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| Stratonovich integral | |
|---|---|
| Name | Stratonovich integral |
| Field | Stochastic calculus |
| Introduced | 1960s |
| Inventor | Ruslan Stratonovich |
| Related | Itô integral, stochastic differential equations, Brownian motion |
Stratonovich integral is a definition of stochastic integration widely used in physics and engineering to model systems driven by noisy signals where ordinary calculus rules are desirable. It was developed in the context of statistical mechanics and signal processing by Ruslan Stratonovich and has connections to foundational work by Kiyosi Itô, Paul Lévy, Norbert Wiener, Andrey Kolmogorov, and Wassily Hoeffding. The integral appears in formulations of stochastic differential equations used in studies associated with Langevin equation, Fokker–Planck equation, Kramers–Moyal expansion, Onsager reciprocal relations, and theories connected to Gibbs measure and Boltzmann equation.
The Stratonovich integral is defined as a limit of symmetric Riemann sums for stochastic processes, conceived to respect the classical chain rule and to align with mid-point discretizations used in physical modeling by researchers such as Norbert Wiener and Paul Langevin. For a semimartingale integrator, the integral definition emerged in the works of Ruslan Stratonovich and was developed alongside parallel formalism by Kiyosi Itô and contributors like Hiroshi Watanabe and Philip Protter. Its formulation uses symmetric sampling points analogous to constructions in numerical analysis by Richard Hamming and Gilbert Strang and is often contrasted with asymmetric sampling used in alternative integrals associated with Kiyosi Itô.
Construction proceeds by taking limits of mid-point Riemann sums for adapted processes relative to filtrations developed in probability theory by Andrey Kolmogorov, Joseph Doob, and Paul Lévy. For continuous semimartingales, existence and uniqueness leverage martingale decomposition techniques introduced by Joseph Doob and extended by Shizuo Kakutani and Meyer; continuity properties are studied in the framework of topology on path space used by Norbert Wiener and Marshall Stone. Key properties include compatibility with ordinary calculus rules as in works by Élie Cartan and Sophus Lie and preservation of Stratonovich integrals under smooth changes of variables, a feature emphasized in applications by Lars Onsager and Ilya Prigogine. Linearity, locality, and limits under uniform convergence on compacts echo analytic foundations articulated by Bernhard Riemann and André Weil.
The relation to the Itô integral was clarified through stochastic calculus identities developed by Kiyosi Itô and refined by H. P. McKean, Daniel Stroock, and S. R. S. Varadhan. Formally, for a continuous semimartingale integrator, the Stratonovich and Itô integrals differ by a correction term involving quadratic covariation, a concept tied to work by Paul Lévy and Norbert Wiener. This correction term appears in asymptotic expansions studied by Freidlin, Wentzell, and Kurtz, and it underpins conversions used in stochastic modeling in publications by Edward Nelson and Mark Kac.
The Stratonovich integral satisfies a chain rule resembling the classical chain rule used since Isaac Newton and generalized by Gottfried Leibniz; this property was emphasized in applications by Lars Onsager and Richard Feynman. Under smooth diffeomorphisms associated with manifolds studied by Élie Cartan and Hermann Weyl, compositions of Stratonovich-integrated processes transform according to ordinary calculus, a feature exploited in stochastic differential geometry by K. D. Elworthy, Mikhael Gromov, and Shigeyoshi Ohta. In contrast, the Itô formula of Kiyosi Itô includes extra second-order terms tied to quadratic variation, a distinction central to comparisons by Daniel Stroock and S. R. S. Varadhan.
The Stratonovich integral is widely used in physics, engineering, and applied mathematics for modeling noisy dynamical systems encountered in research by Paul Langevin, Ludwig Boltzmann, Enrico Fermi, and Lev Landau. Applications include stochastic thermodynamics influenced by Ilya Prigogine and Lars Onsager, control theory linked to Norbert Wiener and Rudolf Kalman, signal processing building on methods by Harry Nyquist and Claude Shannon, and climate modeling related to studies by Edward Lorenz. In neuroscience, biochemical kinetics, and systems biology the Stratonovich interpretation is favored in models following approaches by Alan Hodgkin, Andrew Huxley, Stanley Prusiner, and Sydney Brenner. In geometric mechanics and stochastic differential geometry, connections to work by Élie Cartan, Hermann Weyl, and Michael Atiyah support formulations on manifolds and fiber bundles relevant to Yang–Mills theory.
Canonical examples include integration against Brownian motion originated by Norbert Wiener and computations for the stochastic harmonic oscillator analyzed by Paul Langevin and Richard Feynman. Practical discretizations use midpoint schemes studied by J. C. Butcher and G. W. Stewart and weak/strong convergence results established by Kurtz, Protter, and I. Karatzas with numerical methods developed in texts by Peter Kloeden and Eckhard Platen. Explicit calculations converting between Stratonovich and Itô representations rely on quadratic variation identities attributed to Paul Lévy and on perturbation techniques used by Freidlin and Wentzell.