Preservation of Martingale property

Question

Can someone help me to prove this? If possible I'd like the prove can avoid the use of local martingale.

Prove the Ito integral $\int_0^T \Delta_t(\omega) dW_t(\omega)$ is a martingale if $E[\int_0^T \Delta^2_t(\omega)dt]<+\infty$.

Answer 1

For simple functions $\Delta$, i.e. functions of the form

$$\Delta(s,\omega) = \sum_{j=1}^n \varphi_j(\omega) \cdot 1_{[s_{j-1},s_j)}(s) \tag{1}$$

where $\varphi_j \in L^{\infty}(\mathcal{F}_{s_{j-1}})$ and $0=s_0<\ldots<s_n \leq T$, the Itô integral is defined as

$$\int_0^t \Delta_s dW_s := \sum_{j=1}^n \varphi_j \cdot (B_{s_j \wedge t}-B_{s_{j-1} \wedge t})$$

A straight-forward calculation shows that this process is a martingale. Moreover,

$$ \mathbb{E} \left[ \left( \int_0^T \Delta_t \, dW_t \right)^2 \right] = \mathbb{E} \left(\int_0^T \Delta_t^2 \, dt \right)$$

i.e. Itô's isometry holds for all simple functions of the form $(1)$. This isometry allows us to extend the Itô integral to all progressively measurable functions $\Delta$ satisfying

$$\mathbb{E} \left(\int_0^T \Delta_t^2 \, dt\right) < \infty$$

since any of these functions can be approximated (with respect to the product measure $\mathbb{P} \otimes \lambda_T$) in $L^2$ by simple functions $(\Delta_n)_n$ of the form $(1)$: The isometry shows that

$$\int_0^t \Delta_s \, dW_s := L^2(\mathbb{P})-\lim_{n \to \infty} \int_0^t \Delta_n(s) \, dW_s \tag{2}$$

exists and is well-defined. Since martingales are preserved under $L^2$-limits, we conclude that the integral defined in $(2)$ is a martingale.