Backward Deep BSDE Methods and Applications to Nonlinear Problems
Abstract
:1. Introduction
2. FBSDE for Nonlinear Problems
2.1. Backward TimeStepping
2.1.1. Exact Backward TimeStepping
2.1.2. TimeStepping 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 ${Z}_{t}={\Pi}_{t}^{T}\sigma (t,{X}_{t})$ and $\tilde{f}(t,X,Y,Z)=f(t,X,Y,\sigma {(t,{X}_{t})}^{T}{Z}_{t}^{T})$. Then, (Pardoux 1998, Theorem 2.2) with this ${Z}_{t}$ and $\tilde{f}$ shows that one can construct a solution ${Y}_{t}$ and ${Z}_{t}$ of the BSDE from a classical ${C}^{2,1}$ solution of a corresponding PDE and a solution ${X}_{t}$. Rewriting the PDE and BSDE in terms of ${\Pi}_{t}^{T}$ instead of ${Z}_{t}$ and using function f rather than $\tilde{f}$ gives the form reported below. For the opposite direction, (Pardoux 1998, Theorem 2.4) shows how a solution ${Y}_{t}$ of the BSDE (corresponding to a X that starts at x at time t) gives a continuous function $u(t,x)$, which is a viscosity solution of the corresponding PDE. 
2  In general, the terminal condition could be given as a random variate ${G}_{T}$ that is measurable with respect to the information available of time T (i.e., the sigma algebra generated by ${X}_{t}$ with $t\le T$). The FBSDE approach then will be more general than the PDE approach. If there is an exact (or approximate) Markovianization with a Markov state ${M}_{t}$, the strategy ${\Pi}_{t}$ and the solution ${Y}_{t}$ would in general be functions $\pi (t,{M}_{t})$ and $u(t,{M}_{t})$ of that Markov state. We only treat the usual final value case here. 
3  
4  This is actually the expectation of the conditional variance $\mathsf{var}\left({\mathcal{Y}}_{0}^{\pi}\right{X}_{0})$ over the distribution of ${X}_{0}$. 
5  Here, we consider the 1dimensional case, while Han and Jentzen (2017) consider the 100dimensional case. 
6  Notice that (Forsyth and Labahn 2007, Figure 1) do not give the number of space or timesteps used for their plot. 
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Method  Upper Price  Lower Price 

Results from Forsyth and Labahn101 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 ${Y}_{0}$ (shared)  24.06 (23.99–24.14)  23.10 (23.02–23.17) 
Learned ${Y}_{0}$ (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 ${Y}_{0}$ (shared)  24.06 (23.99–24.14)  23.10 (23.02–23.17) 
Learned ${Y}_{0}$ (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