Let assume we have a portfolio with strategy described by $θ_t = \int_0^t S_udu$ (position in stock) and $ψ_t = -\int_0^t \frac{S_u^2}{B_u}du$ (position in bond). How to prove that this strategy is self financing? The underlying stock process is $dS_t = μS_tdt + σS_tdW_t$ and underlying process in bond market is $dB_t = rB_tdt$.
From the definition, we have $$V = θ_tS_t + ψ_tB_t$$ $$dV = θ_tdS_t + ψ_tdB_t$$.
To prove this, we firstly set $$V = \int_0^t S_udu*S_t -\int_0^t \frac{S_u^2}{B_u}du*B_t$$ and we need to prove that $dV$ of the above process will sattisfy equation $$dV = θ_tdS_t + ψ_tdB_t = (θ_tμS_t + ψ_trB_t)dt + θ_tS_tdW_t$$ that is $$dV = \left(μS_t\int_0^t S_udu - rB_t\int_0^t \frac{S_u^2}{B_u}du \right)dt + \left(S_t\int_0^t S_udu\right)dW_t$$ How to find $dV$ from $V$ described above?
By differentiating $V$, we have $$dV_t=\left(S_td\theta_t+\theta_tdS_t+d<S_t,\theta_t>\right)+\left(B_td\psi_t+\psi_tdB_t+d<B_t,\psi_t>\right)$$
We also have $$d\theta_t=S_tdt$$ $$d\psi_t=-\frac{S_t^2}{B_t}dt$$ and $$d<S_t,\theta_t>=0$$ $$d<B_t,\psi_t>=0$$ because $\theta$ and $\psi$ do not have a Brownian motion component.
Substituting these results to the first equation we have
$$dV_t=\left(S_t^2dt+\theta_tdS_t\right)+\left(-S_t^2dt+\psi_tdB_t\right)$$ Finally, $$dV_t=\theta_tdS_t+\psi_tdB_t$$