Backward Deep BSDE Methods and Applications to Nonlinear Problems
Abstract
:1. Introduction
2. FBSDE for Nonlinear Problems
2.1. Backward Time-Stepping
2.1.1. Exact Backward Time-Stepping
2.1.2. Time-Stepping from Taylor Expansion
3. Deep BSDE Approach
3.1. Forward Approach
3.2. Backward Approach
Algorithm 1 The pathwise Backward deep BSDE Method |
|
4. Results
4.1. Call Combination
4.2. Straddle
5. Conclusions
6. Disclaimer
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Notes
1 | Set and . Then, (Pardoux 1998, Theorem 2.2) with this and shows that one can construct a solution and of the BSDE from a classical solution of a corresponding PDE and a solution . Rewriting the PDE and BSDE in terms of instead of and using function f rather than gives the form reported below. For the opposite direction, (Pardoux 1998, Theorem 2.4) shows how a solution of the BSDE (corresponding to a X that starts at x at time t) gives a continuous function , which is a viscosity solution of the corresponding PDE. |
2 | In general, the terminal condition could be given as a random variate that is measurable with respect to the information available of time T (i.e., the sigma algebra generated by with ). The FBSDE approach then will be more general than the PDE approach. If there is an exact (or approximate) Markovianization with a Markov state , the strategy and the solution would in general be functions and of that Markov state. We only treat the usual final value case here. |
3 | |
4 | This is actually the expectation of the conditional variance over the distribution of . |
5 | Here, we consider the 1-dimensional case, while Han and Jentzen (2017) consider the 100-dimensional case. |
6 | Notice that (Forsyth and Labahn 2007, Figure 1) do not give the number of space or time-steps used for their plot. |
References
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Method | Upper Price | Lower Price |
---|---|---|
Results from Forsyth and Labahn-101 nodes | ||
Fully Implicit HJB PDE (implicit control) | 24.02 | 23.06 |
Crank–Nicolson HJB PDE (implicit control) | 24.05 | 23.09 |
Fully Implicit HJB PDE (pwc policy) | 24.01 | 23.07 |
Crank–Nicolson HJB PDE (pwc policy) | 24.07 | 23.09 |
Forward deep BSDE—20,000 batches, size 256 | ||
Learned (shared) | 24.06 (23.99–24.14) | 23.10 (23.02–23.17) |
Learned (separate) | 24.07 (24.01–24.12) | 23.10 (23.06–23.15) |
Backward deep BSDE—20,000 batches, size 256 | ||
Batch variance/1 (shared) | 24.14 (23.98–24.30) | 23.19 (23.00–23.37) |
Batch variance/100 (shared) | 24.08 (24.06–24.09) | 23.13 (23.09–23.16) |
Batch variance/1 (separate) | 24.06 (23.94–24.19 *) | 23.10 (22.95–23.25) |
Batch variance/100 (separate) | 24.07 (24.06–24.09 *) | 23.12 (23.10–23.13) |
Learned (shared) | 24.06 (23.99–24.14) | 23.10 (23.02–23.17) |
Learned (separate) | 24.06 (24.01–24.11 *) | 23.10 (23.06–23.15) |
Method | Upper Price | Lower Price |
---|---|---|
Results from Forsyth and Labahn—101 nodes | ||
Fully Implicit (implicit control) | 24.02 | 23.06 |
Crank–Nicolson (implicit control) | 24.05 | 23.09 |
Fully Implicit (pwc policy) | 24.01 | 23.07 |
Crank–Nicolson (pwc policy) | 24.07 | 23.09 |
Backward deep BSDE—20,000 batches, size 512, five seeds | ||
Batch variance/1 (shared) | 24.02–24.08 (23.87–24.29) | 23.06–23.12 (22.91–23.33) |
Batch variance/100 (shared) | 24.06–24.07 (24.03–24.09) | 23.11–23.12 (23.09–23.15) |
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Yu, Y.; Ganesan, N.; Hientzsch, B. Backward Deep BSDE Methods and Applications to Nonlinear Problems. Risks 2023, 11, 61. https://doi.org/10.3390/risks11030061
Yu Y, Ganesan N, Hientzsch B. Backward Deep BSDE Methods and Applications to Nonlinear Problems. Risks. 2023; 11(3):61. https://doi.org/10.3390/risks11030061
Chicago/Turabian StyleYu, Yajie, Narayan Ganesan, and Bernhard Hientzsch. 2023. "Backward Deep BSDE Methods and Applications to Nonlinear Problems" Risks 11, no. 3: 61. https://doi.org/10.3390/risks11030061
APA StyleYu, Y., Ganesan, N., & Hientzsch, B. (2023). Backward Deep BSDE Methods and Applications to Nonlinear Problems. Risks, 11(3), 61. https://doi.org/10.3390/risks11030061