Quelques remarques sur un théorème de Yan. (Some remarks on a theorem of Yan). (French) Zbl 0701.60041

Ansel, Jean-Pascal; Stricker, Christophe

Séminaire de probabilités XXIV 1988/89, Lect. Notes Math. 1426, 266-274 (1990).

[For the entire collection see Zbl 0695.00024.]
We begin with improving a theorem of J. Yan [Séminaire de Probabilités XIV, 1978/79, Lect. Notes Math. 784, 148-151 (1980; Zbl 0432.60071)] by extending it to spaces \(L^ p\), \(1\leq p<+\infty\). Thanks to this, the second author [Arbitrage et lois de martingale, Ann. Inst. Henri Poincaré 26, No.3, 451-460 (1990)] has recently shown the equivalence between the existence of a martingale law and the absence of free lunch. Then we prove that an admissible trading strategy for an agent \(\alpha_ 0\) [according to D. Duffie and C. Huang’s definition, J. Math. Econ. 15, 283-303 (1986; Zbl 0608.90006)] is not generally admissible for an agent \(\alpha_ 1\) even if \(\alpha_ 1\) is better informed than \(\alpha_ 0.\)
Finally, we show that if there is no free lunch, classical hypothesis for economic models, this pathology doesn’t occur. This leads us to a detailed study of different notions of free lunch.

Reviewer: Jean-Pascal Ansel

MSC:

60G42

Martingales with discrete parameter

Keywords:

existence of a martingale law; admissible trading strategy; economic models

Citations:

Zbl 0695.00024; Zbl 0432.60071; Zbl 0608.90006