14
The Fundamental Theorem of Asset Pricing
for Unbounded Stochastic Processes (1998)
14.1 Introduction
The topic of the present paper is the statement and proof of the subsequent
Fundamental Theorem of Asset Pricingin ageneral version for not necessarily
locally bounded semi-martingales:
Main Theorem 14.1.1.LetS=(St)t∈R+be anRd-valued semi-martingale
defined on the stochastic base(Ω,F,(Ft)t∈R+,P).
ThenSsatisfies the condition of no free lunch with vanishing riskif and
only if there exists a probability measureQ∼Psuch thatSis a sigma-
martingale with respect toQ.
This theorem has been proved under the additional assumption that the
processSis locally bounded in Chap. 9. Under this additional assumption
one may replace the term “sigma-martingale” above by the term “local mar-
tingale”.
We refer to Chap. 9 for the history of this theorem, which goes back to
the seminal work of Harrison, Kreps and Pliska ([HK 79], [HP 81], [K 81])
and which is of central importance in the applications of stochastic calculus
to Mathematical Finance. We also refer to Chap. 9 for the definition of the
concept ofno free lunch with vanishing riskwhich is a mild strengthening of
the concept ofno-arbitrage.
On the other hand, to the best of our knowledge, the second central concept
in the above theorem, the notion of asigma-martingale(see Definition 14.2.1
below) has not been considered previously in the context of Mathematical
Finance. In a way, this is surprising, as we shall see in Remark 14.2.4 that
this concept is very well-suited for the applications in Mathematical Finance,
where one is interested not so much in the processS itself but rather in
the family (H·S)ofstochastic integralson the processS,whereH runs
[DS 98] The Fundamental Theorem of Asset Pricing for Unbounded Stochastic
Processes.Mathematische Annalen, vol 312, pp. 215–250, (1998).