CONSUMPTION AND PORTFOLIO RULES 375 Markov processes of the first type called Ito Processes are defined as the solution to the stochastic differential equation dP=f(P, tdt+g(P, t)dz where P,, 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).4 The fundamental tool for formal manipulation and solution of stochastic processes of the Ito type is Ito,'s Lemma stated as followss LEMMA. Let F(Pi,, Pn, t be a C2 function defined on R"XIO, co) and take the stochastic integrals Pi(t)=P:(O)+f(P, s)ds+ g(P, s)dzt, i=l,,; then the time-dependent random variable y= f is a stochastic integral and its stochastic diferential is d=∑F dp2++l∑∑ a"F dp, dPi where the product of the differentials dP, dP, are defined by the multi Pu dt, i, j dzi dt=0 Processes are a special case of a more general class of stochastic processes trong diffusion processes(see Kushner [9, p. 22].(1)is a short-hand expression P()=P(O)+f(P, s)ds+ g(P, s)dz, here P()is the solution to(1) with probability one Arigorous discussion of the meaning of a solution to equations like(1)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 Ito Processes, see the seminal paper of Ito [7l, Ito and McKean [8], and McKear [11]. For a short description and some proofs, see Kushner 9, pp. 12-18]. For an euristic discussion of continuous-time Markov processes in general, see Cox and Miller [3, Chap. 51 4 dz is often referred to in the literature as"Gaussian White Noise. There are some imposed on the functions f and g. It is assumed throughout hat such conditions are satisfied. For the detail 5 See McKean [11, pp 32-35 and 44] for proofs of the Lemma in one and n dimen- sionsCONSUMPTION AND PORTFOLIO RULES 375 Markov processes of the first type called It6 Processes are defined as the solution to the stochastic differential equation3 dP = f (P, t) dt + g(P, t) dz, 02 where P, f, and g are n vectors and z(t) is an rz vector of standard normal random variables. Then dz(t) is called a multidimensional Wiener process (or Brownian motion).4 The fundamental tool for formal manipulation and solution of stochastic processes of the It6 type is Ita’s Lemma stated as follow@ LEMMA. Let F(Pl ,..., P, , t) be a C2 function dejned on and take the stochastic integmfs then the time-dependent random variable Y = F is a stochastic integral and its stochastic d@erential is where the product of the difSerentiaIs dPi dPj are defijled by the mukti￾plication rub dz, dzj = ,aij dt, i, j = I,..., n, dzi dt = 0, i = I,.~., y1 9 3 It6 Processes are a special ease of a more general cIass of stochastic processes called Strong diffusion processes (see Kushner 19, p. 221). (1) is a short-hand expression for the stochastic integral P(t) = P(0) + j-‘-f@‘, s) ds + jt g(P, s) dz, Ll 0 where P(t) is the solution to (1) with probability one. A rigorous discussion of the meaning of a solution to equations like (I) 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 Pto Processes, see the seminal paper of It6 173, It6 and McKean IS], and McKean [ll]. For a short description and some proofs, see Kushner [9, pp- 12-IS]. For an heuristic discussion of continuous-time Markov processes in general, see Cox and Miller [3, Chap. 51. * dz is often referred to in the literature as “Gaussian White Noise.” There are some regularity conditions imposed on the functions f and g. It is assumed throughout the paper that such conditions are satisfied. For the details, see [9] or [ll]. 5 See McKean [ll, pp. 32-35 and 441 for proofs of the Lemma in one and IZ dimen￾sions