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In this wikipedia page, we consider the following financial market enter image description here

The formulas for the stocks are given here

enter image description here

And the gain process of a portfolio $\pi$ is defined such that

enter image description here

From what I understand, the first term of the formula of the gain process is due to the riskless asset, meaning that we consider undiscounted quantities (otherwise the riskless asset would not be considered in the expression of the gains I guess). But then the second term comes from the discounted formula of the risky assets. Hence it is a bit unclear for me what exactly are the computations behind this formula and whether we use discounted quantities or not.

I thought that the formula for gain process was roughly given by

\begin{equation} G(t) = \int_0^t \pi_r \frac{dS_r}{S_r} \end{equation} but this doesn't seem to correspond with Wikipedia.

I would be glad if someone could explain a bit more about it, particularly since it is hard to find any reference for this or gain processes in general. Thank you in advance.

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    $\begingroup$ You are right that gains process is not a term that you hear very much. It originated in the writings of Samuleson and Merton. And Wikipdia does not give any explanation. Have you looked at Merton, R.C.: Optimum consumption and portfolio rules in a continuous time model (1971) and was it any help? (TBH I haven't looked at that stuff for years). $\endgroup$
    – nbbo2
    May 13, 2023 at 11:32
  • $\begingroup$ @nbbo2 thanks I will have a look ! $\endgroup$
    – yrual
    May 13, 2023 at 12:42

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@nbbo2 Thank you very much for providing this useful reference, I had a look into it and I think I understand now :) For simplicity, let's take $A \equiv 0$, $\delta \equiv 0$ and $r(s) \equiv r$ (it is not very important anyway). Using undiscounted expression of the price process, one has that

\begin{align} dG(t) &= \sum_i \pi_i(t) \frac{dS_i(t)}{S_i(t)} \\ &= \sum_i \pi_i(t) b_i(t) dt + \sum_i \pi_i(t) \sum_j \sigma_{ij}dW_j(t) \\ &= \sum_i \pi_i(t) (b_i(t) - r) dt + \sum_i \pi_i(t)rdt + \sum_i \pi_i(t) \sum_j \sigma_{ij}dW_j(t) \\ \end{align} But now since $\frac{\pi_i(t)}{G(t)}$ is the proportion of wealth invested in asset i at time t, it is clear that $\sum_i \frac{\pi_i(t)}{G(t)} = 1$ and thus \begin{align} \sum_i \pi_i(t)rdt &= \sum_i \frac{\pi_i(t)}{G(t)} G(t)rdt \\ &= G(t)rdt \end{align} Therefore we finally obtain \begin{align} dG(t) = \sum \pi_i(t) (b_i(t) - r) dt + G(t)rdt + \sum_i \pi_i(t) \sum_j \sigma_{ij}dW_j(t) \end{align} which is what is obtained in the Wikipedia article.

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