Brownian sheet as a martingale measure.

Question

Let $(\Omega, \mathcal{F}, \mathcal{F}_t, P)$ be a stochastic basis. Define a one-dimensional two-parameter and $\mathcal{F}_t$ adapted centered Gaussian process $W(x,t)$ with

$$E[W(x,t)W(y,s)]=\min(x,y)\min(s,t).$$

Is it possible to prove $W(x,t)$ as a martingale measure as defined following way viva J.B. Walsh (Chapter 2, page 287)

Definition. Let $(\mathcal{F}_t)$ be a right continuous filtration. A process $\{M_t(A), \mathcal{F}_t, t>0\}$ is a martingale measure if

• (i) $M_0(A)=0;$
• (ii) if $t>0$, $M_t$ is a $\sigma$-finite $L^{2}$-valued measure;
• (iii) $\{M_t(A), \mathcal{F}_t, t>0\}$ is a martingale.

$\rangle\rangle$ My question is can we prove that $W(x,t)$ is a martingale measure?

Thanks in advance.