Stochastic differential equation

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]

^ 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.
^ Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin.
^ Cite error: The named reference oksendal was invoked but never defined (see the help page).
^ 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
^ 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.
^ Michel Emery (1989). Stochastic calculus in manifolds. Springer Berlin, Heidelberg. Doi https://doi.org/10.1007/978-3-642-75051-9
^ Zdzisław Brzeźniak and K. D. Elworthy, Stochastic differential equations on Banach manifolds, Methods Funct. Anal. Topology 6 (2000), no. 1, 43-84.
^ 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

[rogerswilliams-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.

[musielarutkowski-2] Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin.

[oksendal-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.

[Emery-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.

[sdesjets-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

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