I'm trying to understand existence of solutions to SDPEs of the form $$ dX = AXdt + F(t,X)dt + B(t,X)dW $$ from the Hilbert space point of view, following Da Prato & Zabcyzk. They rely on the standard fixe point method, showing that if the interval $[0,T]$, is sufficiently small, the mapping $$ \mathcal{K}(Y)(t)= S(t)\xi + \int_0^t S(t-s)F(t,Y)dt + S(t-s)B(t,Y)dW $$ has a fixed point in the space $$ C([0,T];L^2(\Omega, \mathbb{P};H)) $$ equipped with the norm $$ \|X\|_T^2= \sup_{0\leq t\leq T}\mathbb{E}\|X(t)\|_H^2. $$ The existence argument is standard, using the sequence $X_0 =\xi$, $X_{n+1} = \mathcal{K}(X_n)$.
What I am curious/unsure about is the adaptedness, both of the sequence and the limiting solution. Assuming $\xi$ is $\mathcal{F}_0$, how can I see that:
- Each $X_n$ is adapted
- The limit, $X$, is also adapted