# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Utility maximization with partial information. (English) Zbl 0834.90022
Summary: We address two maximization problems: the maximization of expected total utility from consumption and the maximization of expected utility from terminal wealth. The price process of the available financial assets is assumed to satisfy a system of functional stochastic differential equations. The difference between this paper and the existing papers on the same subject is that here we require the consumption and investment processes to be adapted to the natural filtration of the price processes. This requirement means that the only available information for agents in this economy at a certain time are the prices of the financial assets up to that time. The underlying Brownian motion and the drift process in the system of equations for the asset prices are not directly observable. Particular details will be worked out for the “Bayesian” example, when the dispersion coefficient is a fixed invertible matrix and the drift vector is an $F_0$-measurable, unobserved random variable with known distribution.

##### MSC:
 91B16 Utility theory 60H10 Stochastic ordinary differential equations
Full Text:
##### References:
 [1] Browne, S.; Whitt, W.: The Bayesian kelly criterion, manuscript. (1994) · Zbl 0867.90010 [2] Cox, J. C.; Huang, C. F.: Optimal consumption and portfolio policies when asset prices follow a diffusion process. J. economic theory 49, 33-83 (1989) · Zbl 0678.90011 [3] Cox, J. C.; Ingersoll, J. E.; Ross, S. A.: An intertemporal general equilibrium model of asset prices. Econometrica 53, 363-384 (1985) · Zbl 0576.90006 [4] Duffie, D.; Zame, W.: The consumption-based capital asset pricing model. Econometrica 57, 1279-1297 (1989) · Zbl 0684.90007 [5] He, H.; Pearson, N.: Consumption and portfolio policies with incomplete markets: the infinite dimensional case. J. econom. Theory 54, 259-305 (1991) · Zbl 0736.90017 [6] Karatzas, I.; Lakner, P.; Lehoczky, J. P.; Shreve, S. E.: Equilibrium in a simplified dynamic, stochastic economy with heterogeneous agents. Stochastic anal, 245-272 (1991) · Zbl 0735.90024 [7] Karatzas, I.; Lehoczky, J. P.; Shreve, S. E.: Optimal portfolio and consumption decisions for a ”small investor” on a finite horizon. SIAM journal of control and optimization 25, 1557-1586 (1987) · Zbl 0644.93066 [8] Karatzas, I.; Lehoczky, J. P.; Shreve, S. E.; Xu: Martingale and duality methods for utility maximization in incomplete markets. SIAM J. Control optim. 29, 702-730 (1991) · Zbl 0733.93085 [9] Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus. (1988) · Zbl 0638.60065 [10] Ocone, D. L.; Karatzas, I.: A generalized Clark representation formula, with application to optimal portfolios. Stochastics and stochastic reports 34, 187-220 (1991) · Zbl 0727.60070 [11] Protter, P.: Stochastic integration and differential equations. (1990) · Zbl 0694.60047