The fundamental theorem of utility maximization (called FTUM hereafter) says that the utility maximization admits solution if and only if there exists an equivalent martingale measure. This theorem is true for discrete market models (where the number of scenarios is finite), andremains valid for general discrete-time market modelswhen the utility is smooth enough. However, this theorem --in this current formulation-- fails in continuous-time framework even with nice utility function, where there might exist arbitrage opportunities and optimal portfolio. This paper addresses the question how far we can weaken the non-arbitrage condition as well as the utility maximization problem to preserve their complete and strong relationship described by the FTUM. As application of our version of the FTUM, we establish equivalence between the No-Unbounded-Profit-with-Bounded-Risk condition  , the existence of  num\'eraire portfolio, and the existence of solution to the utility maximization under equivalent probability measure  . The latter fact can be interpreted as a sort of weak form of market's viability, while this equivalence is established with a much less technical approach. Furthermore, the obtained equivalent probability can be chosen as close to the real-world probability measure as we want (but might not be equal) {\it   .
This paper proposes two approaches that quantify the exact relationship among the viability, the absence of arbitrage, and/or the existence of the num\'eraire portfolio under minimal assumptions and for general continuous-time market models. Precisely, our first and principal contribution proves the equivalence among the No-Unbounded-Profit-with-Bounded-Risk condition (NUPBR hereafter) , the existence of the num\'eraire portfolio, and the existence of the optimal portfolio under an equivalent probability measure for any "nice" utility and positive initial capital. Herein, a 'nice" utility is any smooth von URLenstern utility satisfying Inada's conditions and the elasticity assumptions of Kramkov and Schachermayer . Furthermore, the equivalent probability measure ---under which the utility maximization problems have solutions--- can be chosen as close to the real-world probability measure as we want (but might not be equal) . Without changing the underlying probability measure and under mild assumptions, our second contribution proves that the NUPBR is equivalent to the "{\it local " existence of the optimal portfolio. This constitutes an alternative to the first contribution, if one insists on working under the real-world probability. These two contributions lead naturally to new types of viability that we call weak and local viabilities .
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The fundamental theorem of utility maximization (called FTUM hereafter) says that
This paper proposes two approaches that quantify the exact relationship among the viability, the absence of arbitrage, and/or the existence of the num\'eraire portfolio under minimal assumptions and for general continuous-time market models. Precisely, our first and principal contribution proves the equivalence among
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utility maximization admits solution if and only if there exists an equivalent martingale measure. This theorem is true for discrete market models (where the number of scenarios is finite), andremains valid for general discrete-time market modelswhen the utility is smooth enough. However, this theorem --in this current formulation-- fails in continuous-time framework even with nice utility function, where there might exist arbitrage opportunities and optimal portfolio. This paper addresses the question how far we can weaken the non-arbitrage condition as well as the utility maximization problem to preserve their complete and strong relationship described by the FTUM. As application of our version of the FTUM, we establish equivalence between the
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(NUPBR hereafter)
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the
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solution to the utility maximization under
the optimal portfolio under an
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for any "nice" utility and positive initial capital. Herein, a 'nice" utility is any smooth von URLenstern utility satisfying Inada's conditions and the elasticity assumptions of Kramkov and Schachermayer
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The latter fact can be interpreted as a sort of weak form of market's viability, while this equivalence is established with a much less technical approach.
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obtained equivalent probability
equivalent probability measure ---under which the utility maximization problems have solutions---
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. Without changing the underlying probability measure and under mild assumptions, our second contribution proves that the NUPBR is equivalent to the "
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local
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" existence of the optimal portfolio. This constitutes an alternative to the first contribution, if one insists on working under the real-world probability. These two contributions lead naturally to new types of viability that we call weak and local viabilities