Stochastic differential equation

Differential equations
Scope
Fields
Applied mathematics
Social sciences
List of named differential equations
Classification
Types
By variable type
Features
Relation to processes
Solution
Existence and uniqueness
General topics
Solution methods
People
List

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process,[1] resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices,[2] random growth models[3] or physical systems that are subjected to thermal fluctuations.

SDEs have a random differential that is in the most basic case random white noise calculated as the derivative of a Brownian motion or more generally a semimartingale. However, other types of random behaviour are possible, such as jump processes like Lévy processes[4] or semimartingales with jumps. Random differential equations are conjugate to stochastic differential equations. Stochastic differential equations can also be extended to differential manifolds.[5][6][7][8]

  1. ^ Rogers, L.C.G.; Williams, David (2000). Diffusions, Markov Processes and Martingales, Vol 2: Ito Calculus (2nd ed., Cambridge Mathematical Library ed.). Cambridge University Press. doi:10.1017/CBO9780511805141. ISBN 0-521-77594-9. OCLC 42874839.
  2. ^ Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin.
  3. ^ Cite error: The named reference oksendal was invoked but never defined (see the help page).
  4. ^ Kunita, H. (2004). Stochastic Differential Equations Based on Lévy Processes and Stochastic Flows of Diffeomorphisms. In: Rao, M.M. (eds) Real and Stochastic Analysis. Trends in Mathematics. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2054-1_6
  5. ^ Imkeller, Peter; Schmalfuss, Björn (2001). "The Conjugacy of Stochastic and Random Differential Equations and the Existence of Global Attractors". Journal of Dynamics and Differential Equations. 13 (2): 215–249. doi:10.1023/a:1016673307045. ISSN 1040-7294. S2CID 3120200.
  6. ^ Michel Emery (1989). Stochastic calculus in manifolds. Springer Berlin, Heidelberg. Doi https://doi.org/10.1007/978-3-642-75051-9
  7. ^ Zdzisław Brzeźniak and K. D. Elworthy, Stochastic differential equations on Banach manifolds, Methods Funct. Anal. Topology 6 (2000), no. 1, 43-84.
  8. ^ Armstrong J. and Brigo D. (2018). Intrinsic stochastic differential equations as jets. Proc. R. Soc. A., 474: 20170559, http://doi.org/10.1098/rspa.2017.0559