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) ∈ R 3 is the velocity of the fluid at the point x at time t, then a reasonable mathematical model for the position X t of the particle at time t would be a stochastic differential equation of the form
where W t ∈ R 3 denotes “white noise” and σ(t,x) ∈ R 3×3. The Ito interpretation of this equation is
where B t is 3-dimensional Brownian motion, and similarly (with a correction term added to b) for the Stratonovich interpretation (see (6.3)).
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© 1995 Springer-Verlag Berlin Heidelberg
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Øksendal, B. (1995). Diffusions: Basic Properties. In: Stochastic Differential Equations. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03185-8_7
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DOI: https://doi.org/10.1007/978-3-662-03185-8_7
Publisher Name: Springer, Berlin, Heidelberg
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