Proving adaptedness of $ (\sum_{0 < s \leq t} \Delta X_s 1_{|\Delta X_s| > 1})_{0 \leq t \leq T}$ for RCLL adapted process $(X_t)_{0 \leq t\leq T}$.

Question

Let the probability space $(\Omega, \mathcal{F},P)$ be endowed with a filtration $\mathbb{F}=(\mathcal{F}_t)_{0 \leq t \leq T}$. We are given a RCLL adapted process $X=(X_t)_{0 \leq t\leq T}$. Let me write $\Delta X_s := X_s - \lim_{t \rightarrow s, t < s} X_t$. Defining

$$Z_t := \sum_{0 < s \leq t} \Delta X_t 1_{\{|\Delta X_t| > 1\}}$$

for $0 \leq t \leq T$, I want to show that $Z=(Z_t)_{0 \leq t \leq T}$ is adapted.

I do not now how to prove this. The only thing I see, is that for every $\omega \in \Omega$, the sum above is finite, since the jumps cannot exceed the value $1$ for infinitely many $s \in [0,T]$. So i immediately see, that the trajectories of $Z_t$ are also RCLL, but I do not see why it should be adapted to the filtration $\mathbb{F}$. Does anyone know how to prove adaptedness?

Thanks a lot in advance!

Answer 1

Lemma: Let $(X_t)_{t \geq 0}$ be an adapted cadlag process. If $f$ is a continuous function which has compact support in $\mathbb{R}^d \backslash \{0\}$, then $$\sum_{0 < s \leq t} f(\Delta X_s(\omega)) = \lim_{n \to \infty} \sum_{k=0}^{n-1} f(X_{t_{k+1,n}})(\omega)-X_{t_{k,n}}(\omega)) \tag{1}$$ where $t_{k,n} := \frac{k}{n} t$. In particular, $$\sum_{0 < s \leq t} f(\Delta X_s(\omega)) $$ is adapted.

Proof: Since $(1)$ is a pointwise statement, we fix $\omega \in \Omega$ throughout this proof. By assumption, $f$ has compact support in $\mathbb{R}^d \backslash \{0\}$, and therefore there is $\epsilon>0$ such that $B(0,\epsilon) \cap \text{spt} \, f = \emptyset$. Denote by $J=\{\tau_1,\ldots,\tau_N\}$ the finitely many jumps of size $|\cdot| \geq \epsilon$. We can choose $n \in \mathbb{N}$ sufficiently large such that $$\sharp (J \cap (t_{k,n},t_{k+1,n}]) \leq 1 \quad \text{for all $k=0,\ldots,n-1$}$$ and $$|X_{t_{k+1,n}}-X_{t_{k,n}}| < \epsilon \quad \text{for all $k$ such that $J \cap (t_{k,n},t_{k+1,n}] = \emptyset$}.$$ Hence, $$\sum_{k=0}^{n-1} f(X_{t_{k+1,n}}-X_{t_{k,n}}) = \sum_{\substack{0 \leq k \leq n-1 \\ J \cap (t_{k,n},t_{k+1,n}] \neq \emptyset}} f(X_{t_{k+1,n}}-X_{t_{k,n}}).$$ Using that $J$ is a finite set, $f$ is continuous and $t \mapsto X_t$ is cadlag, it is not difficult to see that the sum on the right-hand side converges to $\sum_{0 < s \leq t} f(\Delta X_s)$ as $n \to \infty$, and this proves the assertion.

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Now let $(f_n)_{n \in \mathbb{N}}$ be a sequence of continuous functions such that $f_n(x) \to x 1_{|x|>1}$ and $\text{spt} f_n \subseteq B(0,1/2)^c$ for all $n \geq 1$. By the above lemma,

$$\sum_{0 < s \leq t} f_n(\Delta X_s)$$

is adapted for each $n \in \mathbb{N}$. Since there are only finitely many jumps of size $|\cdot|>1/2$ we can let $n \to \infty$ to conclude that

$$\sum_{0 < s \leq t} \Delta X_s 1_{\{|\Delta X_s| > 1\}} = \lim_{n \to \infty} \sum_{0 < s \leq t} f_n(\Delta X_s)$$

is adapted.