10-StochDiffEq Page 282 Wednesday, February 4, 2004 12:51 PM
282 The Mathematics of Financial Modeling and Investment Management1
Xt = x 0 exp
μ – ---σ(^2)
t + σBt
^2
SUMMARY
■ Stochastic differential equations give meaning to ordinary differential
equations where some terms are subject to random perturbation.
■ Following Itô and Stratonovich, stochastic differential equations are
defined through their integral equivalent: the differential notation is
just a shorthand.
■ Itô processes are the sum of a time integral plus an Itô integral.
■ Itô processes are closed with respect to smooth maps: a smooth func-
tion of an Itô process is another Itô process defined through the Itô for-
mula.
■ Stochastic differential equations are equations established in terms of
Itô processes.
■ Linear equations can be solved explicitly as the sum of the solution of
the associated deterministic equation plus a stochastic cumulative
term.