## Arbitrage Possibilities In Bessel Processes And Their Relations To Local Martingales (1994)

Venue: | Probab. Theory Related Fields |

Citations: | 24 - 1 self |

### BibTeX

@ARTICLE{Delbaen94arbitragepossibilities,

author = {Freddy Delbaen and Walter Schachermayer},

title = {Arbitrage Possibilities In Bessel Processes And Their Relations To Local Martingales},

journal = {Probab. Theory Related Fields},

year = {1994},

volume = {102},

pages = {357--366}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We show that, if we allow general admissible integrands as trading strategies, the three dimensional Bessel process, Bes 3 , admits arbitrage possibilities. This is in contrast with the fact that the inverse process is a local martingale and hence is arbitrage free. This leads to some economic interpretation for the analysis of the property of arbitrage in foreign exchange rates. This notion (relative to general admissible integrands) does depend on the fact, which of the two currencies under consideration is chosen as num'eraire. The results rely on a general construction of strictly positive local martingales. The construction is related to the Follmer measure of a positive super-martingale. Introduction. In our paper Delbaen-Schachermayer [DS1], we showed that the inverse of the Bes 3 process, an example of a strict local martingale, doesn't allow arbitrage possibilities. In the present paper we investigate the Bes 3 process itself. The methods used in Delbaen-Schachermayer...

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Citation Context ... by the European Community Stimulation Plan for Economic Science contract Number SPES-CT91-0089 Typeset by A M S-T E X We will make use of the notation and definitions as in the book by Revuz and Yor =-=[RY]-=-. Especially for the definition of Bessel processes, the theory of continuous martingales and Girsanov transformations, we will use this book as the basic reference. The authors want to thank Michel '... |

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Citation Context ...has the G \Gammapredictable representation property with respect to R. Consequently 1=M also has the G \Gammapredictable representation property with respect to R. Proof. See Th 12.22 in Jacod's book,=-=[J]-=-.sCorollary. If M is a local martingale with the F \Gammapredictable representation property under P and if R and P are equivalent, i.e. P [M T = 0] = 0, then 1=M has the F \Gammapredictable represent... |

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Citation Context ... precisely we want to make the construction of the preceding section more transparent. The main ingredient for this is the Girsanov transformation for Bessel processes. The theory is developed by Yor =-=[Y]-=- and in particular the formula (2.c) there, relates Bessel processes from dimension ffi ? 2 with Bessel processes of (possibly negative) dimension 4 \Gamma ffi. An application of these results immedia... |

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Citation Context ...agent. In Delbaen-Schachermayer [DS3] we discuss the arbitrage property when the num'eraire is changed. The results given there extend the previous results. The method however is related to our paper =-=[DS2]-=- and is more involved than the construction of strict local martingales given here. We finally remark that Theorem 5 can be proved directly, i.e. without using the projective limit construction. Of co... |

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Citation Context ...e change of currency. So one agent can use admissible strategies (buying and selling) that have no equivalent admissible translation (selling and buying) for the other agent. In Delbaen-Schachermayer =-=[DS3]-=- we discuss the arbitrage property when the num'eraire is changed. The results given there extend the previous results. The method however is related to our paper [DS2] and is more involved than the c... |

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Citation Context ...ng the projective limit construction. Of course proceeding that way does not indicate how strict local martingales arise in a natural way. We also remark that the counterexample constructed in [S] or =-=[DS4]-=- yields a continuous local martingale L and a uniformly integrable strictly positive martingale Z such that LZ is a uniformly integrable martingale. Since Z = (1=L) (LZ) is a martingale, we find that ... |

18 |
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Citation Context ...y on a general construction of strictly positive local martingales. The construction is related to the Follmer measure of a positive super-martingale. Introduction. In our paper Delbaen-Schachermayer =-=[DS1]-=-, we showed that the inverse of the Bes 3 process, an example of a strict local martingale, doesn't allow arbitrage possibilities. In the present paper we investigate the Bes 3 process itself. The met... |

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Citation Context ...e we also need an extension theorem for measures we need a space that is big enough. The construction is an interpretation of the construction of the Follmer measure of a supermartingale.(see Follmer =-=[F]-=-, Az'ema-Jeulin [AJ] and Meyer [M]). So the methods we use are standard. However in our approach the supermartingale is a strictly positive continuous local martingale and this simplifies the construc... |

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Citation Context ...s given. We shall always denote by N the process defined as N t = M t \Gamma Z t 0 (1=M u ) dhM ;Mi u : It follows from Lenglart's extension of the Girsanov formula (see, e.g. Revuz-Yor [RY] p.303 or =-=[L]-=-) that N is an R- local martingale and that dM t = dN t + 1 M t dhN t ; N t i is the Dob-Meyer decomposition of the semi-martingale M under R. The usual setting in the applications of probability theo... |

4 |
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Citation Context ...m for measures we need a space that is big enough. The construction is an interpretation of the construction of the Follmer measure of a supermartingale.(see Follmer [F], Az'ema-Jeulin [AJ] and Meyer =-=[M]-=-). So the methods we use are standard. However in our approach the supermartingale is a strictly positive continuous local martingale and this simplifies the construction and allows us to use a natura... |

3 |
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Citation Context ...out using the projective limit construction. Of course proceeding that way does not indicate how strict local martingales arise in a natural way. We also remark that the counterexample constructed in =-=[S]-=- or [DS4] yields a continuous local martingale L and a uniformly integrable strictly positive martingale Z such that LZ is a uniformly integrable martingale. Since Z = (1=L) (LZ) is a martingale, we f... |

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Citation Context ...known, see for instance Revuz-Yor [RY] p. 294 and Ex 1.2.2, question 2 p. 419. This description is an example of a more general procedure known as the construction of the h\Gammaprocess. In Biane-Yor =-=[BY] this cons-=-truction was used to study properties of the "m'eandre brownien". In order to keep a more general framework and, in particular, to be able to derive results for Bessel processes of dimension... |

2 |
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Citation Context ...utely continuous with respect to P . A local martingale that is not a uniformly integrable martingale will be called a strict local martingale. This terminology was introduced by Elworthy, Li and Yor =-=[ELY]-=-, where an analysis of strict local martingales is given. We shall always denote by N the process defined as N t = M t \Gamma Z t 0 (1=M u ) dhM ;Mi u : It follows from Lenglart's extension of the Gir... |

1 |
et T. Jeulin, Pr'ecisions sur la mesure de
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Citation Context ...xtension theorem for measures we need a space that is big enough. The construction is an interpretation of the construction of the Follmer measure of a supermartingale.(see Follmer [F], Az'ema-Jeulin =-=[AJ]-=- and Meyer [M]). So the methods we use are standard. However in our approach the supermartingale is a strictly positive continuous local martingale and this simplifies the construction and allows us t... |

1 |
Topics in Finance, Talk held during the "6.Tagung Geld, Finanzwirtschaft, Banken und Versicherungen
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Citation Context ...he statistician in fact only looks at survivors. The stocks, mutual funds, investment opportunities that performed very badly did not survive and the sample suffers from survivor bias. (see e.g. Ross =-=[R]-=-). The example given is such an illustration. By looking at the trajectories that survived we were even able to obtain arbitrage with respect to simple integrands. Theorem 3 shows the general case. Ar... |