S0219530511001790
- W. E, M. Hutzenthaler, A. Jentzen, and T. Kruse, On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations, J. Sci. Comput. 79 (2019), no. 3, 1534-1571. https://doi.org/10.1007/s10915-018-00903-0
- L. C. Evans, Partial Differential Equations, Amer. Math. Soc., 2002.
- S. Fallah and F. Mehrdoust, On the calibration of fractional two-factor stochastic volatility model with non-Lipschitz diffusions, Commun. Stat. Simul. Comput. (2020), 1-20.
- P. Guasoni and S. Robertson, Static fund separation of long-term investments, Math. Finance 25 (2015), no. 4, 789-826. https://doi.org/10.1111/mafi.12017
- J. Han, A. Jentzen, and W. E, Solving high-dimensional partial differential equations using deep learning, Proc. Natl. Acad. Sci. USA 115 (2018), no. 34, 8505-8510. https://doi.org/10.1073/pnas.1718942115
- S. L. Hansen, Optimal consumption and investment strategies with partial and private information in a multi-asset setting, Math. Financ. Econ. 7 (2013), no. 3, 305-340. https://doi.org/10.1007/s11579-012-0086-1
- K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Netw. 4 (1991), no. 2, 251-257. https://doi.org/10.1016/0893-6080(91)90009-T
- C. Kuhn and M. Stroh, Optimal portfolios of a small investor in a limit order market: a shadow price approach, Math. Financ. Econ. 3 (2010), no. 2, 45-72. https://doi.org/10.1007/s11579-010-0027-9
- O. A. Ladyzhenskaia, V. A. Solonnikov and N. N. Ural'tseva, Linear and quasi-linear equations of parabolic type, Amer. Math. Soc., 1988.
- I. E. Lagaris, A. C. Likas and D. G. Papageorgiou, Neural-network methods for boundary value problems with irregular boundaries, IEEE Trans. Neural. Netw. 11 (2000), no. 5, 1041-1049. https://doi.org/10.1109/72.870037
- H. Lee and I. S. Kang, Neural algorithm for solving differential equations, J. Comput. Phys. 91 (1990), no. 1, 110-131. https://doi.org/10.1016/0021-9991(90)90007-N
- V. Lemaire and T. Montes, Stationary Heston model: Calibration and pricing of exotics using product recursive quantization, arXiv preprint arXiv:2001.03101.
- Z. Liang and M. Ma, Robust consumption-investment problem under CRRA and CARA utilities with time-varying confidence sets, Math. Finance 30 (2020), no. 3, 1035-1072. https://doi.org/10.1111/mafi.12217
- M. Magliocca, Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term, Nonlinear Anal. 166 (2018), 102-143. https://doi.org/10.1016/j.na.2017.09.012
- A. Malek and R. Shekari Beidokhti, Numerical solution for high order differential equations using a hybrid neural network-optimization method, Appl. Math. Comput. 183 (2006), no. 1, 260-271. https://doi.org/10.1016/j.amc.2006.05.068
- R. C. Merton, Lifetime portfolio selection under uncertainty: The continuous-time case, Rev. Econ. Stat. (1969), 247-257.
- M. Nutz, The opportunity process for optimal consumption and investment with power utility, Math. Financ. Econ. 3 (2010), no. 3-4, 139-159. https://doi.org/10.1007/s11579-010-0031-0
- J. L. Pedersen and G. Peskir, Optimal mean-variance portfolio selection, Math. Financ. Econ. 11 (2017), no. 2, 137-160. https://doi.org/10.1007/s11579-016-0174-8
- M. M. Porzio, Existence of solutions for some "noncoercive" parabolic equations, Discrete Contin. Dynam. Systems 5 (1999), no. 3, 553-568. https://doi.org/10.3934/dcds.1999.5.553
- C. Remani, Numerical Methods for Solving Systems of Nonlinear Equations, Lakehead University Thunder Bay, Ontario, Canada, 2013.
- J. Simon, Compact sets in the space Lp(0, T; B), Ann. di Mat. Pura ed Appl. 146 (1986), no. 1, 65-96. https://doi.org/10.1007/BF01762360
- J. Sirignano and K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differential equations, J. Comput. Phys. 375 (2018), 1339-1364. https://doi.org/10.1016/j.jcp.2018.08.029