In the Deep BSDE method proposed by E et al., (2017) and Han et al., (2018), the backward stochastic differential equation (BSDE):
$\begin{cases} dY_t &= -f(t, \mathbf{X}_t, Y_t, \mathbf{Z}_t)\ dt + \mathbf{Z}_t^\intercal \ d\mathbf{W}_t,\hspace{0.64cm} t\in[0, T] \\ Y_T &= g(\mathbf{X}_T) \end{cases}$
is discretized in a forward manner:
$\begin{cases} Y_t = Y_0 - \displaystyle\int_0^t f(s, \mathbf{X}_s, Y_s, \mathbf{Z}_s) ds + \displaystyle\int_0^t \mathbf{Z}_s^\intercal d\mathbf{W}_s, \ t \in [0, T] \\ Y_T = g(\mathbf{X}_T), \end{cases}$
$Y_{t_{n+1}} \approx Y_{t_n} - f \bigl( t_n, \mathbf{X}_{t_n}, Y_{t_n}, \mathbf{Z}_{t_n}) \bigr) \Delta t_n + \mathbf{Z}^\intercal_{t_n} \Delta \mathbf{W}_n,\hspace{0.5cm} Y_T = g(\mathbf{X}_T).$
Subsequently, the solution of the BSDE is approximated using a deep neural network.
My question is does this forward discretization change the "backward" nature of BSDE? I ask this because my lecturer in the field of analysis questioned how a BSDE can be solved in a forward manner despite its name suggesting otherwise.
E, W., Han, J., & Jentzen, A. (2017). Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Communications in mathematics and statistics, 5 (4), 349–380.
Han, J., Jentzen, A. and E, W., 2018. Solving high-dimensional partial differential equations using deep learning. Proceedings of the National Academy of Sciences, 115(34), pp.8505-8510.