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Diffusions: Basic Properties

  • Bernt Øksendal
Chapter
  • 407 Downloads
Part of the Universitext book series (UTX)

Abstract

Suppose we want to describe the motion of a small particle suspended in a moving liquid, subject to random molecular bombardments. If b(t,x) ∈ ℝ3 is the velocity of the fluid at the point x at time t, then a reasonable mathematical model for the position Xt of the particle at time t would be a stochastic differential equation of the form
$$\frac{{d{X_t}}}{{dt}} = b(t,{X_t}) + \sigma (t,{X_t}){W_t}$$
(7.1)
where Wt ∈ ℝ3 denotes “white noise” and σ(t, x) ∈ ℝ3×3. The Ito interpretation of this equation is
$$d{X_t} = b(t,{X_t})dt + \sigma (t,{X_t})dB$$
(7.2)
where Bt is 3-dimensional Brownian motion, and similarly (with a correction term added to b) for the Stratonovich interpretation (see (6.2)).

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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Bernt Øksendal
    • 1
  1. 1.Department of MathematicsUniversity of OsloBlindern, Oslo 3Norway

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