Mathematics Branches, Topics, and Sub-Topics

A structured visual guide to the major mathematical areas and their relationships.

Search by code, branch, topic, subtopic, or a keyword from the descriptions.

60Hxx Stochastic analysis

This subtopic introduces the core ideas in stochastic analysis, including foundational concepts, standard methods, and the main questions used to organize the area. Typical uses include building mathematical background, framing related research problems, and supporting applications in neighboring fields where these concepts provide useful structure.

Specific topics

60H05 Stochastic integrals

Overview

Stochastic integrals. This topic covers stochastic analysis, including stochastic calculus, stochastic differential equations, and analytic tools for random systems.

Related Wikipedia Page

Wikipedia: Stochastic calculus

Useful Links

Key Ideas

  • Ito and Stratonovich integration
  • stochastic differential equations
  • martingale and analytic frameworks

Typical Uses

Used to analyze continuous-time random dynamics and develop rigorous stochastic PDE/SDE methods.

Applications

  • Quantitative finance
  • Random dynamical models
  • Stochastic control and filtering

References

Recommended Textbooks

60H07 Stochastic calculus of variations and the Malliavin calculus

Overview

Stochastic calculus of variations and the Malliavin calculus. This topic covers stochastic analysis, including stochastic calculus, stochastic differential equations, and analytic tools for random systems.

Related Wikipedia Page

Wikipedia: Stochastic calculus

Useful Links

Key Ideas

  • Ito and Stratonovich integration
  • stochastic differential equations
  • martingale and analytic frameworks

Typical Uses

Used to analyze continuous-time random dynamics and develop rigorous stochastic PDE/SDE methods.

Applications

  • Quantitative finance
  • Random dynamical models
  • Stochastic control and filtering

References

Recommended Textbooks

60H10 Stochastic ordinary differential equations

Overview

Stochastic ordinary differential equations. This topic covers stochastic analysis, including stochastic calculus, stochastic differential equations, and analytic tools for random systems.

Related Wikipedia Page

Wikipedia: Stochastic calculus

Useful Links

Key Ideas

  • Ito and Stratonovich integration
  • stochastic differential equations
  • martingale and analytic frameworks

Typical Uses

Used to analyze continuous-time random dynamics and develop rigorous stochastic PDE/SDE methods.

Applications

  • Quantitative finance
  • Random dynamical models
  • Stochastic control and filtering

References

Recommended Textbooks

60H15 Stochastic partial differential equations

Overview

Stochastic partial differential equations. This topic covers stochastic analysis, including stochastic calculus, stochastic differential equations, and analytic tools for random systems.

Related Wikipedia Page

Wikipedia: Stochastic calculus

Useful Links

Key Ideas

  • Ito and Stratonovich integration
  • stochastic differential equations
  • martingale and analytic frameworks

Typical Uses

Used to analyze continuous-time random dynamics and develop rigorous stochastic PDE/SDE methods.

Applications

  • Quantitative finance
  • Random dynamical models
  • Stochastic control and filtering

References

Recommended Textbooks

60H20 Stochastic integral equations

Overview

Stochastic integral equations. This topic covers stochastic analysis, including stochastic calculus, stochastic differential equations, and analytic tools for random systems.

Related Wikipedia Page

Wikipedia: Stochastic calculus

Useful Links

Key Ideas

  • Ito and Stratonovich integration
  • stochastic differential equations
  • martingale and analytic frameworks

Typical Uses

Used to analyze continuous-time random dynamics and develop rigorous stochastic PDE/SDE methods.

Applications

  • Quantitative finance
  • Random dynamical models
  • Stochastic control and filtering

References

Recommended Textbooks

60H25 Random operators and equations

Overview

Random operators and equations. This topic covers stochastic analysis, including stochastic calculus, stochastic differential equations, and analytic tools for random systems.

Related Wikipedia Page

Wikipedia: Stochastic calculus

Useful Links

Key Ideas

  • Ito and Stratonovich integration
  • stochastic differential equations
  • martingale and analytic frameworks

Typical Uses

Used to analyze continuous-time random dynamics and develop rigorous stochastic PDE/SDE methods.

Applications

  • Quantitative finance
  • Random dynamical models
  • Stochastic control and filtering

References

Recommended Textbooks

60H30 Applications of stochastic analysis

Overview

Applications of stochastic analysis. This topic covers stochastic analysis, including stochastic calculus, stochastic differential equations, and analytic tools for random systems.

Related Wikipedia Page

Wikipedia: Stochastic calculus

Useful Links

Key Ideas

  • Ito and Stratonovich integration
  • stochastic differential equations
  • martingale and analytic frameworks

Typical Uses

Used to analyze continuous-time random dynamics and develop rigorous stochastic PDE/SDE methods.

Applications

  • Quantitative finance
  • Random dynamical models
  • Stochastic control and filtering

References

Recommended Textbooks

60H35 Computational methods for stochastic equations

Overview

Computational methods for stochastic equations. This topic covers stochastic analysis, including stochastic calculus, stochastic differential equations, and analytic tools for random systems.

Related Wikipedia Page

Wikipedia: Stochastic calculus

Useful Links

Key Ideas

  • Ito and Stratonovich integration
  • stochastic differential equations
  • martingale and analytic frameworks

Typical Uses

Used to analyze continuous-time random dynamics and develop rigorous stochastic PDE/SDE methods.

Applications

  • Quantitative finance
  • Random dynamical models
  • Stochastic control and filtering

References

Recommended Textbooks

60H40 White noise theory

Overview

White noise theory. This topic covers stochastic analysis, including stochastic calculus, stochastic differential equations, and analytic tools for random systems.

Related Wikipedia Page

Wikipedia: Stochastic calculus

Useful Links

Key Ideas

  • Ito and Stratonovich integration
  • stochastic differential equations
  • martingale and analytic frameworks

Typical Uses

Used to analyze continuous-time random dynamics and develop rigorous stochastic PDE/SDE methods.

Applications

  • Quantitative finance
  • Random dynamical models
  • Stochastic control and filtering

References

Recommended Textbooks

60H99 None of the above

Overview

None of the above. This topic covers stochastic analysis, including stochastic calculus, stochastic differential equations, and analytic tools for random systems.

Related Wikipedia Page

Wikipedia: Stochastic calculus

Useful Links

Key Ideas

  • Ito and Stratonovich integration
  • stochastic differential equations
  • martingale and analytic frameworks

Typical Uses

Used to analyze continuous-time random dynamics and develop rigorous stochastic PDE/SDE methods.

Applications

  • Quantitative finance
  • Random dynamical models
  • Stochastic control and filtering

References

Recommended Textbooks