I'm doing my first research project on Stochastic Analysis and in order to prove something which is crucial, I need to prove the following claim:
LEMMA:
Denote by $(C_{0}[0,\,T],\,||\centerdot||_{\infty},\,\mathcal{B})$ the classical Wiener space of continuous paths defined on $[0,\,T]$ and started at zero, equipped the supremum norm $||\centerdot||_{\infty}$ and the corresponding Borel sigma algebra $\mathcal{B}$.
For an $m-$dimensional Brownian motion, consider the n-dimensional SDE: $ X_{t}=X_{0}+\int_{0}^{t}a(s,\,X_{s})ds+\int_{0}^{t}b(s,\,X_{s})dB_{s},\,0\leq t\leq T$
where the random variable $X_{0}$ and the coefficients $a,\,b$ have a very good regularity (as good as we want).
For each Brownian path $B_{.}:=\{B_{t}\}_{0\leq t\leq T}$ and realization $x$ of the random variable $X_{0}$, denote by $f(B_{.},\,x)$ the corresponding path of the pathwise unique strong solution to the above SDE, which is given by Ito's theorem.
Then the map $f$ which maps the product space $(C_{0}[0,\,T])^{m}\times\mathbb{R}^{n}$ to the product space $(C_{0}[0,\,T])^{n}$ is measurable with the product topologies.
. . . . .
I have searched a lot, but I have been unable to find something relevant. Working with Rough path theory, we find that this result is true if we replace the supremum norm with the a-Holer norm. However this does not help, because I want the preimages to be measurable with the Wiener probability measure, and again this is something I'm unable to prove or disprove. Can anyone help?
Thank you in advance.