The Elementary Stochastic Integral Process is a Square-Integrable Martingale

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P)$ be a filtered probability space, let $(M,\rho)$ be an It^{o} integrator of intensity type with respect to $(\mathcal{F}_t)_{t\ge0}$, let $T>0$ be real, and let $H$ be a simple adapted process on $(0,T]$.

For $t\in(0,T]$, the restriction $(H_u)_{u\in(0,t]}$ is a simple adapted process on $(0,t]$; write

for its elementary stochastic integral, and set $I_0=0$.

Then the process $(\tilde I_s)_{s\ge0}$ defined by $\tilde I_s=I_{\min(s,T)}$ is a square-integrable martingale with respect to $(\mathcal{F}_s)_{s\ge0}$. In particular, for all real $0\le s\le t\le T$ and every $A\in\mathcal{F}_s$,