数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (4): 1681-1689.doi: 10.1007/s10473-022-0423-2

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THE METRIC GENERALIZED INVERSE AND ITS SINGLE-VALUE SELECTION IN THE PRICING OF CONTINGENT CLAIMS IN AN INCOMPLETE FINANCIAL MARKET

王紫1, 王筱凌2,3, 王玉文4   

  1. 1. School of Mathematics Sciences, Harbin Normal University, Harbin, 150025, China;
    2. Institute of Finance, Heilongjiang University of Finance and Economics, Harbin, 150025, China;
    3. International School, Krirk University, Bangkok, 10220, Thailand;
    4. School of Mathematics Science, Harbin Normal University, Harbin, 150025, China
  • 收稿日期:2021-05-09 修回日期:2021-06-17 出版日期:2022-08-25 发布日期:2022-08-23
  • 通讯作者: Xiaoling WANG,E-mail:wangxiaoling1985@aliyun.com E-mail:wangxiaoling1985@aliyun.com
  • 基金资助:
    The first author is supported by the National Science Foundation (12001142) and Harbin Normal University doctoral initiation Fund (XKB201812); The second author is supported by the Science Foundation Grant of Heilongjiang Province (LH2019A017).

THE METRIC GENERALIZED INVERSE AND ITS SINGLE-VALUE SELECTION IN THE PRICING OF CONTINGENT CLAIMS IN AN INCOMPLETE FINANCIAL MARKET

Zi WANG1, Xiaoling WANG2,3, Yuwen WANG4   

  1. 1. School of Mathematics Sciences, Harbin Normal University, Harbin, 150025, China;
    2. Institute of Finance, Heilongjiang University of Finance and Economics, Harbin, 150025, China;
    3. International School, Krirk University, Bangkok, 10220, Thailand;
    4. School of Mathematics Science, Harbin Normal University, Harbin, 150025, China
  • Received:2021-05-09 Revised:2021-06-17 Online:2022-08-25 Published:2022-08-23
  • Contact: Xiaoling WANG,E-mail:wangxiaoling1985@aliyun.com E-mail:wangxiaoling1985@aliyun.com
  • Supported by:
    The first author is supported by the National Science Foundation (12001142) and Harbin Normal University doctoral initiation Fund (XKB201812); The second author is supported by the Science Foundation Grant of Heilongjiang Province (LH2019A017).

摘要: This article continues to study the research suggestions in depth made by M.Z. Nashed and G.F. Votruba in the journal "Bull. Amer. Math. Soc." in 1974. Concerned with the pricing of non-reachable "contingent claims" in an incomplete financial market, when constructing a specific bounded linear operator $A: l_1^n\rightarrow l_2$ from a non-reflexive Banach space $l_1^n$ to a Hilbert space $l_2$, the problem of non-reachable "contingent claims" pricing is reduced to researching the (single-valued) selection of the (set-valued) metric generalized inverse $A^\partial$ of the operator $A$. In this paper, by using the Banach space structure theory and the generalized inverse method of operators, we obtain a bounded linear single-valued selection $A^\sigma=A^+$ of $A^\partial$.

关键词: Incomplete financial market, bounded linear operator, metric generalized inverse, single-value selection, Moore-Penrose generalized inverse

Abstract: This article continues to study the research suggestions in depth made by M.Z. Nashed and G.F. Votruba in the journal "Bull. Amer. Math. Soc." in 1974. Concerned with the pricing of non-reachable "contingent claims" in an incomplete financial market, when constructing a specific bounded linear operator $A: l_1^n\rightarrow l_2$ from a non-reflexive Banach space $l_1^n$ to a Hilbert space $l_2$, the problem of non-reachable "contingent claims" pricing is reduced to researching the (single-valued) selection of the (set-valued) metric generalized inverse $A^\partial$ of the operator $A$. In this paper, by using the Banach space structure theory and the generalized inverse method of operators, we obtain a bounded linear single-valued selection $A^\sigma=A^+$ of $A^\partial$.

Key words: Incomplete financial market, bounded linear operator, metric generalized inverse, single-value selection, Moore-Penrose generalized inverse

中图分类号: 

  • 47A05