In this exercise, (F) is only assumed continuous on the right, and (Gleft(t^{-} ight))is the left-limit of

In this exercise, \(F\) is only assumed continuous on the right, and \(G\left(t^{-}\right)\)is the left-limit of \(G\) at point \(t\). Prove that the process \(\left(M_{t}, t \geq 0\right)\) defined as

\[M_{t}=D_{t}-\int_{0}^{\tau \wedge t} \frac{d F(s)}{G(s-)}=D_{t}-\int_{0}^{t}\left(1-D_{s-}\right) \frac{d F(s)}{G(s-)}\]

is a \(\mathbf{D}\)-martingale.