CONSUMPTION AND PORTFOLIO RULES the function G(P, t)and as such is the natural generalization of the ordinary time derivative for deterministic functions 3. ASSET PRICE DYNAMICS AND THE BUDGET EqUATION e: Throughout the paper, it is assumed that all assets are of the limited ability type, that there exist continuously-trading perfect markets with no transactions costs for all assets, and that the prices per share, P(t)) are generated by Ito Processes, i.e. pi-a e, t)d e, t) dz where a is the instantaneous conditional expected percentage change in price per unit time and o 2 is the instantaneous conditional variance per unit time. In the particular case where the "geometric Brownian motion hypothesis is assumed to hold for asset prices, ai and o, will be constants For this case, prices will be stationarily and log-normally distributed and it will be shown that this assumption about asset prices simplifies the continuous-time model in the same way that the assumption of normality of prices simplifies the static one-period portfolio model c. To derive the correct budget equation, it is necessary to examine the rete-time formulation of the model and then to take limits carefully to obtain the continuous-time form. Consider a period model with periods of length h, where all income is generated by capital gains, and wealth w(t) and Pit) are known at the beginning of period t. Let the decision variables be indexed such that the indices coincide with the period in which the decisions are implemented. Namely, let M(t)= number of shares of asset i purchased during period t, i.e., between t and t+h C(t)=amount of consumption per unit time during period t. A heuristic method for finding the differential generator is to take the conditional xpectation of dG (found by Ito,s Lemma)and"divide"by dt. The result of this opera- tion will be LpIGl, i.e,, formally dt EdG)=G=sPIGJ The"p"operator is often called a Dynkin operator and is often written as"Dp 642/3/4-3CONSUMPTION AND PORTFOLIO RULES 377 the function G(P, t) and as such is the natural generalization of the ordinary time derivative for deterministic functions.* 3. ASSET PRICE DYNAMICS AND THE BUDGET EQUATION Throughout the paper, it is assumed that all assets are of the limite liability type, that there exist continuously-trading perfect markets with no transactions costs for all assets, and that the prices per share, (p,(t)>, are generated by Ito Processes, i.e., dP. -2 D = q(P, t) dt + ai(P, t) dzizi , where 01~ is the instantaneous conditional expected percentage change in price per unit time and oi2 is the instantaneous conditional variance per unit time. In the particular case where the “geometric Brownian motion hypothesis is assumed to hold for asset prices, 01~ and gi will be constants. For this case, prices will be stationarily and log-normally distributed and it will be shown that this assumption about asset prices simplifies the continuous-time model in the same way that the assumption of normality of prices simplifies the static one-period portfolio model. To derive the correct budget equation, it is necessary to examine t discrete-time formulation of the model and then to take limits carefu to obtain the continuous-time form. Consider a period model with periods of length h, where all income is generated by capital gains, and wealth, W(t) and Pi(t) are known at the beginning of period t. Let the de variables be indexed such that the indices coincide with the per which the decisions are implemented. Namely, let iVi(t) = number of shares of asset i purchased during period t, i.e., between t and t + h and (61 C(l) = amount of consumption per unit time during period t. 8 A heuristic method for finding the differential generator is to take the conditional expectation of dG (found by ItUs Lemma) and “divide” by &. The result of this opera￾tion will be 2$[G], i.e., formally, ; &(dG) = d = 6pp[G]. The “2&” operator is often called a Dynkin operator and is often writterr as “DP”. 642/3/4-3