In case $d=1$ we are only trading $d+1=2$ assets $\bar{S}{t}=\left(S{t}^{(0)}, S_{t}^{(1)}\right)$ and the portfolio allocation reads $\bar{\xi}{t}=\left(\xi{t}^{(0)}, \xi_{t}^{(1)}\right)$. In this case, the self-financing condition means that:

• In the event of an increase in the stock position $\xi_{t}^{(1)}$, the corresponding cost of purchase $\left(\xi_{t+1}^{(1)}-\xi_{t}^{(1)}\right) S_{t}^{(1)}>0$ has to be deducted from the savings account value $\xi_{t}^{(0)} S_{t}^{(0)}$, which becomes updated as
$$\xi_{t+1}^{(0)} S_{t}^{(0)}=\xi_{t}^{(0)} S_{t}^{(0)}-\left(\xi_{t+1}^{(1)}-\xi_{t}^{(1)}\right) S_{t}^{(1)}$$
• In the event of a decrease in the stock position $\xi_{t}^{(1)}$, the corresponding sale profit $\left(\xi_{t}^{(1)}-\xi_{t+1}^{(1)}\right) S_{t}^{(1)}>0$ has to be added to the savings account value $\xi_{t}^{(0)} S_{t}^{(0)}$, which becomes updated as
$$\xi_{t+1}^{(0)} S_{t}^{(0)}=\xi_{t}^{(0)} S_{t}^{(0)}+\left(\xi_{t}^{(1)}-\xi_{t+1}^{(1)}\right) S_{t}^{(1)}$$Clearly, the chosen unit of time may not be the day and it can be replaced by weeks, hours, minutes, or fractions of seconds in high-frequency trading.

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