Deep_BSDE_solver

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Deep_BSDE_solver

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Deep BSDE solver in Pytorch

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# Deep BSDE solver in Pytorch

## Usage

This package provides a solver for the following FBSDE:
```math
 \left\{
    \begin{aligned}
        X_{t} & = x + \int_{0}^{t}b(s,X_{s},Y_{s})\mathop{}\mathrm{d} s+ \int_{0}^{t}\langle\sigma(s,X_{s}),\mathop{}\mathrm{d} W_{s}\rangle \\
        Y_{t} & =g(X_{T})+\int_{t}^{T}f(s,X_{s},Y_{s},Z_{s})\mathop{}\mathrm{d} s-\int_{t}^{T}\langle Z_{s},\mathop{}\mathrm{d} W_{s}\rangle.
    \end{aligned}
    \right.
```

`solver.py` contains the main solver based on the Deep BSDE method proposed in [1].

`cir_bond.py` and `multi_cir_bond.py` are codes for the examples in [2].

One can easily apply this solver to other FBSDEs by redefining the coefficients and parameters.

The training data will be saved in `loss_data.npy` and `y0_data.npy` which record the loss $\mathbb{E}[g(X_{T})-Y_{T}]^{2}$ and the initial value $y_0$ after each iteration.

## Dependencies

* Pytorch >= 1.4.0
* Numpy

## References

[1] E, W., Han, J., and Jentzen, A. Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations, Communications in Mathematics and Statistics, 5, 349–380 (2017).

[2] Jiang, Y., Li, J. Convergence of the deep BSDE method for FBSDEs with non-lipschitz coefficients, Probability, Uncertainty and Quantitative Risk, 6 (4), 391-408 (2021).

Deep_BSDE_solver

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