I am currently studying the existence and uniqueness of strong solutions of SDEs of type $$\left[\begin{array}{l} \, dX_t=\mu(t,X_t)\,\mathrm{d}t+\sigma(t,X_t)\,\mathrm{d}W_t\\ X_0=\xi\end{array}\right.$$ where certain Lipschitz and growth conditions on the coefficients are imposed (see e.g. Theorem 5.2.9 in [Karatzas,Shreve - Brownian Motion And Stochastic Calculus]). Another condition that is assumed to hold in many textbooks is that the initial variable $\xi$ should be independent of the natural filtration of the involved Brownian motion $W$. Can somebody tell me where I need this assumption in the prove of this result? (By 'proof', I mean the standard, Picard-Lindelöf-type approach.)
2026-04-03 04:01:33.1775188893
Strong solution SDE - independence of initial conditiion
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As I understand it, if the initial condition $\xi$ is dependent on the natural filtration $\{\mathcal{F}_{t}\}_{t>0}$ then $\xi \in \mathcal{F}_{s}$ for some $s>0.$ This implies that process $X$ is anticipating -- and not adapted to the natural filtration -- since in order to determine $X_{s-1}$ you need information from the future time $s$.