# Self-financing portfolio under $Q$-dynamics

I know what given stocks $S_1, ..., S_N$ with SDE's, a portfolio must have a particular value dynamics shape (which depends on the dynamics of $S_1,...,S_N$), if that portfolio is to be self-financing.

However, does this have to hold under every probability measure? Usually we are given dynamics of stocks in P-world, but we also later study the martingale measure $Q$, and we know that the stocks have some known $Q$-dynamics as well.

So, given a portfolio, do the same restrictions on its value dynamics apply in $Q$-world?

Infinitesimally, a self-financing portfolio is a portfolio

$$V_t = \sum_{i=1}^N \phi_i(t) S_{t,i}$$

whose value changes only because the values of the assets in which it invests change (no in/out exogenous cash flows, hence its name), i.e.

$$dV_t = \sum_{i=1}^N \phi_i(t) dS_{t,i}$$

Whether these price changes $dS_{t,i}$ are described under this or that probability measures does not matter.

• The Self Financing condition reminds me of the Conservation of Mass in Chemistry or Physics. It is there to make sure that the mathematics describes a physically or financially realistic situation. It has to be true in every scenario that occurs, hence it does not really matter what kinds of probabilities you assign to the different scenarios. – Alex C Mar 4 '17 at 22:22