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A process $H$ is progressively measurable when, for every $T$, its restriction to $[0,T]\times \Omega$ is measurable for $\mathcal{B}([0,T])\otimes {\mathcal{F}}_T$. Continuous adapted processes are progressively measurable, and a Borel function of such a process remains so. A progressively measurable square-integrable Brownian integrand has an equivalent predictable integrand for the Itô integral, up to equality for $dtd\mathbb{P}$.

The spot variance is the instantaneous squared noise coefficient $\sigma (t{)}^2$ of an Itô integral. It can be estimated by averaging the weighted realized variance over a shrinking time window. A Hölder continuity assumption bounds the smoothing bias.

A martingale is square integrable when its value at every time belongs to L2 space. This implies ordinary integrability by Cauchy-Schwarz inequality. The condition is separate from a uniform bound on these second moments over an infinite horizon. An Itô integral is square integrable on a finite interval when its predictable integrand has finite expected integrated square, by Itô isometry.

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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Stochastic analysis develops calculus and analytic tools for stochastic processes, especially when their sample paths are too irregular for ordinary differentiation. Brownian motion is the central example: its increments have size of order $\sqrt{\Delta t}$ rather than $\Delta t$, and its paths are almost surely nowhere differentiable.

The Itô integral is defined through an appropriate limit of sums whose integrands use information available before each increment. Its Itô isometry gives $\mathbb{E}({\int }_0^t{H}_{s}d{B}_s{)}^2=\mathbb{E}{\int }_0^t{H}_s^{2}ds$ for square-integrable adapted integrands. The resulting Itô formula includes a second-derivative correction:Taking expectations recovers $\mathbb{E}{B}_t^2=t$. Ordinary calculus would miss the extra term and predict the wrong variance. This is why stochastic calculus is needed for stochastic differential equations and their stochastic partial differential equation counterparts, rather than treating random trajectories as differentiable curves.^[ref]

Stochastic calculus studies integration and change of variables for stochastic processes such as Brownian motion and semimartingales. A Brownian motion path is continuous. The nowhere differentiability of Brownian motion prevents treating its increment $dW$ as an ordinary velocity times $dt$.

The crucial replacement is quadratic variation: $[W{]}_t=t$. The Itô integral uses information available before each increment. Consequently the Itô formula gives $d(W_t^2)=2W_{t}d{W}_t+dt$, and henceThe Stratonovich integral absorbs half of the quadratic covariation into its definition and gives ${\int }_0^t{W}_s\circ d{W}_s=W_t^2/2$. These conventions describe different limiting sums and must be specified in a stochastic differential equation.

Stochastic differential equations model the combination of drift and fluctuating forces in Langevin dynamics, diffusion and mathematical finance. Martingales control their conditional means, while the Feynman-Kac formula translates suitable expectations into partial differential equations.

A stochastic convolution propagates random forcing through a linear evolution. Let $S(t)$ be a strongly continuous semigroup on a Hilbert space $H$, let $W$ be a Cylindrical Wiener process on a Hilbert space $U$, and let $Q:U\to H$ be a bounded linear operator. The expressionis an Itô integral whenever its integrand is square integrable in the Hilbert-Schmidt operator norm. In particular,This Itô isometry tests whether the accumulated forcing is actually $H$-valued. The factor $S(t-s)$ evolves the contribution arriving at time $s$ through the remaining time $t-s$. For $du=Audt+Qd{W}_t$ with deterministic initial value $u_0$, the mild solution is $u(t)=S(t)u_0+Z(t)$.^[ref]

For $S(t)=e^{-at}$, $Q=\sigma$ and real Brownian motion $B$, with $a>0$,This is the zero-initial-value Ornstein-Uhlenbeck process: damping limits the variance despite continued forcing. With $a=0$, the limit is ${\sigma }^{2}t$ instead. These formulas follow by integrating ${\sigma }^2{e}^{-2a(t-s)}$ in the Itô isometry.

The word convolution describes the dependence on the time difference $t-s$. It does not mean an ordinary integral against a differentiable noise function; Brownian motion requires a stochastic integral.

A stochastic differential equation specifies a stochastic process through a combination of ordinary drift and random increments. In the scalar Itô integral convention, the notationmeans the stochastic integral equation $X_t=X_0+{\int }_0^{t}b(s,X_s)ds+{\int }_0^t\sigma (s,X_s)d{W}_s$, with the required measurability and local integrability. The driver $W$ is a Brownian motion; $b$ is the drift coefficient and $\sigma$ the diffusion amplitude.

For example, the Ornstein-Uhlenbeck process satisfies $d{X}_t=-\lambda X_{t}dt+\sigma d{W}_t$: linear restoring drift competes with persistent noise. A strong solution of a stochastic differential equation uses a prescribed driver, whereas a weak solution of a stochastic differential equation may construct the probability space and driver. Pathwise uniqueness, uniqueness in law and the lifetime of a maximal local solution of a stochastic differential equation are separate questions. If the equation uses a Stratonovich integral, its convention changes the drift conversion and must be stated.

A stochastic Fubini theorem permits interchange of a parameter integral with a stochastic integral. One sufficient version uses a measure space $(U,\mu )$ with $\mu (U)<\infty$ and a function $H(s,\omega ,u)$ measurable for the product of the predictable sigma-algebra and the parameter sigma-algebra and satisfiesThen both sides below exist in L2 space and are equal:For elementary parameter functions the identity is linearity of the Itô integral. Approximation in the displayed squared norm, the Itô isometry and the Cauchy-Schwarz inequality pass to the limit. Finite parameter measure is part of this sufficient version; more general versions require different integrability bounds. For example, inserting the deterministic indicator $1_{\{s\le u\}}$ on $[0,T{]}^2$ gives ${\int }_0^T{W}_{u}du={\int }_0^T(T-s)d{W}_s$.

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.

Fix a common Brownian motion $W$ and a finite horizon $T$. Let positive zero-rate stock prices satisfywith predictable processes $\sigma ,{\sigma }^{\prime }$ bounded in absolute value by a deterministic constant. Their stochastic exponentials are martingales with finite second moments.

If $S_T=a{S}_T^{\prime }$ for a deterministic constant $a$, equality of the expected values forces $a=1$. The martingale identity $S_t=\mathbb{E}[S_T\mid {\mathcal{F}}_t]$ then gives $S_t=S_t^{\prime }$ at each $t$. Continuity of the paths makes this equality hold simultaneously for all times outside a single null set.

Subtract the two Itô integrals. The Itô isometry givesSince $S_t>0$, the coefficients are equal almost everywhere for the product measure $dt\otimes \mathbb{P}$. The common driving Brownian motion is part of the statement.

Perfect positive correlation gives an affine terminal relation that may have a nonzero intercept. It does not supply the proportionality used here; the terminal correlation does not identify adapted stock volatility example demonstrates that distinction.

For observations of an Itô integral on a partition of an interval, the statistic ${\sum }_{i}g(t_{i-1})(X_{t_i}-X_{t_{i-1}}{)}^2$ estimates $\int g(t)\sigma (t{)}^{2}dt$. With a deterministic noise coefficient the squared increments are independent, and the fluctuation has variance $2{\sum }_{i}g(t_{i-1}{)}^2({\int }_{t_{i-1}}^{t_i}{\sigma }^2{)}^2$.