Backward time-stepping for BSDEs is a numerical technique where the backward stochastic differential equation is propagated from the known terminal condition Y_T = g(X_T) to the initial time t_0, in contrast to the forward deep BSDE method which propagates Y_t forward from an unknown Y_0. At each backward step, one must solve the implicit equation Y_{t_i} - f(t_i, X_{t_i}, Y_{t_i}, Pi_{t_i}) Delta t = Y_{t_{i+1}} - Pi_{t_i}^T sigma(t_i, X_{t_i}) Delta W_i for Y_{t_i}, given known values of Y_{t_{i+1}}, X_{t_i}, Pi_{t_i}, and the Brownian increment.
For nonlinear generators f(t, x, y, z), solving this implicit equation is non-trivial. Yu, Hientzsch, Ganesan (2020) present two approaches. The exact analytical approach exploits the structure of specific generators: for the differential rates problem, the max operator creates two linear regimes that can be solved in closed form by case-splitting on whether the cash position is positive or negative. The condition for which case applies can be evaluated using the right-hand side quantities (Y_{t_{i+1}}, Pi, sigma, Delta W), avoiding circularity. The Taylor expansion approach linearises f around the known value Y_{t_{i+1}}: f(t_i, X, Y_{t_i}, Z) is approximated as f(t_i, X, Y_{t_{i+1}}, Z) - (df/dY)(t_i, X, Y_{t_{i+1}}, Z) * Delta Y, yielding the explicit formula Y_{t_i} = [Y_{t_{i+1}} + f(t_i, X, Y_{t_{i+1}}, Z) Delta t - Pi^T sigma Delta W] / [1 - (df/dY) Delta t].
The Taylor approach is attractive for general nonlinear problems because it requires only f and its Y-derivative, not problem-specific analytical solutions. Yu et al. demonstrated that for the differential rates problem, both approaches yield identical results up to differences of order 10^{-5}.
Key Details
- Exact backward step: requires problem-specific derivation; for differential rates, two closed-form expressions depending on borrowing vs lending regime
- Taylor backward step: generic first-order approximation; Y_{t_i} = [Y_{t_{i+1}} + f Delta t - Z^T Delta W] / [1 - (df/dY) Delta t]; works for any generator with computable Y-derivative
- Condition resolution: for differential rates, the borrowing/lending condition can be checked using Y_{t_{i+1}} and known quantities, avoiding the circular dependence on the unknown Y_{t_i}
- Error: exact-Taylor difference ~ O(10^{-5}) for differential rates with 50-100 time steps
- Advantage over forward: terminal condition is satisfied exactly (no terminal mismatch error); loss function measures how well the backward roll-back produces a consistent initial value
- Limitation: each backward step requires solving a potentially nonlinear equation, adding per-step computational cost compared to the explicit forward propagation