I know that the Ito integral is defined in general for continuous semimartingales. But it can also be defined only for Ito processes. My question is if every process $X_t$ satisying a SDE of the form $dX_t=f(X_t)dt+g(X_t)dB_t$ where $f,g$ are "well-behaved" functions is a semimartingale. If so, why is that true?
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Let $dX_t = \sigma_t \, dB_t+b_t \, dt$ the solution of the SDE, i.e. $$X_t= X_0 + \int_0^t \sigma_s dB_s + \int_0^t b_s \, ds$$ Then
Hence you obtain the decomposition $X_t = M_t+A_t$ which implies that $X_t$ is a semimartingale.
Because the integration with respect to Brownian motion preserves the local martingale property and the first term gives you a finite variation process. This is precisely the canonical decomposition of a semimartingale, for which the finite variation property must hold only locally.
You can also define stochastic integrals for processes involving jumps!