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3.3.2 Itô lemma

[ SLIDE
Rules -
Ito's lemma ||
VIDEO
modem -
LAN -
DSL]

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:

After substitution of the value of the stochastic differential (3.3.1#eq.1),
this leads directly to *Itô's lemma*,
here given for the function of only one stochastic variable

In words, the Itô lemma states that the differential of a stochastic
function is the superposition of a deterministic component
proportional to the time step
, and a random component
proportional to the Wiener increment
.
Remember that the factor
= 0 or 1 here chooses between a normal
or log-normal distribution of the price increments
and can also take
other values if this is found to be appropriate.
**SYLLABUS** ** Previous:** 3.3.1 Wiener process and
**Up:** 3.3 Improved model using
**Next:** 3.3.3 Evaluate an expectancy