Deep solver for FBSDE with jumps. In this article we elaborate on the…

We use the superscript Y0, Z, R to emphasize that we interpret the scalar Y0 and processes Z and R as controls to be learned by the neural network. In this regard, the setup is similar to the solver of E et al. we referenced in the beginning, with the difference that now we also have the control process R accounting for the jumps. Note that our system is a forward-backward system, which means that we know the initial value of the forward process X and terminal value of the backward process Y. Thus, following the idea from E et al., we set some random initial value for the initial value Y0 of the backward process, and treat it as a trainable parameter, which will be learned by backpropagation. Therefore, knowing X0 and Y0 makes the above equation into a forward scheme for X and Y. We sample N paths of Brownian motion and Poisson process, and employ Euler–Maruyama forward scheme to evaluate the equation (1) forward in time.