Abstract
Coupled forward-backward stochastic differential equations (FBSDEs) are closely related to financially important issues such as optimal investment. However, it is well known that obtaining solutions is challenging, even when employing numerical methods. In this paper, we propose new methods that combine an algorithm recently developed for coupled FBSDEs and an asymptotic expansion approach to those FBSDEs as control variates for learning of the neural networks. The proposed method is demonstrated to perform better than the original algorithm in numerical examples, including one with a financial implication. The results show that the proposed method exhibits not only faster convergence but also greater stability in computation.
Summary
The key idea is to use asymptotic expansions of coupled FBSDEs as control variates for the deep BSDE method. Instead of learning Y and Z from scratch, the neural networks learn the residual: replace phi^1 (network for Y) with chi * Y^{AE,l} + phi^1 and phi^2 (network for Z) with chi * Z^{AE,l} + phi^2, where chi in {0,1} toggles the expansion. When chi=1, the networks only need to learn the small correction Y - Y^{AE} and Z - Z^{AE}.
Asymptotic expansion construction: Introduce a perturbation parameter epsilon in the FBSDE coupling (eqs. 12-13). Taylor expand (X^eps, Y^eps, Z^eps) in powers of epsilon. The leading order (X^0, Y^0) are deterministic (eqs. 25-26): X_t^0 solves an ODE, Y_t^0 = g(X_T^0) + integral f^0 ds. The first-order (X^1, Y^1, Z^1) are Gaussian (eqs. 16-18). Crucially, Z_t^0 = 0 identically and Z_t^1 is deterministic. These deterministic control variates are cheap to compute and dramatically reduce the learning burden.
Algorithm 1 (Algorithm 2 of Ji et al. + asymptotic expansion): outer loop over SGD steps k, inner loop over M Monte Carlo paths. For each path: set Y_0 = chi * Y_0^{AE,l} + phi^1(X_0; theta), Z_0 = chi * Z_0^{AE,l} + phi^2(X_0; theta). Propagate forward X and backward Y using discretized FBSDE dynamics. Loss = (1/M) sum_m [|Y_T - g(X_T)|^2 + (T/N) sum_i |Y_{t_{i+1}} - u_{t_{i+1}}|^2]. SGD update on theta. Network architecture (following Ji et al.): one n-dim input layer, two hidden (n+10)-dim layers, m-dim output for phi^1 and (m x d)-dim output for phi^2. Batch norm, Adam optimizer. Batch size 256, learning rate 0.005, N=25 time steps.
Critical stability result: with random initialization from [-2,2], 90 out of 100 trials fail to update the network within the first 100 learning steps for the original algorithm. With the asymptotic expansion as control variate (chi=1), 0 out of 100 fail. The expansion provides a warm start that is always in the basin of attraction.
Financial application (Example 3): portfolio optimization in incomplete markets with d_1=5 risk factors, 1 non-traded risk source, d=6 total dimension. The FBSDE is fully coupled through Z in the forward equation for wealth X. The proposed method achieves better expected utility than both the original deep solver and the asymptotic expansion alone. Code: https://github.com/Makot0922/Python-Code (PyTorch).
Key Contributions
- First application of asymptotic expansion as control variate for coupled (not just uncoupled) FBSDEs
- Order-of-magnitude improvement in loss function values
- Eliminates initialization instability (90% failure rate → 0%)
- Both Y and Z expansions needed simultaneously (using only one is insufficient)
- Open-source PyTorch implementation
Methodology
The control variate for deep BSDE approach works because the asymptotic expansion Y^{AE,l} captures the dominant structure of the solution. The neural network only needs to learn the residual, which is small when the expansion is accurate. This is equivalent to variance reduction in Monte Carlo: the control variate removes the “known” part of the variance, leaving the network to learn only the “unknown” correction.
Key Findings
- Loss function improved by 1-2 orders of magnitude vs. original algorithm
- Y_0 error improved by 1-3 orders of magnitude
- Convergence speed: reaches target accuracy in ~10% of the computational time
- Using only Y or only Z expansion as control variate gives marginal improvement; both together are needed
- Higher-order expansions (l=2) could further improve results but computational cost grows
Important References
- Three Algorithms for Solving High-Dimensional Fully-Coupled FBSDEs Through Deep Learning — Ji, Peng, Peng, Zhang (2020), the base algorithms enhanced here
- Asymptotic Expansion for Optimal Investment Problems — Takahashi, Yoshida (2004), the asymptotic expansion methodology
- Forward-Backward Systems for Expected Utility Maximization — Horst, Hu, Imkeller, Reveillac, Zhang (2014), the financial FBSDE in Example 3