Backward Deep BSDE Methods and Applications to Nonlinear Problems
We present a pathwise deep Backward Stochastic Differential Equation (BSDE) method for Forward Backward Stochastic Differential Equations with terminal conditions that time-steps the BSDE backwards and apply it to the differential rates problem as a prototypical nonlinear problem of independent fina...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Risks |
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| Online Access: | https://www.mdpi.com/2227-9091/11/3/61 |
| _version_ | 1797609165509623808 |
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| author | Yajie Yu Narayan Ganesan Bernhard Hientzsch |
| author_facet | Yajie Yu Narayan Ganesan Bernhard Hientzsch |
| author_sort | Yajie Yu |
| collection | DOAJ |
| description | We present a pathwise deep Backward Stochastic Differential Equation (BSDE) method for Forward Backward Stochastic Differential Equations with terminal conditions that time-steps the BSDE backwards and apply it to the differential rates problem as a prototypical nonlinear problem of independent financial interest. The nonlinear equation for the backward time-step is solved exactly or by a Taylor-based approximation. This is the first application of such a pathwise backward time-stepping deep BSDE approach for problems with nonlinear generators. We extend the method to the case when the initial value of the forward components X can be a parameter rather than fixed and similarly to also learn values at intermediate times. We present numerical results for a call combination and for a straddle, the latter comparing well to those obtained by Forsyth and Labahn with a specialized PDE solver. |
| first_indexed | 2024-03-11T05:57:44Z |
| format | Article |
| id | doaj.art-e70a98cd5681459eba099b20d76795f9 |
| institution | Directory Open Access Journal |
| issn | 2227-9091 |
| language | English |
| last_indexed | 2024-03-11T05:57:44Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Risks |
| spelling | doaj.art-e70a98cd5681459eba099b20d76795f92023-11-17T13:41:49ZengMDPI AGRisks2227-90912023-03-011136110.3390/risks11030061Backward Deep BSDE Methods and Applications to Nonlinear ProblemsYajie Yu0Narayan Ganesan1Bernhard Hientzsch2Corporate Model Risk, Wells Fargo, New York, NY 10017, USACorporate Model Risk, Wells Fargo, New York, NY 10017, USACorporate Model Risk, Wells Fargo, New York, NY 10017, USAWe present a pathwise deep Backward Stochastic Differential Equation (BSDE) method for Forward Backward Stochastic Differential Equations with terminal conditions that time-steps the BSDE backwards and apply it to the differential rates problem as a prototypical nonlinear problem of independent financial interest. The nonlinear equation for the backward time-step is solved exactly or by a Taylor-based approximation. This is the first application of such a pathwise backward time-stepping deep BSDE approach for problems with nonlinear generators. We extend the method to the case when the initial value of the forward components X can be a parameter rather than fixed and similarly to also learn values at intermediate times. We present numerical results for a call combination and for a straddle, the latter comparing well to those obtained by Forsyth and Labahn with a specialized PDE solver.https://www.mdpi.com/2227-9091/11/3/61differential ratesFBSDEsnonlinear pricingdeep learning for pricing |
| spellingShingle | Yajie Yu Narayan Ganesan Bernhard Hientzsch Backward Deep BSDE Methods and Applications to Nonlinear Problems Risks differential rates FBSDEs nonlinear pricing deep learning for pricing |
| title | Backward Deep BSDE Methods and Applications to Nonlinear Problems |
| title_full | Backward Deep BSDE Methods and Applications to Nonlinear Problems |
| title_fullStr | Backward Deep BSDE Methods and Applications to Nonlinear Problems |
| title_full_unstemmed | Backward Deep BSDE Methods and Applications to Nonlinear Problems |
| title_short | Backward Deep BSDE Methods and Applications to Nonlinear Problems |
| title_sort | backward deep bsde methods and applications to nonlinear problems |
| topic | differential rates FBSDEs nonlinear pricing deep learning for pricing |
| url | https://www.mdpi.com/2227-9091/11/3/61 |
| work_keys_str_mv | AT yajieyu backwarddeepbsdemethodsandapplicationstononlinearproblems AT narayanganesan backwarddeepbsdemethodsandapplicationstononlinearproblems AT bernhardhientzsch backwarddeepbsdemethodsandapplicationstononlinearproblems |