self financing strategy

Question

how could one prove the following proposition from stochastic calculus applied to finance?

Proposition : Let $\Phi$ a trading strategy. Then, $\Phi$ is self financing if and only if $D(0,t)V_t(\Phi)=V_0(\Phi)+\int_0^t{\Phi_ud(D(0,u)S_u)}$

for the context, here are some some definitions :

$D(t,T)=e^{-\int_t^Tr(s)ds}$ with $r(s)$ is a stochastic process

$\Phi$ is a vector whose components $\phi^0$, $\phi^1$,...$\phi^K$ are locally bounded and predictable.

$V_t(\Phi)=\Phi_tS_t=\sum_{k=0}^{K}\Phi_t^kS_t^k$

the definition for a self financing strategy $\Phi$ is the following : $\Phi$ is self financing if $V_t(\Phi)=V_0(\Phi)+\int_0^t \Phi_udS_u$

Thank you in advance!

Answer 1

Note that \begin{align*} d(D(0, t)) = -r_t D(0, t) dt. \end{align*} Then, from the integral \begin{align*} D(0, t) V_t(\Phi) = V_0(\Phi) +\int_0^t\Phi_u d(D(0, u)S_u), \end{align*} we obtain that \begin{align*} D(0, t)dV_t(\Phi) - V_t(\Phi)r_t D(0, t) dt&=-\Phi_t S_t D(0, t) r_t dt +\Phi_t D(0, t) dS_t\\ &=-V_t(\Phi) D(0, t) r_t dt +\Phi_t D(0, t) dS_t. \end{align*} That is, \begin{align*} dV_t(\Phi) &= \Phi_t dS_t. \end{align*} Consequently, \begin{align*} V_t(\Phi) = V_0(\Phi) + \int_0^t\Phi_u dS_u. \end{align*}

For the other side, we go backward each step above.