How to show that the stochastic integral of a jump process w.r.t. Brownian motion is a local martingale by using this special localizing sequence

Question

Suppose that $Y$ is a pure jump process with $N_t$ jumps in $(0,t]$ and $E[N_t]<\infty$. Denote the jump times by $T_i$. Let $W$ be a Brownian motion. If $T_0=0$, then
\begin{equation}
M_t=\int_0^t Y_s\,dW_s=\sum_{i=0}^{N_t-1} Y_{T_i}(W_{T_{i+1}}-W_{T_i})+Y_t(W_t-W_{T_{N_t}}).
\end{equation}
The process $M$ is continuous. I want to show that $M$ is a local martingale by using the sequence $\tau_n=\inf\{t: |M_t|>n\}$. That is, I have to show that $\{M_{\tau_n\wedge t}\}$ is a martingale. I have read that one can use the above sequence of stopping times for continuous local martingales. But why or how?

Answer 1

I guess you could use the sequence $\tau_n=\inf\{t\,:\, N_t=n\}$ and then Corollary 6.14 (or 7.14 depending on the edition, it is called "martingale transforms") from Kallenberg's Foundations of Modern Probability yields the result.

The corollary states that $Y_t(W_t-W_{T_{N_i}})$ and $$\sum_{i=0}^{n}Y_{T_i}(W_{T_{i+1}}-W_{T_i})$$ are martingales for all (fixed / deterministic) $n\in \mathbb N$.

But in your case, you need that $$\sum_{i=0}^{N_{t-1}}Y_{T_i}(W_{T_{i+1}}-W_{T_i})$$ is a local martingale for random $N_{t-1}$. This is why you need to stop the process. With the stopping time suggested above, you obtain $$M_t^{T^n}=\sum_{i=0}^{n-1}Y_{T_i}(W^{T_{i+1}}_t-W^{T_i}_t)$$ which is a martingale.

Of course, you could also use the well known fact, that a stochastic integral with respect to continuous local martingale is always a local martingale. Then you still need to show, that stopping at $\tau_n=\inf\{t\,;\,|M_t|>n\}$ yields a uniformly integrable martingale. Because of the continuity of the process $M_t$, you get that $M_t^{\tau_n}$ is bounded by $n$ and any bounded local martingale is a uniformaly integrable martingale. Hence the result follows.