# 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)
A secret to create a complete market from an incomplete market. (English) Zbl 1193.91060
Summary: The Martingale method has been given increasing attention since it was conducted by Cox and Huang in 1989. Martingale method allows us to solve the problems of utility maximization in a very elegant manner. However, the Martingale method is not omnipotent. When the market is incomplete, traditional Martingale method will be problematic. To overcome the problem of incompleteness, {\it I. Karatzas, J. P. Lehoczky, S. E. Shreve} and {\it G.-L. Xu} [SIAM J. Control Optimization 29, No. 3, 702--730 (1991; Zbl 0733.93085)] developed a way to complete the market by introducing additional fictitious stocks and then making them uninteresting to the investor. Nevertheless, to find such fictitious stocks is not straightforward. In particular, when the number of such stocks needed in order to complete the market were very large, it would be very computational, and even may not be possible to be expressed explicitly. To make life easier, we provide an alternative method by directly creating a complete market from the incomplete one such that the dimension of the underlying Brownian motion equals the number of available stocks. Our approach is ready to be used.

##### MSC:
 91B26 Market models (auctions, bargaining, bidding, selling, etc.)
Full Text:
##### References:
 [1] Karatzas, I.; Lehoczky, J. P.; Shreve, S. E.; Xu, G. L.: Martingale and duality for utility maximization in an incomplete market. SIAM J. Contr. optim. 29, 702-730 (1991) · Zbl 0733.93085 [2] Korn, R.; Korn, E.: Option pricing and portfolio optimization. Graduate studies in mathematics 31 (2000) · Zbl 1039.91028 [3] Mas-Colell, A.; Whinston, M. D.; Green, J. R.: Microeconomic theory. (1995) · Zbl 1256.91002 [4] Shreve, S. E.: Stochastic calculus for finance II: Continuous-time models. (2004) · Zbl 1068.91041