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Numerical Methods for Stochastic Differential Equations

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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...

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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

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  • Kloeden PE, Platen E (1992) The numerical solution of stochastic differential equations. Springer, Berlin (3rd revised printing 1999)

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© 2011 Springer-Verlag Berlin Heidelberg

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Kloeden, P.E. (2011). Numerical Methods for Stochastic Differential Equations. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_424

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