A backward stochastic differential equation (BSDE) of the form \[ -dY_t = f (s, Y_s, Z_s) ds - Z_t dB_t, \quad Y_T = \xi \] is considered as a functional differential equation running forward in time, i.e., \[ V_T - V_t = \int_t^T f(s, Y_s, Z_s)\, ds; \] when we require \(V_0 = 0\), then this formula is uniquely determined by \(Y\) and \(Z\). Define \[ M_t = \operatorname{E}[ \xi + V_T | \mathcal{F}_t]. \] The authors take the opposite point of view. To every \(V\), it is possible to associate \((Y(V),Z(V))\) by setting \[ Y(V)_t = M(V)_t - V_t = \operatorname{E}[\xi + V_T | \mathcal{F}_t] - V_t, \] and, in martingale representation, \[ \int_t^T Z(V)_s dB_s = M(V)_T - M(V)_t. \] The BSDE can then be written in the equivalent form \[ \frac{dV}{dt} = f(t, Y(V)_t, Z(V)_t), \quad V_0 = 0. \] This is a functional differential equation which can be solved forward in time. The authors further generalize this, by noting that \(V \mapsto Z(V)\) can be written as \(V \mapsto L(M(V))\), where \(M(V)\) is given as above, and \(L\) is an operator from the space of martingales to the space of predictable processes. This can then be treated independently of the martingale representation property. This leads to functional equations of the form \[ \frac{dV}{dt} = f(t, Y(V)_t, L(M(V))_t), \quad V_0 = 0. \] The authors prove uniqueness and local existence of a generalized version of this equation under rather mild (Lipschitz) assumptions. They point out that global uniqueness is not expected to hold without stronger assumptions on the operator \(L\). Under a so called local-in-time property of \(L\), they are then able to establish the existence of global solutions.
This allows to give probabilistic representations of certain nonlinear nonlocal partial differential equations.