In Chap. 2 we only dealt with finite probability spaces. This was mainly done because of technical difficulties. As soon as the probability space (Ω,ℱ T ,P) is no longer finite, the corresponding function spaces such as L1(Ω,ℱ T ,P) or L∞(Ω,ℱ T ,P) are infinite dimensional and we have to fall back on functional analysis. In this chapter we will present a proof of the Fundamental Theorem of Asset Pricing, Theorem 2.2.7, in the case of general (Ω,ℱ T ,P), but still in finite discrete time. Since discounting does not present any difficulty, we will suppose that the d-dimensional price process S has already been discounted as in Sect. 2.1. Also the notion of the class ℋ of trading strategies does not present any difficulties and we may adopt Definition 2.1.4 verbatim also for general (Ω,ℱ T ,P) as long as we are working in finite discrete time. In this setting we can state the following beautiful version of the Fundamental Theorem of Asset Pricing, due to Dalang, Morton and Willinger [DMW 90].
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© 2006 Springer-Verlag Berlin Heidelberg
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Delbaen, F., Schachermayer, W. (2006). The Dalang-Morton-Willinger Theorem. In: The Mathematics of Arbitrage. Springer Finance. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31299-4_6
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DOI: https://doi.org/10.1007/978-3-540-31299-4_6
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