A stochastic differential equation (SDE)
$$d{X}_{t} = f(t,{X}_{t})\,dt + g(t,{X}_{t})\,d{W}_{t}$$
is, in fact, nota differential equation at all, but only a symbolic representation for the stochastic integralequation
$${X}_{t} = {X}_{{t}_{0}} +{ \int\nolimits \nolimits }_{{t}_{0}}^{t}f(s,{X}_{s})\,ds +{ \int\nolimits \nolimits }_{{t}_{0}}^{t}g(s,{X}_{s})\,d{W}_{s},$$
where the first integral is a deterministic Riemann integral for eachsample path. The second integral is an Itô stochastic integral, which isdefined as the mean-square limit of sums of products of the integrand\(g\) evaluatedat the start of each discretization subinterval times the increment of the Wienerprocess \({W}_{t}\)(which is often called a Brownian motion, seeBrownian Motion and Diffusions). Itis not possible to define this stochastic integral pathwise as a Riemann–Stieltjesintegral, because the sample paths of a Wiener process, although continuous, arenowhere differentiable and not even of bounded variation on...
References and Further Reading
Jentzen A, Kloeden PE, Neuenkirch A (2009) Convergence of numerical approximations of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients. Numerische Mathematik, 112(1):41–64
MATH
MathSciNet
Google Scholar
Kloeden PE, Platen E (1992) The numerical solution of stochastic differential equations. Springer, Berlin (3rd revised printing 1999)
Google Scholar
Download references
Author information
Authors and Affiliations
Professor,
Goethe-Universität, Frankfurt, Germany
Peter E. Kloeden
Editor information
Editors and Affiliations
Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia
Miodrag Lovric
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg