Geodesic
On the Existence of an Equivalent Supermartingale Density for a Fork-Convex Family of Stochastic Processes
Matematičeskie zametki, Tome 87 (2010) no. 4, pp. 594-603.

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

We prove that a fork-convex family $\mathbb W$ of nonnegative stochastic processes has an equivalent supermartingale density if and only if the set $H$ of nonnegative random variables majorized by the values of elements of $\mathbb W$ at fixed instants of time is bounded in probability. A securities market model with arbitrarily many main risky assets, specified by the set $\mathbb W(\mathbb S)$ of nonnegative stochastic integrals with respect to finite collections of semimartingales from an arbitrary indexed family $\mathbb S$, satisfies the assumptions of this theorem.
Keywords: stochastic process, fork-convex family, supermartingale, semimartingale, securities market, probability space, convergence in probability, stochastic integral. 
@article{MZM_2010_87_4_a10,
     author = {D. B. Rokhlin},
     title = {On the {Existence} of an {Equivalent} {Supermartingale} {Density} for a {Fork-Convex} {Family} of {Stochastic} {Processes}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {594--603},
     publisher = {mathdoc},
     volume = {87},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_87_4_a10/}
}
TY  - JOUR
AU  - D. B. Rokhlin
TI  - On the Existence of an Equivalent Supermartingale Density for a Fork-Convex Family of Stochastic Processes
JO  - Matematičeskie zametki
PY  - 2010
SP  - 594
EP  - 603
VL  - 87
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2010_87_4_a10/
LA  - ru
ID  - MZM_2010_87_4_a10
ER  - 
%0 Journal Article
%A D. B. Rokhlin
%T On the Existence of an Equivalent Supermartingale Density for a Fork-Convex Family of Stochastic Processes
%J Matematičeskie zametki
%D 2010
%P 594-603
%V 87
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2010_87_4_a10/
%G ru
%F MZM_2010_87_4_a10
D. B. Rokhlin. On the Existence of an Equivalent Supermartingale Density for a Fork-Convex Family of Stochastic Processes. Matematičeskie zametki, Tome 87 (2010) no. 4, pp. 594-603. http://geodesic.mathdoc.fr/item/MZM_2010_87_4_a10/

[1] J. Jacod, A. N. Shiryaev, Limit Theorems for Stochastic Processes, Grundlehren Math. Wiss., 288, Springer-Verlag, Berlin, 2003 | MR | Zbl

[2] G. Žitković, “A filtered version of the bipolar theorem of Brannath and Schachermayer”, J. Theoret. Probab., 15:1 (2002), 41–61 | DOI | MR | Zbl

[3] D. Kramkov, W. Schachermayer, “The asymptotic elasticity of utility functions and optimal investment in incomplete markets”, Ann. Appl. Probab., 9:3 (1999), 904–950 | DOI | MR | Zbl

[4] Yu. Kabanov, C. Stricker, “Remarks on the true no-arbitrage property”, Séminaire de Probabilités XXXVIII, Lecture Notes in Math., 1857, Springer-Verlag, Berlin, 2005, 186–194 | MR | Zbl

[5] M. M. Christensen, K. Larsen, “No arbitrage and the growth optimal portfolio”, Stoch. Anal. Appl., 25:1 (2007), 255–280 | DOI | MR | Zbl

[6] I. Karatzas, C. Kardaras, “The numéraire portfolio in semimartingale financial models”, Finance Stoch., 11:4 (2007), 447–493 | DOI | MR | Zbl

[7] F. Delbaen, W. Schachermayer, The Mathematics of Arbitrage, Springer Finance, Springer-Verlag, Berlin, 2006 | MR | Zbl

[8] Yu. M. Kabanov, D. O. Kramkov, “Bolshie finansovye rynki: asimptoticheskii arbitrazh i kontigualnost”, TVP, 39:1 (1994), 222–229 | MR | Zbl

[9] Yu. M. Kabanov, D. O. Kramkov, “Asymptotic arbitrage in large financial markets”, Finance Stoch., 2:2 (1998), 143–172 | DOI | MR | Zbl

[10] M. De Donno, P. Guasoni, M. Pratelli, “Super-replication and utility maximization in large financial markets”, Stochastic Process. Appl., 115:12 (2005), 2006–2022 | DOI | MR | Zbl

[11] F. Delbaen, W. Schachermayer, “A general version of the fundamental theorem of asset pricing”, Math. Ann., 300:3 (1994), 463–520 | DOI | MR | Zbl

[12] F. Delbaen, W. Schachermayer, “The fundamental theorem of asset pricing for unbounded stochastic processes”, Math. Ann., 312:2 (1998), 215–250 | DOI | MR | Zbl

[13] E. Platen, D. Heath, A Benchmark Approach to Quantitative Finance, Springer Finance, Springer-Verlag, Berlin, 2006 | MR | Zbl

[14] C. Zălinescu, Convex Analysis in General Vector Spaces, World Sci. Publ., Singapore, 2002 | MR | Zbl

[15] M. Pratelli, “A minimax theorem without compactness hypothesis”, Mediterr. J. Math., 2:1 (2005), 103–112 | DOI | MR | Zbl

[16] D. Becherer, “The numéraire portfolio for unbounded semimartingales”, Finance Stoch., 5:3 (2001), 327–341 | DOI | MR | Zbl

[17] O. Kallenberg, Foundations of Modern Probability, Probab. Appl. (N. Y.), Springer-Verlag, New York, NY, 1997 | MR | Zbl

[18] I. Karatzas, G. Žitković, “Optimal consumption from investment and random endowment in incomplete semimartingale markets”, Ann. Probab., 31:4 (2003), 1821–1858 | DOI | MR | Zbl