Given the standart stochastic differential equation setting (i.e. a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in > [0,\infty]},P)$ with complete filtration and ($\mathcal{F}_t$)-Brownian motion $B$), we consider $$X_t=X_0+\int_0^tb(s,X_s)ds+\int_0^t \sigma(s,X_s)dB_s$$ It is well known, that in the Lipschitz case (i.e. $\sigma(t,x),\ b(t,x)$ are continuous on $\mathbb{R}_+ \times \mathbb{R}^d$ and Lipschitz continuous in $x$ uniformly for $t\ge 0$) the solution exists uniquely for $\mathcal{F}_0$-measurable $X_0$.
My Question: Many sources use solutions to this SDE for initial conditions $X_0$, distributed according to some distribution (e.g. uniform or normal). How to overcome the following problem:
Can we even define a random variable $X_0$ distributed to a given distribution, which is $\mathcal{F}_0$-measurable?
- If one just augments the filtration by $\tilde{\mathcal{F}}_t=\sigma({\mathcal{F}_t,X_0})$, we would lose, that $B$ is a $(\mathcal{F}_t)$-Brownian motion.
- If $X_0$ is not $\mathcal{F}_0$-measurable, then the solution cannot be adapted.