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You can do what we always do and take logs and Itô's Lemma: $$\text{d}\ln(X_t)= \left( b(t)-\frac{1}{2}\sigma^2(t)\right)\text{d}t+\sigma(t)\text{d}B_t.$$ Then, by definition, $$\ln(X_t)=\ln(X_0)+\int_0^t\left( b(s)-\frac{1}{2}\sigma^2(s)\right)\text{d}s +\int_0^t \sigma(s)\text{d}B_s$$ or X_t=X_0\exp\left(\int_0^t\left( b(s)-\frac{1}{2}\sigma^2(s)\right)\...

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Just a few notes How to make sense of $\text dW_t$ is the entire point of stochastic calculus. It's far beyond the scope of any answer here. You should read some introductory lecture notes/books on stochastic calculus. You could start here. The idea: Riemann-Stieltjes integrals are of the form $\int_0^t f(s)\mathrm{d}g(s)$ and are well-defined if $f$ is ...

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An explicit reference could be helpful. It seems to me like an independence statement. For if $Y$ is independent of $\mathcal{F}_{0}$, then $\mathbb{E}[Y|\mathcal{F}_{0}]=\mathbb{E}[Y]$.

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