Despite the short-hand differential notation that has been used so far,
the stochastic differential equation (3.3.1#eq.1) is formally defined
only in its integral form: in other words, a probability weighted
average has to be carried out before the random sampling from the Wiener
process acquires any significance.
The stochastic or Itô calculus dealing properly with the extensions of
the usual Riemann integrals to non-smooth differentials
leads to
the Itô lemma and draws on mathematics that goes beyond the scope of
this course.
The same result can however be derived from a Taylor expansion in multiple
dimensions, keeping terms up to
and
and applying the special rules for stochastic calculus:
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