STOCHASTIC

CONSUMPTION AND PORTFOLIO RULES 375
Markov processes of the first type called ltd Processes are defined as the
solution to the stochastic differential equation 3
dP=f(P,t)dt + g(P,t)dz, (1)
where P, f, and g are n vectors and z{t) is an n vector of standard normal
random variables. Then dz(t) is called a multidimensional Wiener process
(or Brownian motion). 1
The fundamental tool for formal manipulation and solution of stochastic
processes of the ltd type is It6's Lemma stated as follows 5
LEMMA. Let F^ ,..., Pn ,t) be a C 2 function defined on R n X[0, oo)
and take the stochastic integrals
Pit) = PM + (MP, s) ds + ( gi(P, s) dz,, i=l,...,n;
then the time-dependent random variable Y = F is a stochastic integral
and its stochastic differential is
dY = \ W i d P i + -W d ' + 2?? -8p-BP; dPidP '"
where the product of the differentials dPt dPj are defined by the multiplication
rule
dz( dzj — ptj dt, i,j = 1,..., n,
dZidt = 0, i = 1,..., n,
3 Ito Processes are a special case of a more general class of stochastic processes
called Strong diffusion processes (see Kushner [9, p. 22]). (1) is a short-hand expression
for the stochastic integral
P(t) = P(0) + f /(P, s) ds + f g(P, s) dz,
Jo Jo
where Pit) is the solution to (1) with probability one.
A rigorous discussion of the meaning of a solution to equations like(l) is not presented
here. Only those theorems needed for formal manipulation and solution of stochastic
differential equations are in the text and these without proof. For a complete discussion
of ltd Processes, see the seminal paper of ltd [7], ltd and McKean [8], and McKean
[11]. For a short description and some proofs, see Kushner [9, pp. 12-18]. For an
heuristic discussion of continuous-time Markov processes in general, see Cox and
Miller [3, Chap. 5].
* dz is often referred to in the literature as "Gaussian White Noise." There are some
regularity conditions imposed on the functions / and g. It is assumed throughout the
paper that such conditions are satisfied. For the details, see [9] or [11].
s See McKean [11, pp. 32-35 and 44] for proofs of the Lemma in one and n dimen­
sions.
4. THE CAPITAL GROWTH CRITERION AND CONTINUOUS-TIME MODELS 623