Brownian Motion, 2nd Edition

19 Stochastic differential equations

The ordinary differential equation , x(0) = x₀, describes the position x_t 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

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

(19.1)

Since (19.1) is a formal ...