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
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