Itô isometry

For a square-integrable predictable process $H$ and Brownian motion $W$,More generally, for a continuous $L^2$-bounded martingale $M$,

For a zero-starting L2-bounded continuous martingale $M$, the space $L^2(M)$ consists of predictable $H$ with $\mathbb{E}{\int }_0^{\infty }{H}_s^{2}d[M{]}_s<\infty$, identifying processes equal almost everywhere for the quadratic-variation measure. The finite-horizon version stops the integral at $T$. The Itô integral is the isometric extension from elementary predictable integrands.

// Target: probability-and-statistics.bigb

The Cauchy-Schwarz inequality and Jensen inequality give the displayed estimate, with $C=(\mathbb{E}|N{|}^{-1/2}{)}^{1/2}<\infty$. No independence between $R_{\epsilon }$ and $B_{\epsilon }$ is required. If $R_{\epsilon }={\int }_0^{\epsilon }(H_s-H_0)d{B}_s$ for a bounded continuous adapted process, the Itô isometry and the dominated convergence theorem make the right side tend to zero. Time shifting proves recovery of $H_t$ in probability from the ratio of a short stochastic integral to its Brownian increment.

For an L2-bounded continuous martingale and ordered possibly infinite stopping times $\sigma \le \tau$, interpret the terminal stopped value as $M_{\infty }$. The Doob L2 maximal inequality bounds the expected squared path supremum. Localization of $M^2-[M]$ then gives $\mathbb{E}[M{]}_{\infty }<\infty$, so this square-minus-bracket local martingale has integrable supremum and is uniformly integrable. Apply the optional stopping theorem to it and to $M$. Expanding $(M_{\tau }-M_{\sigma }{)}^2$ makes the cross term vanish conditionally, proving the identity. Bounded ${\mathcal{F}}_{\sigma }$-measurable factors can be multiplied into either side.

For a L2-bounded continuous martingale $M$, define a finite measure on the predictable sigma-algebra by ${\nu }_M(A)=\mathbb{E}{\int }_0^{\infty }{1}_{A}d[M]$. Its total mass is $\mathbb{E}[M{]}_{\infty }=\mathbb{E}{M}_{\infty }^2-\mathbb{E}{M}_0^2$. The Lebesgue space $L^2({\nu }_M)$ identifies predictable processes that agree ${\nu }_M$-almost everywhere. Integration against $M$ is an isometry from this space into L2-bounded continuous martingales starting at zero by the Itô isometry. This gives the correct integrand space even when the quadratic variation is random or is not absolutely continuous in time.