# Adaptedness of Solution to SPDE

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:

1. Each $$X_n$$ is adapted
2. The limit, $$X$$, is also adapted
• Is it possible that you forgot an integral in the definition of $\mathcal{K}$? Dec 22, 2018 at 9:53
• do you mean against the stochastic term? I had meant that single integral to be against both the $dt$ and the $dW$. I will edit if you think that would be helpful. Dec 22, 2018 at 17:31
• I think it follows inductively. $X_0$ is measurable as a constant. The stochastic integral is a martingale and hence $X_n$ is adapted. Dec 22, 2018 at 23:57
• I can see the merit to an inductive argument, but I could use some tips on filling in the details. Dec 28, 2018 at 14:31