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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
P.S : Here's the title of the article I'm currently studying
"Duality Theory for Portfolio Optimisation under Transaction Costs".
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
P.S : Here's the title of the article I'm currently studying
"Duality Theory for Portfolio Optimisation under Transaction Costs".