STOCHASTIC

376 MERTON
where pit is the instantaneous correlation coefficient between the Wiener
processes dzt and dij."
Armed with Ito's Lemma, we are now able to formally differentiate
most smooth functions of Brownian motions (and hence integrate
stochastic differential equations of the Ito type). 7
Before proceeding to the discussion of asset price behavior, another
concept useful for working with Ito Processes is the differential generator
(or weak infinitesimal operator) of the stochastic process P(t). Define
the function G(P, t) by
G(P> t) s jim Et [W + ft. + AI-WO^ (2)
when the limit exists and where "E" is the conditional expectation
operator, conditional on knowing P(t). If the Pt(t) are generated by Ito
Processes, then the differential generator of P, SCP , is defined by
d *
Y'spt ' at ' 2 ^ ^ " ept dPj '
^^i/iA+4-+^ii^-
where f = (fx ,...,/„), g = (g,,..., £„), and au = gigip(J. Further, it can
be shown that
6{P, t) = &PIG(P,«)]. (4)
G can be interpreted as the "average" or expected time rate of change of
•This multiplication rule has given_rise to the formalism of writing the Wiener
process differentials as dzt = ^ Vdt where the ^ are standard normal variates
{e.g., see [3]).
7 Warning: derivatives (and integrals) of functions of Brownian motions are similar
to, but different from, the rules for deterministic differentials and integrals. For example,
if
then dP = Pdz. Hence
p(t) = p(0) eSl*-i> = P(0) ««i«-ioi-i