A backward stochastic differential equation with random horizon is a BSDE of the form:
-dY(t) = f(t, Y(t), Z(t), U_1(t), U_2(t)) dt - Z(t)^T dW(t) - U_1(t) dM_1(t) - U_2(t) dM_2(t), t in [0, tau_1 ^ tau_2 ^ T]
with terminal condition Y(tau_1 ^ tau_2 ^ T) = phi_1(tau_1) 1_{tau_1 < tau_2 ^ T} + phi_2(tau_2) 1_{tau_2 < tau_1 ^ T} + xi_T 1_{T < tau_1 ^ tau_2}, where tau_1 and tau_2 are random times (e.g., default times) and T is a fixed maturity.
The key technical feature is that the horizon tau_1 ^ tau_2 ^ T is not deterministic: it depends on the jump times of the martingales M_1, M_2. The solution (Y, Z, U_1, U_2) lives on the progressively enlarged filtration G = F v H, where F is the Brownian filtration and H is generated by the default indicator processes.
Existence and uniqueness (Theorem 1 of Sekine and Tanaka, 2020): Under standard Lipschitz and integrability conditions on the driver f, the BSDE admits a unique solution in S^2_{beta,T} x H^{2,n+2}_{beta,T} for sufficiently large beta > 0. The proof uses a Picard iteration / contraction mapping argument.
Reduction to reference filtration (Theorem 2): Under bounded hazard rates, the solution has the representation Y(t) = Y_bar(t) 1_{t < tau_1 ^ tau_2 ^ T} + (terminal payoff) 1_{t = tau_1 ^ tau_2 ^ T}, Z(t) = Z_bar(t), U_i(t) = phi_i(t) - Y_bar(t), where (Y_bar, Z_bar) solves a standard BSDE on the smaller Brownian filtration F with a modified driver that absorbs the jump components. This reduction is the foundation for connecting to semilinear PDEs via the nonlinear Feynman-Kac formula.
The random horizon BSDE naturally arises in pricing defaultable derivatives where the contract terminates at the first default of either party. See also Darling and Pardoux (1997) for the single random time case.