# Discounted self-financing portfolio still a self-financing portfolio?

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?