Stochastic integral ($\int HdX$)

The stochastic integral $\int HdX$ integrates a predictable process against a semimartingale. It extends pathwise integration against finite-variation processes and the Itô integral against local martingales.

For càdlàg semimartingales, the product rule uses left-limit integrands and quadratic covariation including jump products. For continuous semimartingales it reduces to the Itô product rule without left-limit notation. This rule permits an induction proof of the Itô formula for polynomial functions.

If uniformly bounded predictable processes $H_n$ converge pointwise to $H$, then their integrals against a fixed semimartingale converge to $H\cdot X$ uniformly on compact intervals in probability. Localized domination gives corresponding extensions. This continuity permits a functional monotone-class argument from elementary predictable integrands to all bounded predictable integrands.

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For a left-continuous locally bounded adapted process $H$ and a continuous semimartingale $X$, the elementary predictable left-endpoint approximations tend pointwise to $H$. Decompose $X$ into a continuous local martingale and a continuous finite-variation process. After localization, dominated convergence with respect to the variation measure handles the finite-variation integral. The Itô isometry and Doob L2 maximal inequality handle the martingale integral. The omitted partial last interval is bounded by the local bound on $H$ times the modulus of continuity of $X$. This proves uniform convergence on compacts in probability and is the grid-sum foundation of the Itô product rule.

For a finite stopping time $T$, use the shifted filtration ${\mathcal{G}}_s={\mathcal{F}}_{T+s}$. An martingale with a bounded $L^2$ norm shifts to the martingale $M_{T+s}-M_T$; a continuous bounded adapted integrand shifts to a predictable integrand. Their quadratic variation and integration intervals shift by subtracting the value at $T$. Both integrals have identical left-endpoint sums, and the Itô isometry passes the equality to the limit. The starting jump at $T$ is excluded.

If the entire integrand path is independent of the integrating Brownian motion, conditioning on that path gives a deterministic square-integrable integrand. The Itô isometry and Gaussian approximation by simple integrands show that its stochastic integral is conditionally centered normal with the stated variance. A strictly positive finite conditional variance implies that the unconditional distribution has no atoms.

For continuous semimartingales, the Stratonovich integral is defined by ${\int }_0^t{Y}_s\circ d{X}_s={\int }_0^t{Y}_{s}d{X}_s+\frac{1}{2}[Y,X{]}_t$, where the first term is an Itô integral and the correction is quadratic covariation. It obeys the ordinary chain rule and is the limit in probability of symmetric endpoint sums.

For a continuous semimartingale $X$ and a sufficiently smooth $f$, the Stratonovich integral obeys $f(X_t)-f(X_0)={\int }_0^t{f}^{\prime }(X_s)\circ d{X}_s$. In general $f^{\prime }(X)$ is a semimartingale, so the definition must allow this larger class of integrands.

The Itô integral ${\int }_0^t{H}_{s}d{W}_s$ integrates a predictable square-integrable process against Brownian motion. It is defined first for predictable step processes and then completed using the Itô isometry.

For deterministic square-integrable $h$, approximate by step functions. Their Itô integrals are sums of independent normal Brownian increments. The Itô isometry gives convergence in $L^2$, and the variances converge to $\int h^2$; convergence of characteristic functions proves the normal limit. Each linear combination of finitely many such integrals is another deterministic stochastic integral, so they are jointly Gaussian.

Deterministic Itô integrals are jointly centered Gaussian by approximation from step functions and the Itô isometry. Their covariance is the L2 inner product of their integrands. Orthonormal integrands therefore produce independent standard-normal coordinates, with independence of a countable family understood through all finite subfamilies.

For a bounded continuous adapted process $H$ and a Brownian motion $B$, ${\int }_t^{t+h}{H}_{s}d{B}_s/(B_{t+h}-B_t)$ converges in probability to $H_t$ as $h\downarrow 0$. The Itô isometry makes the error numerator small relative to $\sqrt{h}$; the normal distribution of the denominator controls its chance of being too close to zero.

Whenever the integrands are admissible,

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.

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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.

For a continuous local martingale $M$,

For a continuous local martingale and a locally bounded predictable process, stop at a localizing time, an integrand bound, a time bound, and level bounds for the centered martingale and its bracket. The stopped martingale is bounded and its bracket and integrand are deterministically bounded. The Itô isometry, applied to $1_F{1}_{(s,t]}H$ for $F\in {\mathcal{F}}_s$, gives the conditional second-moment identity. Expanding the square proves that $(H\cdot M{)}^2-H^2\cdot [M]$ is a martingale after stopping. Uniqueness of an increasing square compensator identifies the bracket, and increasing stopping times patch the identities into the displayed local result.