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**Optimal consumption and portfolio policies when asset prices follow a diffusion process.**
*(English)*
Zbl 0678.90011

Summary: We consider a consumption-portfolio problem in continuous time under uncertainty. A martingale technique is employed to characterize optimal consumption-portfolio policies when there exist nonnegativity constraints on consumption and on final wealth. We also provide a way to compute and verify optimal policies. Our verification theorem for optimal policies involves a linear partial differential equation, unlike the nonlinear partial differential equation of dynamic programming. The relationship between our approach and dynamic programming is discussed. We demonstrate our technique by explicitly computing optimal policies in a series of examples. In particular, we solve the optimal consumption-portfolio problem for hyperbolic absolute risk aversion utility functions when the asset prices follow a geometric Brownian motion. The optimal policies in this case are no longer linear when nonnegativity constraints on consumption and on final wealth are included. By these examples, one can see that our approach is much easier than the dynamic programming approach.

### MSC:

91G10 | Portfolio theory |

49L20 | Dynamic programming in optimal control and differential games |

60J60 | Diffusion processes |

90C39 | Dynamic programming |

### Keywords:

consumption-portfolio problem; continuous time; uncertainty; martingale technique; linear partial differential equation; hyperbolic absolute risk aversion utility functions; geometric Brownian motion; continuous-time
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\textit{J. C. Cox} and \textit{C.-f. Huang}, J. Econ. Theory 49, No. 1, 33--83 (1989; Zbl 0678.90011)

### References:

[1] | Apostol, T, Mathematical analysis, (1974), Addison-Wesley Reading, MA |

[2] | Black, F; Scholes, M, The pricing of options and corporate liabilities, J. polit. econ., 81, 637-654, (1973) · Zbl 1092.91524 |

[3] | Breeden, D, An intertemporal capital asset pricing model with stochastic investment opportunities, J. finan. econ., 7, 265-296, (1979) · Zbl 1131.91330 |

[4] | Brennan, M; Solanski, R, Optimal portfolio insurance, J. finan. quant. anal., 16, 279-300, (1981) |

[5] | Chung, K; Williams, R, Introduction to stochastic integration, (1983), Birkhäuser Boston, MA · Zbl 0527.60058 |

[6] | Cox, J; Huang, C, A variational problem arising in financial economics, (1985), Sloan School of Management, Massachusetts Institute of Technology, mimeo |

[7] | Cox, J; Leland, H, Notes on intertemporal investment policies, (1982), Graduate School of Business, Stanford University, mimeo |

[8] | Dellacherie, C; Meyer, P, Probabilities and potential B: theory of martingales, (1982), North-Holland New York |

[9] | Duffie, D; Huang, C, Implementing Arrow-Debreu equilibria by continuous trading of few long-lived securities, Econometrica, 53, 1337-1356, (1985) · Zbl 0576.90014 |

[10] | Dybvig, P, A positive wealth constraint precludes arbitrage in the Black-Scholes model, (1980), Economics Department, Princeton University, mimeo |

[11] | {\scP. Dybvig and C. Huang}, Nonnegative wealth, absence of arbitrage, and feasible consumption plans, to appear in Review of Finan. Stud. |

[12] | Fisk, D, Quasi-martingales, Trans. amer. math. soc., 120, 369-389, (1965) · Zbl 0133.40303 |

[13] | Fleming, W; Rishel, R, Deterministic and stochastic optimal control, (1975), Springer-Verlag New York/Berlin · Zbl 0323.49001 |

[14] | Friedman, A, () |

[15] | Gihman, I; Skorohod, A, Stochastic differential equations, (1972), Springer-Verlag New York/Berlin · Zbl 0242.60003 |

[16] | Gihman, I; Skorohod, A, Controlled stochastic processes, (1979), Springer-Verlag New York/Berlin · Zbl 0404.60061 |

[17] | Harrison, M; Kreps, D, Martingales and multiperiod securities markets, J. econ. theory, 20, 381-408, (1979) · Zbl 0431.90019 |

[18] | Harrison, M; Pliska, S, Martingales and stochastic integrals in the theory of continuous trading, Stochastic process appl., 11, 215-260, (1981) · Zbl 0482.60097 |

[19] | Hestenes, M, Optimization theory: the finite dimensional case, (1975), Wiley New York · Zbl 0327.90015 |

[20] | Holmes, R, Geometric functional analysis and its applications, (1975), Springer-Verlag New York/Berlin · Zbl 0336.46001 |

[21] | Huang, C, Information structures and viable price systems, J. math. econ., 14, 215-240, (1985) · Zbl 0606.90012 |

[22] | Huang, C, An intertemporal general equilibrium asset pricing model: the case of diffusion information, Econometrica, 55, 117-142, (1987) · Zbl 0611.90035 |

[23] | Jacod, J, Calcul stochastique et problèmes de martingales, () · Zbl 0414.60053 |

[24] | Karatzas, I; Lehoczky, J; Sethi, S; Shreve, S, Explicit solution of a general consumption/investment problem, Math. oper. res., 11, 613-636, (1986) · Zbl 0622.90018 |

[25] | Karatzas, I; Lehoczky, J; Shreve, S, Optimal portfolio and consumption decisions for a “small investor” on a finite horizon, SIAM J. control optim., 25, 1557-1586, (1987) · Zbl 0644.93066 |

[26] | Kreps, D, Three essays on capital markets, () |

[27] | Kreps, D, Arbitrage and equilibrium in economies with infinitely many commodities, J. math. econ., 8, 15-35, (1981) · Zbl 0454.90010 |

[28] | Krylov, N, Controlled diffusion processes, (1980), Springer-Verlag New York/Berlin · Zbl 0459.93002 |

[29] | Liptser, R; Shiryayev, A, Statistics of random processes I: general theory, (1977), Springer-Verlag New York/Berlin · Zbl 0364.60004 |

[30] | Merton, R, Optimum consumption and portfolio rules in a continuous time model, J. econ. theory, 3, 373-413, (1971) · Zbl 1011.91502 |

[31] | Merton, R, Continuous-time finance, (1989), Blackwell Oxford |

[32] | Pliska, S, A stochastic calculus model of continuous trading: optimal portfolios, Math. oper. res., 11, 371-382, (1986) · Zbl 1011.91503 |

[33] | Rockafellar, R, Integral functionals, normal integrands and measurable selections, () · Zbl 0374.49001 |

[34] | Royden, H, Real analysis, (1968), Macmillan Co New York · Zbl 0197.03501 |

[35] | Sethi, S; Taksar, M, A note on Merton’s optimum consumption and portfolio rules in a continuous-time model, J. econ. theory, 46, 395-401, (1988) · Zbl 0657.90028 |

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