Geodesic
A~criterion for the absence of arbitrage in a~discrete model of a~securities market under convex portfolio constraints
Sibirskij žurnal industrialʹnoj matematiki, Tome 7 (2004) no. 1, pp. 95-108.

Voir la notice de l'article provenant de la source Math-Net.Ru

In the case of a finite probability space, we obtain a criterion for the absence of arbitrage in a model of a securities market assuming only that the set of constraints on investment strategies (portfolios) is closed and convex. We use the language of nonstandard analysis to formulate the corresponding version of the first fundamental theorem of asset pricing. We show that two topological versions of the no-arbitrage condition reduces to it by a change of constraints. 
@article{SJIM_2004_7_1_a8,
     author = {D. B. Rokhlin},
     title = {A~criterion for the absence of arbitrage in a~discrete model of a~securities market under convex portfolio constraints},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {95--108},
     publisher = {mathdoc},
     volume = {7},
     number = {1},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2004_7_1_a8/}
}
TY  - JOUR
AU  - D. B. Rokhlin
TI  - A~criterion for the absence of arbitrage in a~discrete model of a~securities market under convex portfolio constraints
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2004
SP  - 95
EP  - 108
VL  - 7
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2004_7_1_a8/
LA  - ru
ID  - SJIM_2004_7_1_a8
ER  - 
%0 Journal Article
%A D. B. Rokhlin
%T A~criterion for the absence of arbitrage in a~discrete model of a~securities market under convex portfolio constraints
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2004
%P 95-108
%V 7
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2004_7_1_a8/
%G ru
%F SJIM_2004_7_1_a8
D. B. Rokhlin. A~criterion for the absence of arbitrage in a~discrete model of a~securities market under convex portfolio constraints. Sibirskij žurnal industrialʹnoj matematiki, Tome 7 (2004) no. 1, pp. 95-108. http://geodesic.mathdoc.fr/item/SJIM_2004_7_1_a8/

[1] Harrison J. M., Kreps D. M., “Martingales and arbitrage in multiperiod securities markets”, J. Econom. Theory, 20 (1979), 381–408 | DOI | MR | Zbl

[2] Harrison J. M., Pliska S. R., “Martingales and stochastic integrals in the theory of continuous trading”, Stochastic Process. Appl., 11:3 (1981), 215–260 | DOI | MR | Zbl

[3] Taqqu M. S., Willinger W., “The analysis of finite security markets using martingales”, Adv. in Appl. Probab., 19 (1987), 1–25 | DOI | MR | Zbl

[4] Melnikov A. V., Feoktistov K. M., “Voprosy bezarbitrazhnosti i polnoty diskretnykh rynkov i raschety platezhnykh obyazatelstv”, Obozrenie prikl. i promyshl. matematiki, 8:1 (2001), 28–40 | MR

[5] Shiryaev A. N., Osnovy stokhasticheskoi finansovoi matematiki, Fazis, M., 1998

[6] Rokafellar R. T., Vypuklyi analiz, Mir, M., 1973

[7] Shiryaev A. N., Veroyatnost, Nauka, M., 1980 | MR | Zbl

[8] Schachermayer W., “A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time”, Insurance: Mathematics and Economics, 11 (1992), 249–257 | DOI | MR | Zbl

[9] Dalang R. C., Morton A., Willinger W., “Equivalent martingale measures and no-arbitrage in stochastic securities market models”, Stochastics Stochastic Rep., 29:2 (1990), 185–201 | MR | Zbl

[10] Jacod J., Shiryaev A. N., “Local martingales and the fundamental asset pricing theorems in the discrete-time case”, Finance Stochastics, 2:3 (1998), 259–273 | DOI | MR | Zbl

[11] Kabanov Yu., Stricker C., “A teacher's note on no-arbitrage criteria”, Lecture Notes in Math., 1755, Springer, Berlin; Heidelberg, 2001, 149–152 | MR | Zbl

[12] Brannath W., No arbitrage andmartingalemeasures in option pricing, Doct. diss. Wien Univ., 1997

[13] Pham H., Touzi N., “The fundamental theorem of asset pricingwith cone constraints”, J. Math. Econom., 31 (1999), 265–279 | DOI | MR | Zbl

[14] Carassus L., Pham H., Touzi N., “No arbitrage in discrete time under portfolio constraints”, Math. Finance, 11:3 (2001), 315–329 | DOI | MR | Zbl

[15] Evstigneev I. V., Shürger K., Taksar M. I., On the fundamental theorem of asset pricing: random constraints and bang-bang no-arbitrage criteria, Discussion paper 24/2002. Univ. Bonn, 2002

[16] Napp C., “The Dalang-Morton-Willinger theorem under cone constraints”, J. Math. Econom., 39 (2003), 111–126 | DOI | MR | Zbl

[17] Rokhlin D. B., “Kriterii otsutstviya asimptoticheskogo besplatnogo lencha na konechnomernom rynke pri vypuklykh ogranicheniyakh na portfel i vypuklykh operatsionnykh izderzhkakh”, Sib. zhurn. industr. matematiki, 5:1(9) (2002), 133–144 | MR | Zbl

[18] Rokhlin D. B., “Rasshirennaya versiya pervoi fundamentalnoi teoremy finansovoi matematiki pri konicheskikh ogranicheniyakh na portfel”, Obozrenie prikl. i promyshl. matematiki, 9:1 (2002), 131–132 | MR

[19] Uspenskii V. A., Chto takoe nestandartnyi analiz?, Nauka, M., 1987 | MR

[20] Albeverio S., Fenstad I., Kheeg-Kron R., Lindstrem T., Nestandartnye metody v stokhasticheskom analize i matematicheskoi fizike, Mir, M., 1990 | MR | Zbl

[21] Gordon I. E., Kusraev A. G., Kutateladze S. S., Infinitezimalnyi analiz, Izd-vo In-ta matematiki, Novosibirsk, 2001

[22] Kreps D. M., “Arbitrage and equilibrium in economies with infinitely many commodities”, J. Math. Econom., 8 (1981), 15–35 | DOI | MR | Zbl

[23] Clark S. A., “Arbitrage approximation theory”, J. Math. Econom., 33 (2000), 167–181 | DOI | MR | Zbl

[24] Jouini E., Kallal H., “Viability and equilibrium in securities markets with frictions”, Math. Finance, 9:3 (1999), 275–292 | DOI | MR | Zbl

[25] Klein I., “A fundamental theorem of asset pricing for large financial market”, Math. Finance, 10:4 (2000), 443–458 | DOI | MR | Zbl