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There is an amazing answer on mathematics stackexchange which defines what a self-financing strategy is—both in the discrete and continuous sense. Please check out this short answer to better understand my question.

I have a short follow-up question: Baxter and Rennie—while defining a self-financing strategy—requires the portfolio process to be previsible. The way the answer in the link arrives at the definition of self-financing, which is also the way I derive it, doesn't seem to require previsiblilty.

We are holding $(\Delta_t, E_t)$ over $t$ to $t+1$, which will be know to us at time $t$, even if the process is just adapted; previsiblilty is not required to know $(\Delta_t, E_t)$ at time $t$.

Extra for those who are interested: As always, things get muddier in the continuous time setting.

I think I have an intuition for what previsibility means when dealing with continuous processes: if a process $\phi$ is left-continuous, then we can know it's value $\phi(t)$ at a particular time $t$ with arbitrary precision by pushing the inputs close enough to $t$ from below, without actually having to reach $t$; this makes the value of $\phi(t)$ predictable with information upto but not including time $t$.

But it isn't clear to me why this previsibility is required while arriving at a sensible definition of a self-financing strategy —as the answer in the link succeeds to do.

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A self-financing strategy needs to be previsible (aka predictable) since at time $t$, you need to decide (with the information from $\mathcal{F}_t$) how much you want to be invested in the different assets at time $t+1$. So, you need to decide in advance which makes the trading strategy predictable.

Of course, the asset prices (and hence the value process of your strategy) remain adapted and are not previsible.

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  • $\begingroup$ I do understand the basic argument: we need to know at time $t$ itself what needs to be held over the next time-tick; if we don't know that, we simply won't be able to execute our strategy in time. My question is based on the technical definition of previsibility and how it applies to the notation in the answer I have linked. If the portfolio held over $t$ to $t+1$ is $(\Delta_t, E_t)$, then we only DO NOT require $\Delta$ and $E$ to be previsible - we just need them to be adapted so that we know their values at $t$, which can be held over $t$ to $t+1$. $\endgroup$ – Dhruv Gupta Aug 26 at 9:24
  • $\begingroup$ We could change the notation slightly, and use $(\Delta_{t+1}, E_{t+1})$ to denote the portfolio held over $t$ to $t+1$. If we do this, we would indeed require both processes to be previsible; but the concept of previsibility being dependent on notation would mean that it's not a 'universal' requirement—we could just stick to the notation under which the process is adapted and not previsible. $\endgroup$ – Dhruv Gupta Aug 26 at 10:04
  • $\begingroup$ It is not only a notational but also conceptional necessity that you have predictability. In the notation of the answer you have linked, the OP said that the portfolio with payoff $\Delta_1S_1+E_1B_1$ needs to be purchasable at time zero, i.e. $\Delta_1$ and $E_1$ need to be known at time zero, i.e. $(\Delta_t)$ and $(E_t)$ are previsible. Of course, you can change notation but it does not alter the concept that you need to know one period in advance what your position in the stock/bond/etc. is. $\endgroup$ – KeSchn Aug 26 at 14:02
  • $\begingroup$ I read the answer again just to recheck: $(\Delta_1, E_1)$ does not have to be purchasable at time zero. We only need the values of $\Delta_0$ and $E_0$ at time 0, because we are holding $(\Delta_0, E_0)$ over time zero to one. PS - I am really thankful for you taking out the time to answer my question and follow up on it. $\endgroup$ – Dhruv Gupta Aug 26 at 14:53
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When you think in continuous time, for continuous processes, the distinction does not matter much. But now consider a jump process. You want the strategy to be predictable because adapted won’t do- did you change your holding at the time of the jump? Predictable removes any ambiguity.

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    $\begingroup$ Yes, even I thought that previsibility is mostly a sort of 'regularity condition' in the continuous case. $\endgroup$ – Dhruv Gupta Aug 26 at 15:47

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