A progressively enlarged filtration is constructed by augmenting a reference filtration F with the information generated by one or more random times. Given a Brownian filtration F = (F_t){t >= 0} and default indicator processes N_i(t) = 1{t >= tau_i} generating H = (H_t)_{t >= 0}, the progressively enlarged filtration is:
G_t := F_t v H_t, t >= 0.
This is the smallest right-continuous filtration containing both F and H. It carries the combined information about the driving noise and the occurrence (or non-occurrence) of defaults.
Key structural results (Lemma 1 of Sekine and Tanaka, 2020, following Pham 2010):
- Any G-predictable process P(t) decomposes as P(t) = p_0(t) 1_{t ⇐ tau_1 ^ tau_2} + p_1(tau_1) 1_{tau_1 < t ⇐ tau_2} + p_2(tau_2) 1_{tau_2 < t ⇐ tau_1} + p_{1,2}(tau_1, tau_2) 1_{t > tau_1 v tau_2}, where p_0 is F-predictable and the others are parametrised F-predictable processes.
- The Brownian motion W and the compensated default martingales M_i(t) = N_i(t) - integral_0^t (1 - N_i(s)) h_i(s) ds remain martingales under G, and (W, M_1, M_2) are mutually independent.
This decomposition is essential for reducing BSDEs on G to BSDEs on F: the jump components U_i are determined by the pre-default value process, and the Brownian component Z is unchanged. The theory originates from the work of Aksamit and Jeanblanc (2017, “Enlargement of Filtration with Finance in View”) and is applied extensively in credit risk modelling.