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Brownian Motion, 2nd Edition by Björn Böttcher, Lothar Partzsch, René L. Schilling

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19 Stochastic differential equations

The ordinary differential equation e9783110307290_i4256.jpg, x(0) = x0, describes the position xt of a particle which moves with speed b(s, x) depending on time and on the current position. One possibility to take into account random effects, e. g. caused by measurement errors or hidden parameters, is to add a random perturbation which may depend on the current position. This leads to an equation of the form

e9783110307290_i4257.jpg

Letting ∆t → 0 we get the following equation for stochastic differentials, cf. Section 17.1,

Since (19.1) is a formal ...

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