Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case. (English) Zbl 0736.90017

Summary: We employ a martingale approach to study a dynamic consumption-portfolio problem in continuous time with incomplete markets and short-sale constraints. We introduce a notion of minimax local martingale and transform the dynamic problem into a static problem of maximizing expected utility over the consumption bundles that satisfy a single budget constraint formed using that measure. We establish the existence of and characterize the minimax local measure, provide sufficient conditions for the dynamic consumption-portfolio problem to have a solution, and relate the optimal policies to the solution of quasi-linear partial differential equation.


91B62 Economic growth models
91B28 Finance etc. (MSC2000)
Full Text: DOI


[1] Ames, W. F., Numerical Method for Partial Differential Equations (1977), Academic Press: Academic Press New York · Zbl 0219.35007
[2] Aumann, R. J.; Perles, M., A variational problem arising in economics, J. Math. Anal. Appl., 11, 488-503 (1965) · Zbl 0137.39201
[3] Bismut, J. M., Convex functions in optimal stochastic control, J. Math. Anal. Appl., 44, 384-404 (1973)
[4] Chamberlain, G., Asset priving in multiperiod securities markets, Econometrica, 56, 1283-1300 (1987) · Zbl 0657.90012
[5] Chung, K. L.; Williams, R. J., An Introduction to Stochastic Integration (1983), Birkhäuser: Birkhäuser Boston · Zbl 0527.60058
[6] Clark, J. M., The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Statist., 41, 1282-1296 (1970) · Zbl 0213.19402
[7] Cox, J. C.; Huang, C.-f, A variational problem arising in financial economics, (Sloan School of Management. Sloan School of Management, J. Math. Econ. (1989), MIT), to appear · Zbl 0734.90009
[8] Cox, J. C.; Huang, C.-f, Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Econ. Theory, 49, 33-83 (1989) · Zbl 0678.90011
[9] Dellacherie, C.; Meyer, P.-A, Probabilities and Potential (1978), North-Holland: North-Holland New York
[10] Dellacherie, C.; Meyer, P.-A, Probabilities and Potential B: Theory of Martingales (1982), North-Holland: North-Holland New York
[11] Duffie, D.; Huang, C.-F, Implementing Arrow-Debreu equilibria by continuous trading of a few long-lived securities, Econometrica, 53, 1337-1356 (1985) · Zbl 0576.90014
[12] Dunford, N. H.; Schwartz, J. T., Linear Operators (1958), Interscience: Interscience New York
[13] Dybvig, P. H.; Huang, C.-f, Nonnegative wealth, absence of arbitrage, and feasible consumption plans, Rev. Financial Stud., 1, 377-401 (1988)
[14] Fleming, W. H.; Rishel, R. W., Deterministic and Stochastic Optimal Control (1975), Springer-Verlag: Springer-Verlag Berling · Zbl 0323.49001
[15] Fleming, W. H.; Vermes, D., Convex duality approach to the optimal control of diffusions, SIAM J. Control Optimization, 27, 115-1136 (1989) · Zbl 0693.93082
[17] Friedman, A., (Stochastic Differential Equations and Applications, Vol. 1 (1975), Academic Press: Academic Press New York)
[18] Harrison, J. M.; Kreps, D. M., Martingales and multiperiod markets, J. Econ. Theory, 20, 381-408 (1979) · Zbl 0431.90019
[19] Harrison, J. M.; Pliska, S. R., Martingales and stochastic integrals in the theory of continuous trading, Stochastic Processes Appl., 11, 215-260 (1981) · Zbl 0482.60097
[20] He, H.; Pearson, N. D., Consumption and portfolio policies with incomplete markets and short-sale constraints: The finite dimensional case (1989), Simon Graduate School of Business Administration, University of Rochester, mimeo
[21] Holmes, R., Geometric Functional Analysis and Its Applications (1975), Springer-Verlag: Springer-Verlag New York · Zbl 0336.46001
[22] Huang, C.-f, An intertemporal general equilibrium asset price model: The case of diffusion information, Econometrica, 55, 117-142 (1987) · Zbl 0611.90035
[23] Huang, C.-f, Information structure and viable price systems, J. Math. Econ., 14, 215-240 (1985) · Zbl 0606.90012
[24] Karatzas, I.; Lehoczky, J. P.; Shreve, S. E., Optimum portfolio and consumption decisions for a “small investor” on a finite horizon, SIAM J. Control Optimization, 25, 1557-1586 (1987) · Zbl 0644.93066
[25] Karatzas, I.; Lehoczky, J. P.; Shreve, S. E.; Xu, G.-l, Optimality conditions for utility maximization in an incomplete market (1989), Carnegie Mellon University, mimeo
[26] Kreps, D. M., Arbitrage and equilibrium in economies with infinitely many commodities, J. Math. Econ., 8, 15-35 (1981) · Zbl 0454.90010
[27] Litpser, R. S.; Shiryayev, A. N., Statistics of Random Process I: General Theory (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0364.60004
[28] Merton, R. C., An intertemporal capital asset pricing model, Econometrica, 41, 867-887 (1973) · Zbl 0283.90003
[29] Merton, R. C., Optimum consumption and portfolio rules in a continuous time model, J. Econ. Theory, 3, 373-413 (1971) · Zbl 1011.91502
[30] Merton, R. C., Lifetime portfolio under certainty: The continuous time case, Rev. Econ. Statist., 247-257 (1969), LI
[31] Mossin, J., Optimal multiperiod portfolio policies, J. Bus., 41, 215-229 (1968)
[32] Pagès, H., Optimal consumption and portfolio policies when markets are incomplete (1987), Massachusetts Institute of Technology, mimeo
[33] Pliska, S. R., A stochastic calculus model of continuous trading: Optimal portfolios, Math. Operations Res., 11, 371-382 (1986) · Zbl 1011.91503
[34] Rockafellar, R. T., Integral functionals, normal integrands, and measurable selections, (Gossez, J.; etal., Non-Linear Operators and the Calculus of Variations (1975), Springer-Verlag: Springer-Verlag New York) · Zbl 0374.49001
[35] Rockafellar, R. T., Conjugate Duality and Optimization (1974), SIAM: SIAM Philadelphia · Zbl 0326.49008
[36] Samuelson, P., Lifetime portfolio selection by dynamic stochastic programming, Rev. Econ. Statist., 239-246 (1969), LI
[37] Svensson, L. E.O, Portfolio Choice and Asset Pricing with Non-Traded Assets, NBER Working Paper No. 2774 (1988)
[38] Xu, G.-l, A Duality Approach to a Stochastic Portfolio/Consumption Decision Problem in A Continuous Time Market with Short-Selling Restrictions, (unpublished Ph.D. thesis (1990), Carnegie Mellon University)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.