Price Operators Analysis in L p -Spaces
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Springer 2005
Price Operators Analysis in Lp-Spaces SERGIO ALBEVERIO1, GIULIA DI NUNNO2 and YURI A. ROZANOV3 1
Abteilung fu¨r Wahrscheinlichkeitstheorie und Mathematische Statistik, Wegelerstrasse 6, D-53115 Bonn, Germany; SFB611; BiBoS; IZKS; CERFIM (Locarno); Acc. Arch. (Mendrisio). e-mail: [email protected] 2 Centre of Mathematics for Applications (CMA), Matematisk Institutt, Universitetet i Oslo, Postboks 1053 Blindern, N-0316 Oslo, Norway. e-mail: [email protected] 3 IMATI-CNR, Via E. Bassini 15, I-20133 Milano, Italy. e-mail: [email protected] (Received: 16 June 2003; accepted: 22 September 2005) Abstract. An integral type representation and various extension theorems for monotone linear operators in Lp-spaces are considered in relation to market price modelling. As application, a characterization of the existence of a risk-neutral probability measure equivalent to the applied underlying one is provided in terms of the given prices. These results are in the line of the fundamental theorem of asset pricing. Here, in particular, the risk-neutral probability measure considered has the advantage of having its density laying in pre-considered upper and lower bounds. Mathematics Subject Classifications (2000): Primary: 91B28, Secondary: 91B70, 60Hxx. Key words: risk-neutral probability measure, price operator, Ho¨lder equality, HahnYBanach extension theorem, Ko¨nig sandwich theorem.
1. Introduction In this first section, we shortly recall the modelling of market prices. Here we refer in particular to [3] and [12] among the vast literature in mathematical finance, as these papers are the ones originally mostly related to our considerations. In the course of time, the stochastic behaviour of the market is associated with some increasing -algebras of events At ;
0 t T;
considered in the probability space ð; A; PÞ:
ð0:1Þ
For any t, the -algebra At represents the information of the market events up to time t. The probability measure P :¼ PðAÞ;
A 2 A;
is considered on the -algebra A ¼ AT of the market events up to time T, for T > 0 fixed. We will refer to this measure as the Funderlying_ probability measure or
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the Fapplied_ probability measure. In fact it is the probability measure actually applied in the large part of the market analysis. The prices in the market model concern some riskless security, for which the initial 1-unit investment will bring, in time t, the deterministic value Rt > 0, and to some risky securities. For any of these market risky securities, the price Xt at time t is a real random variable measurable with respect to At . The 1-unit investment made at time s in this security will bring, in time t > s, the return Xt Xs ; Xs
0 s < t T:
A probability measure P0 equivalent to P, i.e., P0 P
ð0:2Þ
on the -algebra A ¼ AT , is known as being risk-neutral if the conditional expectation E0
hX X i R R t s t s ; As ¼ Xs Rs
0 s < t T;
ð0:3Þ
is the same for all the security returns, including the riskless security return Rt
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