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I'm having a hard time trying to understand a formula about self financing strategy trading. Let's suppose you have two assets, $\phi=(\phi_0,\phi_1)$ is the vector that represents the quantity you have for each one of them and $S=(1,S_1)$ is the price (the first asset isn't risky so we suppose it has a constant price).

In the discrete model, without transaction cost : The self financing hypotheses would mean having $\phi_0(t)+\phi_1(t)S_1(t)=\phi_0(t+1)+\phi_1(t+1)S_1(t)$ which means that when you are in the moment $t$, you see the prices and you take decisions to build a portfolio for the moment $t+1$ so you don't lose money and you don't need money from the outside.

I figured that for discrete time with transaction cost $\lambda$, this would become (correct me if I'm wrong) : $\phi_0(t)+\phi_1(t)(1-\lambda)S_1(t)=\phi_0(t+1)+\phi_1(t+1)S_1(t)$

Now I'm reading this article with a continuous model and transaction cost and it says that a trading strategy would be self financing if we have : $d\phi^{0}_t=(1-\lambda)S^{1}_d\phi^{1,\downarrow}_t-S^{1}_td\phi^{1,\uparrow}_t$ where $\phi^{1}_t=\phi^{1,\uparrow}_t-\phi^{1,\downarrow}_t$ using the Jordan-Hahn decomposition into two no-decreasing function. Now I don't know much about this decomposition, tried to look it up but didn't understand how is it relevant here. I can get intuitively the meaning in the discrete model but I can't establish something between the continuous model and the discrete one.

Can someone give me some help ?

Thanks :)

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