Assume a self-financing portfolio $V_{t}=\theta_{t}^{0}S_{t}^{0}+\theta_{t}S_{t}$ with $S_{t}^{0}$ the value of the non-risky asset at time $t$ and $\theta_{t}^{0}$ the amount of shares of the non-risky asset at time $t$. Similarly, $S_{t}$ and $\theta_{t}$ are the value and the amount of shares of the risky asset at time $t$. We use a tilde, to indicate the discounted form, i.e. $\tilde{V}_{t}$ is the discounted portfolio at time $t$.

I want to proof that for a self-financing portfolio, $V_{t}$, the following holds $$\tilde{V}_{t}=V_{0}+\int^{t}_{0}\theta_{u}dS_{u}$$ My attempt was as following, however throughout this proof I am not sure if my assumptions are justified:

First, we have $\theta^{0}_{t}S^{0}_{t}=\theta^{0}_{0}S^{0}_{0}+\int^{t}_{0}\theta_{u}^{0}dS_{u}^{0}$ and $\theta_{t}S_{t}=\theta_{0}S_{0}+\int_{t}\theta_{u}dS_{u}$ (1). Second, if we discount $V_{0}$ we find $\tilde{V}_{0}=V_{0}$ (2). Third, $S^{0}_{t}=S^{0}_{0}e^{rt}$, thus $\tilde{S}^{0}_{t}=e^{-rt}S^{0}_{0}e^{rt}$ for any risk-free rate $r>0$, which implies that $d\tilde{S}^{0}_{t}=0$ for any $t$ (3).

Now, by (1) $V_{t}=\theta_{t}^{0}S_{t}^{0}+\theta_{t}S_{t}=\theta_{0}^{0}S_{0}^{0}+\theta_{0}S_{0}+\int^{t}_{0}\theta_{u}^{0}dS_{u}^{0}+\int^{t}_{0}\theta_{u}dS_{u}$. If we discount our portfolio we expect to maintain the dynamics, thus by (2) and (3) $\tilde{V}_{t}=\theta_{t}^{0}\tilde{S}_{t}^{0}+\theta_{t}\tilde{S}_{t}=\theta_{0}^{0}\tilde{S}_{0}^{0}+\theta_{0}\tilde{S}_{0}+\int^{t}_{0}\theta_{u}^{0}d\tilde{S}_{u}^{0}+\int^{t}_{0}\theta_{u}d\tilde{S}_{u}=V_{0}+\int^{t}_{0}\theta_{u}^{0}d\tilde{S}_{u}^{0}+\int^{t}_{0}\theta_{u}d\tilde{S}_{u}=V_{0}+\int^{t}_{0}\theta_{u}d\tilde{S}_{u}$.

Are there problems with the assumption that the dynamics of a discounted self-financing portfolio stays exactly the same, or with other assumptions in this proof?


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