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Trudy Mat. Inst. Steklova, 2002, Volume 237, Pages 12–56 (Mi tm323)  

This article is cited in 27 scientific papers (total in 27 papers)

Vector Stochastic Integrals and the Fundamental Theorems of Asset Pricing

A. N. Shiryaeva, A. S. Chernyb

a Steklov Mathematical Institute, Russian Academy of Sciences
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: This paper deals with the foundations of stochastic mathematical finance and has three main purposes: (1) To present a self-contained construction of the vector stochastic integral $H\bullet X$ with respect to a $d$-dimensional semimartingale $X=(X_t^1,…,X_t^d)$. This notion is more general than the componentwise stochastic integral $\sum _{i=1}^d H^i\bullet X^i$. (2) To show that vector stochastic integrals are important in mathematical finance. To be more precise, the notion of componentwise stochastic integral is insufficient in the First and Second Fundamental Theorems of Asset Pricing. (3) To prove the Second Fundamental Theorem of Asset Pricing in the general setting, i.e. in the continuous-time case for a general multidimensional semimartingale. The proof is based on the martingale techniques and, in particular, on the properties of the vector stochastic integral.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 237, 6–49

Bibliographic databases:
UDC: 519.216.8
Received in April 2001

Citation: A. N. Shiryaev, A. S. Cherny, “Vector Stochastic Integrals and the Fundamental Theorems of Asset Pricing”, Stochastic financial mathematics, Collected papers, Trudy Mat. Inst. Steklova, 237, Nauka, MAIK Nauka/Inteperiodika, M., 2002, 12–56; Proc. Steklov Inst. Math., 237 (2002), 6–49

Citation in format AMSBIB
\Bibitem{ShiChe02}
\by A.~N.~Shiryaev, A.~S.~Cherny
\paper Vector Stochastic Integrals and the Fundamental Theorems of Asset Pricing
\inbook Stochastic financial mathematics
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2002
\vol 237
\pages 12--56
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm323}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1975582}
\zmath{https://zbmath.org/?q=an:1034.60058}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2002
\vol 237
\pages 6--49


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