Discrete time

Assume we are given a discrete filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t=0}^{T},P)}$, and let ${\displaystyle K_{t}}$ be the solvency cone (with or without transaction costs) at time t for the market. Denote by ${\displaystyle L_{d}^{p}(K_{t})=\{X\in L_{d}^{p}({\mathcal {F}}_{T}):X\in K_{t}\;P-a.s.\}}$. Then a portfolio ${\displaystyle (H_{t})_{t=0}^{T}}$ (in physical units, i.e. the number of each stock) is self-financing (with trading on a finite set of times only) if

for all ${\displaystyle t\in \{0,1,\dots ,T\}}$ we have that ${\displaystyle H_{t}-H_{t-1}\in -K_{t}\;P-a.s.}$ with the convention that ${\displaystyle H_{-1}=0}$.^[1]

If we are only concerned with the set that the portfolio can be at some future time then we can say that ${\displaystyle H_{\tau }\in -K_{0}-\sum _{k=1}^{\tau }L_{d}^{p}(K_{k})}$.

If there are transaction costs then only discrete trading should be considered, and in continuous time then the above calculations should be taken to the limit such that ${\displaystyle \Delta t\to 0}$.

Continuous time

Let ${\displaystyle S=(S_{t})_{t\geq 0}}$ be a d-dimensional semimartingale frictionless market market and ${\displaystyle h=(h_{t})_{t\geq 0}}$ a d-dimensional predictable stochastic process such that the stochastic integrals ${\displaystyle h^{i}\cdot S^{i}}$ exist ${\displaystyle \forall \,i=1,\dots ,d}$. The process ${\displaystyle h_{t}^{i}}$ denote the number of shares of stock number ${\displaystyle i}$ in the portfolio at time ${\displaystyle t}$, and ${\displaystyle S_{t}^{i}}$ the price of stock number ${\displaystyle i}$. Denote the value process of the trading strategy ${\displaystyle h}$ by

${\displaystyle V_{t}=\sum _{i=1}^{n}h_{t}^{i}S_{t}^{i}.}$

Then the portfolio/the trading strategy ${\displaystyle h=\left((h_{t}^{1},\dots ,h_{t}^{d})\right)_{t}}$ is called self-financing if

${\displaystyle V_{t}=\sum _{i=1}^{n}\left\{h_{0}^{i}S_{0}^{i}+\int _{0}^{t}h_{u}^{i}\mathrm {d} S_{u}^{i}\right\}=h_{0}\cdot S_{0}+\int _{0}^{t}h_{u}\cdot \mathrm {d} S_{u}}$.^[2]

The term ${\displaystyle h_{0}\cdot S_{0}}$ corresponds to the initial wealth of the portfolio, while ${\displaystyle \int _{0}^{t}h_{u}\cdot \mathrm {d} S_{u}}$ is the cumulated gain or loss from trading up to time ${\displaystyle t}$. This means in particular that there have been no infusion of money in or withdrawal of money from the portfolio.