A nonlinear backward stochastic differential equation (BSDE) with jumps is a BSDE whose driver is nonlinear and whose dynamics include compensated jump martingale terms in addition to the Brownian motion term. In the context of XVA pricing, the jumps arise from the default events of the trader and counterparty.
The general form appearing in Bichuch et al. (2015a) is:
-dV_t = f(t, V_t, Z_t, Z_t^I, Z_t^C; V_hat_t) dt - Z_t dW_t^Q - Z_t^I d_t^{C,Q}
with terminal/stopping-time condition V_tau = theta_tau(V_hat) 1_{tau<T} + Phi(S_T) 1_{tau=T}, where tau = tau_I ^ tau_C ^ T is the first default or maturity.
The key features are:
- Nonlinear driver: f^+ depends on (v, z, z^I, z^C) through positive/negative parts, capturing the asymmetric borrowing/lending rates. When rf+ = rf-, the driver linearises and the BSDE admits a closed-form solution.
- Jump terms: Z_t^I and Z_t^C multiply compensated Poisson processes associated with trader and counterparty defaults. The process Z_t^j = theta_j(V_hat) - V_t represents the jump in portfolio value at default of party j.
- Random terminal time: tau is a stopping time (not a fixed time T), requiring extensions of standard BSDE theory. The terminal condition encodes the closeout value at default or the payoff at maturity.
- Existence and uniqueness: follows from Lipschitz continuity of the driver and the comparison theorem (Theorem A.2 and A.3 in Bichuch et al. (2015a)), extending results of Delong (2013).
The nonlinear BSDE can equivalently be represented as a semilinear PDE (Bichuch et al. (2015b)) via the nonlinear Feynman-Kac connection. This connection is what enables both the deep BSDE method and classical finite-difference numerical approaches.