Let
- $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space
- $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
- $\xi$ be an $\mathcal F_0$-measurable square-integrable random variable on $(\Omega,\mathcal A,\operatorname P)$
- $W$ be an $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$
- $b,\sigma:[0,\infty)\times\mathbb R\to\mathbb R$ be Borel measurable with $$|b(t,x)|^2+|\sigma(t,x)|^2\le C_1(1+|x|^2)\;\;\;\text{for all }t\ge0\text{ and }x\in\mathbb R\tag1$$ for some $C_1\ge0$ and $$|b(t,x)-b(t,y)|^2+|\sigma(t,x)-\sigma(t,y)|^2\le C_2|x-y|^2\;\;\;\text{for all }t\ge0\text{ and }x,y\in\mathbb R\tag2$$ for some $C_2\ge0$
Note that $$\mathcal C^T_b:=\left\{X:X\text{ is a continuous }\mathcal F\text{-adapted process on }(\Omega,\mathcal A,\operatorname P)\text{ with }\left\|\sup_{t\in[0,\:T]}|X_t|\right\|_{L^2(\operatorname P)}<\infty\right\}$$ equipped with $$\left\|X\right\|_{\mathcal C_b^T}:=\left\|\sup_{t\in[0,\:T]}|X_t|\right\|_{L^2(\operatorname P)}\;\;\;\text{for }X\in\mathcal C_b^T$$ is a complete semi-normed space for all $T>0$. Now, let $$\Xi_T(X):=\xi+\left(\int_0^tb(s,X_s)\:{\rm d}s\right)_{t\in[0,\:T]}+\left(\int_0^t\sigma(s,X_s)\:{\rm d}W_s\right)_{t\in[0,\:T]}\;\;\;\text{for }X\in C^T_b$$ for $T>0$. We can show that, for all $T>0$, there is a $n\in\mathbb N$ such that the $n$-fold composition of $\Xi_T$ is a contraction on $C_b^T$ and hence there is a unique $X^{(T)}\in\mathcal C_b^T$ with $$\Xi_T\left(X^{(T)}\right)=X^{(T)}\tag3.$$ Clearly, $$X_t:=X^{(N)}_t\;\;\;\text{for }t\in[0,N]\text{ and }N\in\mathbb N$$ is well-defined.
By definition, $X$ is $\mathcal F$-adapted. Assume $\xi$ is independent of $W$. Are we able to show that $X$ is even $(\sigma(\xi)\vee\mathcal F^W_t)_{t\ge0}$-adapted, where $\mathcal F^W$ denotes the filtration generated by $W$?
Fix $T>0$. The idea is to show that $(X_t)_{t\in[0,\:T]}$ is $(\sigma(\xi)\vee\mathcal F^W_t)_{t\in[0,\:T]}$-adapted. Let $Y^0:=\xi$ and $$Y^n:=\Xi_T(Y^{n-1})=\Xi_T^n(\xi)\;\;\;\text{for }n\in\mathbb N.$$ Now, $W$ is a $(\sigma(\xi)\vee\mathcal F^W_t)_{t\ge0}$-Brownian motion and we can show $$\left\|(X_t)_{t\in[0,\:T]}-Y^n\right\|_T\xrightarrow{n\to\infty}0\tag5.$$ The desired claim follows.
However, what I don't get is the following: Why didn't we work out the construction of $X$ with $\mathcal F$ replaced by $(\sigma(\xi)\vee\mathcal F^W_t)_{t\ge0}$ in the first place? Since the latter filtration is smaller, we would still be able to conclude $\mathcal F$-adaptedness of $X$. There must be something crucial I'm missing here ...
EDIT: As the result seems to be wrong, in general: How can we show it, if $\xi$ is non-random?