Set of simple predictable processes is a vector space

Question

I have a question, which is probably very easy for you to answer.

How can I show that the set of simple predictable processes a vector space is? It's clear that I only have to show that the sum of two such processes again a simple predictable process is, which seems to be easy to show but is quite tricky in my point of view cause I'm not sure how to build a suitable sequence of stopping times.

Can anyone help me with this?

Answer 1

Let $(X_t)_{t \geq 0}$ and $(Y_t)_{t \geq 0}$ simple predictable processes, i.e.

$$\begin{align*} X_t &= 1_{\{t=0\}} A_0 + \sum_{k=1}^m 1_{\{S_k<t \leq T_k\}} A_k \\ Y_t &= 1_{\{t=0\}} B_0 + \sum_{j=1}^n 1_{\{U_j<t \leq V_j\}} B_j \end{align*}$$

where $S_k<T_k$, $U_k<V_k$ are stopping times and $A_k$ are $\mathcal{F}_{S_k}$, $B_k$ are $\mathcal{F}_{U_k}$-measurable bounded random variables. Obviously, this implies

$$X_t + Y_t = 1_{\{t=0\}} (A_0+B_0) + \sum_{k=1}^m 1_{\{S_k<t \leq T_k\}} A_k+ \sum_{j=1}^n 1_{\{U_k<t \leq V_k\}} B_k.$$

Note that this already shows that $(X_t+Y_t)_{t \geq 0}$ is a simple process. In fact, we can choose

$$\begin{align*} P_k &:= \begin{cases} S_k, & k=1,\ldots,m \\ U_k, & k=m+1,\ldots,m+n \end{cases} \\ R_k &:= \begin{cases} T_k, & k=1,\ldots,m \\ V_k, & k=m+1,\ldots,m+n \end{cases} \\ C_k &:= \begin{cases} A_k & k=1,\ldots,m \\ B_k & k=m+1,\ldots,m+n \end{cases}\end{align*}$$

and $C_0 := A_0+B_0$. Then

$$X_t+Y_t = 1_{\{t=0\}} C_0 + \sum_{k=1}^{m+n} C_k 1_{\{P_k<t \leq R_k\}}.$$

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Edit: To make the sequence of stopping times increasing, we can argue as follows: Let

$$X_t = \sum_{k=1}^m 1_{\{S_k<t \leq S_{k+1}\}} A_k$$

and

$$Y_t = \sum_{j=1}^n 1_{\{U_j<t \leq U_{j+1}\}} B_j.$$

For $k \in \{1,\ldots,m\}$ and $j \in \{1,\ldots,n\}$ set

$$V_{k,j} := \min\{S_k, U_j\}.$$ Note that $V_{k,j} \leq V_{k',j'}$ for $k \leq k'$ and $j \leq j'$. Define iteratively (with $T_0:=0$)

$$T_i(\omega) := \inf\left\{ V_{k,j}(\omega); V_{k,j}(\omega)>T_{i-1}(\omega), k \in \{1,\ldots,m\}, j \in \{1,\ldots,n\} \right\}$$

for $i \leq mn$. This defines a sequence of non-decreasing stopping times and we can write

$$X_t+ Y_t = \sum_{i=1}^{mn} C_i 1_{\{T_i<t \leq T_{i+1}\}}$$

where

$$C_i := (X_t+Y_t) 1_{\{T_i<t \leq T_{i+1}\}} = \sum_{k=1}^m A_k 1_{\{S_k<t \leq S_{k+1}\}} 1_{\{T_i<t \leq T_{i+1}\}} + \sum_{j=1}^n B_j 1_{\{U_j<t \leq U_{j+1}\}} 1_{\{T_i<t \leq T_{i+1}\}}.$$