Is next question right? And why?

Question

Let $X(t)=(X^1(t),...X^d(t)$ be a d-dimensional ($\mathcal{F_t}$)-semi-martingale such that

(1) $M^i(t) =X^i(t)-X^i(0) \in M_2^{c\ loc}$

and

(2)$\langle M^i,M^j\rangle(t)$=$\delta_{ij}t$ i,j=1,2...,d.

(Where $M_2^{c\ loc}$ is the set of continuous (locally) square integrable martingale,and $\langle\cdot\rangle$ is the quadratic (co)variation.)

On Ikeda-Watanabe's Theorem II-6.1. the authors state that condition $(2)$ implies that $M^i$ is continuous (globally) square integrable martingale. How to see this?

Answer 1

I think (please correct me if I am wrong) that you can see it by using the Doob-Meyer decomposition.

Let $M\in M_2^{c,loc}$ hence defining a localizing sequence of stopping times $\{\tau_k\}$ we have $(M_{t\wedge\tau_k})$ is in $M_2^c$ for all $k\geq 0$.

Then we use Doob-Meyer decomposition to see that $$M_{t\wedge\tau_k}^2= A_{t\wedge\tau_n}+\langle M\rangle_{t\wedge\tau_n}.$$

Then we have that $$E\left(M_{t\wedge\tau_k}^2\right)=E(\langle M\rangle_{t\wedge\tau_n})=E(t\wedge\tau_k)\leq t.$$

By letting $k\to \infty$ you conclude that

$$E(M_t^2)<\infty.$$

Answer 2

From Protter (2003) Corrolary II.6.3 you get the statement, that a local martingale $M^i$ with $\forall t\geq0:E\{\langle M^i,M^i\rangle(t)\}<\infty$ is martingale with $\forall t\geq0: E\{(M^i_t)^2\}<\infty$.