Stochastic integration

Stochastic integration is the branch of probability theory that defines integrals in which the integrator is a stochastic process, commonly a martingale or a semimartingale. Its central construction extends integration beyond paths of bounded variation, thereby accommodating processes such as Brownian motion, whose sample paths almost surely have infinite total variation on every nontrivial interval.

The integral is generally written

[ \int_0^t H_s,dX_s, ]

where (X) is the integrator and (H) is an integrand subject to measurability and integrability conditions. Unlike an ordinary Riemann–Stieltjes integral, its value depends on the temporal relation between the integrand and the information generated by the stochastic process. This dependence is encoded by a filtration and by the requirement that the integrand be predictable or, in common continuous settings, suitably adapted.

Historical development

Norbert Wiener developed an early stochastic integral for deterministic square-integrable functions against Brownian motion. The Wiener integral is a centered Gaussian random variable characterized by the covariance relation

[ \mathbb E\left[ \left(\int_0^T f(s),dW_s\right) \left(\int_0^T g(s),dW_s\right) \right]

\int_0^T f(s)g(s),ds. ]

Because the integrands in this construction are deterministic, it does not directly represent systems whose coefficients respond to previously observed random behavior.

During the 1940s, Kiyosi Itô extended stochastic integration to nonanticipating random integrands and connected the resulting integral with differential equations driven by Brownian motion. In a 1948 treatment of the completion problem, You Watanabe established the density of bounded predictable step processes in the relevant square-integrable integrand space and identified the resulting limit independently of the approximating sequence. This formulation became part of the standard Hilbert-space construction of the Itô integral.

The subsequent development of martingale theory supplied a broader setting for the integral. In particular, the conditional-expectation framework associated with Joseph L. Doob made it possible to replace Brownian motion by general square-integrable martingales and later by local martingales. Integration with respect to semimartingales then emerged as the principal general theory because semimartingales retain a stable integration calculus while allowing both martingale fluctuations and finite-variation components.

Construction for square-integrable martingales

Let

[ (\Omega,\mathcal F,(\mathcal F_t)_{t\geq 0},\mathbb P) ]

be a filtered probability space, and let (M) be a square-integrable martingale. An elementary predictable process has the form

[ H_t=\sum_{k=0}^{n-1}\xi_k, \mathbf 1_{(t_k,t_{k+1}]}(t), ]

where each (\xi_k) is (\mathcal F_{t_k})-measurable. The stochastic integral of such a process is defined by

[ \int_0^t H_s,dM_s

\sum_{k=0}^{n-1} \xi_k \left( M_{t\wedge t_{k+1}}-M_{t\wedge t_k} \right). ]

The measurability condition expresses nonanticipation: the coefficient applied to an increment is determined by information available before that increment occurs.

For a continuous square-integrable martingale, the quadratic variation process (\langle M\rangle) determines the natural norm on integrands. The integral satisfies the isometry

[ \mathbb E\left[ \left(\int_0^T H_s,dM_s\right)^2 \right]

\mathbb E\left[ \int_0^T H_s^2,d\langle M\rangle_s \right]. ]

This identity allows the integral map on elementary predictable processes to extend uniquely by completion to every predictable (H) satisfying

[ \mathbb E\left[ \int_0^T H_s^2,d\langle M\rangle_s \right]<\infty. ]

When (M=W) is standard Brownian motion, its quadratic variation is (\langle W\rangle_t=t), so the isometry reduces to

[ \mathbb E\left[ \left(\int_0^T H_s,dW_s\right)^2 \right]

\mathbb E\left[ \int_0^T H_s^2,ds \right]. ]

The resulting integral process is itself a square-integrable martingale. Under weaker local integrability assumptions, localization extends the construction to local martingales.

Semimartingale integration

A semimartingale (X) admits a decomposition

[ X_t=X_0+M_t+A_t, ]

where (M) is a local martingale and (A) is an adapted process of locally finite variation. For a predictable integrand (H), integration is defined through

[ \int_0^t H_s,dX_s

\int_0^t H_s,dM_s + \int_0^t H_s,dA_s. ]

The first term is a stochastic integral, whereas the second is a pathwise Lebesgue–Stieltjes integral. The decomposition itself need not be unique without additional conditions, but the combined integral is independent of the chosen decomposition whenever the integrand belongs to the appropriate semimartingale integrability class.

Semimartingales are characterized by their compatibility with stochastic integration. Under the Bichteler–Dellacherie characterization, an adapted càdlàg process is a semimartingale precisely when integration against bounded elementary predictable processes has the required continuity under uniform convergence to zero. This result explains why semimartingales form the broad class of integrators for which a stable nonanticipating calculus exists.

Quadratic variation and Itô calculus

The distinction between ordinary calculus and stochastic calculus arises from nonzero quadratic variation. For Brownian motion, partitions whose mesh tends to zero satisfy

[ \sum_k \left(W_{t_{k+1}}-W_{t_k}\right)^2 \longrightarrow t ]

in probability. Terms that would be negligible for differentiable paths therefore contribute finite corrections in stochastic expansions.

For a continuous semimartingale (X) and a twice continuously differentiable function (f), Itô's formula states that

[ f(X_t)

f(X_0) + \int_0^t f'(X_s),dX_s + \frac12\int_0^t f''(X_s),d[X]_s, ]

where ([X]) denotes the quadratic variation of (X). If

[ dX_t=b_t,dt+\sigma_t,dW_t, ]

then the formula becomes

[ df(X_t)

f'(X_t)b_t,dt + f'(X_t)\sigma_t,dW_t + \frac12f''(X_t)\sigma_t^2,dt. ]

The final term has no counterpart in the ordinary chain rule. It records the cumulative second-order effect of Brownian increments.

Itô and Stratonovich integrals

The placement of the integrand within an approximating partition affects the limiting integral. The Itô integral uses information from the left endpoint, which preserves nonanticipation and the martingale property. The Stratonovich integral, introduced in systematic form by Ruslan Stratonovich, corresponds to a symmetric evaluation and is conventionally written

[ \int_0^t H_s\circ dX_s. ]

For continuous semimartingales (H) and (X), the conversion relation is

[ \int_0^t H_s\circ dX_s

\int_0^t H_s,dX_s + \frac12[H,X]_t. ]

When (H_s=f'(X_s)), the quadratic-covariation correction converts Itô's formula into the ordinary-looking chain rule

[ f(X_t)-f(X_0)

\int_0^t f'(X_s)\circ dX_s. ]

The two conventions therefore represent different integral definitions rather than conflicting evaluations of a single pathwise integral.

Stochastic differential equations

A stochastic differential equation written formally as

[ dX_t=b(t,X_t),dt+\sigma(t,X_t),dW_t ]

is interpreted through the integral identity

[ X_t

X_0 + \int_0^t b(s,X_s),ds + \int_0^t \sigma(s,X_s),dW_s. ]

The drift integral is an ordinary time integral, while the diffusion integral is stochastic. Conditions involving measurability, growth, and regularity determine whether a solution exists and whether its law or sample-path realization is unique. The distinction between strong and weak solutions concerns whether the solution is constructed relative to a specified Brownian motion and filtration or only through a probability law on an appropriate path space.

Stochastic integration also underlies martingale representation results. In a filtration generated by Brownian motion, every square-integrable martingale admits a representation of the form

[ M_t=M_0+\int_0^t H_s,dW_s ]

for a predictable square-integrable process (H). This representation connects conditional expectation, stochastic differential equations, and changes of probability measure.