# Concatenation property of a set of semimartingales

Consider as in (1, Definition 2.1) a convex subset $\mathcal{X}_1$ of the set of semimartingales $\mathbb{S}$ satisfying the following properties:

1. $X_0=0$
2. $X_t\geq -1$ for all $t\geq 0$
3. for all bounded predictable strategies $H,G\geq 0$, $X,Y\in\mathcal{X}_1$ with $H_tG_t=0$ for all $t\geq 0$ and $Z=H\dot\ X + G\dot\ Y\geq -1$, it holds that $Z\in\mathcal{X}_1$
4. being closed in the Emery topology

This set represents some wealth processes, as stated in the remark after the definition: Let $S$ be a semi-martingale and set $\mathcal{X}_1:=\{\varphi\dot\ S|\varphi\text{ is }S\text{-integrable and } (\varphi\dot\ S)\geq -1)\}$

Condition 1 states, that the accumulated income of the portfolio at time zero is zero and condition 2 states, that the accumulated loss of the portfolio is bounded from below by $-1$. But what is the financial motivation of condition 3?

• It looks like Z is a strategy that holds variable amounts of security X or security Y, but not both at the same time (since $H_t G_t = 0$) and has limited liability in the sense that you cannot lose more than you put in (i.e. $Z_t \ge -1$ (-1 would be a loss of 100%). HTH – noob2 Jan 3 '17 at 21:06
• @noob2 But we don't invest any money according to point 1, do we? It would be good for me to first clarify what the meaning of $H\dot\ (\varphi\dot\ S)$ is. My understanding is, that $S$ is supposed to be for example the price of a stock dependent on the time, so $(\varphi\dot\ S)_t$ describes the profit for a trading strategy $\varphi$. But what is now $H\dot\ (\varphi\dot\ S)$? – peer Jan 3 '17 at 23:36