TITLE: Definition of self-financing strategy
QUESTION [9 upvotes]: Consider a portfolio of two assets with prices $S_t$, $B_t$ and holdings $\Delta_t$ and $E_t$ respectively. So the portfolio value is
$$ \Pi_t = \Delta_t S_t + E_t B_t$$
The portfolio is defined to be self-financing if we also have
$$ d \Pi_t = \Delta_t d S_t + E_t d B_t $$
I'm trying to figure out an intuitive derivation if this definition. Here's my attempt:
Let $m_t$ be the amount of money moved from your $B_t$ holdings to $S_t$ over a short period of time. Then we have
$$ d \Delta_t = \frac{m_t}{S_t} dt $$
$$ d E_t = - \frac{m_t}{B_t} dt $$
A linear combination of these equations is
$$ S_t d \Delta_t + B_t d E_t = 0 $$
So we can now apply the Ito product rule and then cancel the first terms:
$$ d \Pi_t  = d(\Delta_t S_t) + d(E_t B_t) \\
=  S_t d \Delta_t + \Delta_t dS_t + d \langle \Delta, S \rangle_t \\
+ B_t d E_t + E_t dB_t + d \langle E, B \rangle_t \\
= \Delta_t d S_t + E_t d B_t +  d \langle \Delta, S \rangle_t +d \langle E, B \rangle_t $$
So it remains only to show that the two cross variation terms cancel:
$$ d \langle \Delta, S \rangle_t + d \langle E, B \rangle_t = 0$$
but I don't have a lot of intuition dealing with quadratic variation.

REPLY [3 votes]: While Luigi Scorzato  gives the best possible explanation of a self-financing portfolio, the issue of the residual quadratic terms in variation is missing there.   I believe you can resolve it by claiming that $dΔ_tSt+Δ_tdSt+dE_tBt+E_tdB_t$ converges to $d(Δ_tS_t)+d(E_tB_t)$  in probability.  In the case of Ito processes, $dE_t$  and $dB_t$  are distributed around their means (which are $O(dt)$ ) with the standard deviations $O(\sqrt{dt})$. The quadratic variation terns have the means and the standard deviations $O(dt)$. That is, the probability that the ratio of quadratic variation term to a linear term tends to zero, that probability tends to 1.  For independent jump processes, the quadratic variation terms are non-zero only when both jumps are non-zero, and the probability of that compared to the probability of  a non-zero jump of either process tends to zero with $dt$. Of course, this is nowhere near of a rigorous argument, but you asked for an intuitive answer. I hope this helps.