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67
On The Distribution And Asymptotic Results For Exponential Functionals Of Lévy Processes
, 1997
"... . The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional A t := R t 0 e s ds, where ( s ; s 0) denotes a L'evy process. When A1 ! 1, we show that in most cases, the law of A1 is a solution of an integrodifferential equation ; moreover, this ..."
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Cited by 65 (8 self)
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. The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional A t := R t 0 e s ds, where ( s ; s 0) denotes a L'evy process. When A1 ! 1, we show that in most cases, the law of A1 is a solution of an integrodifferential equation ; moreover, this law is characterized by its integral moments. When the process is asymptotically ffstable, we prove that t \Gamma1=ff log A t converges in law, as t !1, to the supremum of an ffstable L'evy process ; in particular, if E [ 1 ] ? 0, then ff = 1 and (1=t) log A t converges almost surely to E [ 1 ]. Eventually, we use Girsanov's transform to give the explicit behavior of E \Theta (a +A t ()) \Gamma1 as t ! 1, where a is a constant, and deduce from this the rate of decay of the tail of the distribution of the maximum of a diffusion process in a random L'evy environment. 1. Introduction We first describe three different sources of interest for exponential functionals of Brownian mot...
qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation ..."
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Cited by 64 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
The NoArbitrage Property under a change of numéraire
 Stochastics and Stochastic Reports
, 1995
"... Abstract. For a price process that has an equivalent risk neutral measure, we investigate if the same property holds when the numeraire is changed. We give necessary and su cient conditions under which the price process of a particular asset which should be thought of as a di erent currency can be ..."
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Cited by 29 (11 self)
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Abstract. For a price process that has an equivalent risk neutral measure, we investigate if the same property holds when the numeraire is changed. We give necessary and su cient conditions under which the price process of a particular asset which should be thought of as a di erent currency can be chosen as new numeraire. The result is related to the characterization of attainable claims that can be hedged. Roughly speaking: the asset representing the new currency is a reasonable investment (in terms of the old currency) if and only if the market does not permit arbitrage opportunities in terms of the new currency as numeraire. This rough but economically meaningful idea is given a precise content in this paper. The main ingredients are a duality relation as well as a result on maximal elements. The paper also generalizes results previously obtained by Jacka, AnselStricker and the authors.
The VarianceOptimal Martingale Measure for Continuous Processes
 Bernoulli
, 1996
"... Abstract. We prove that for continuous stochastic processes S based on ( � F�P) for which there is an equivalent martingale measure Q 0 with squareintegrable density dQ 0 =dPwe have that the socalled "variance optimal " martingale measure Qopt for which the density dQopt =dPhas minimal L2 (P)nor ..."
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Cited by 25 (2 self)
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Abstract. We prove that for continuous stochastic processes S based on ( � F�P) for which there is an equivalent martingale measure Q 0 with squareintegrable density dQ 0 =dPwe have that the socalled "variance optimal " martingale measure Qopt for which the density dQopt =dPhas minimal L2 (P)norm is automatically equivalenttoP. The result is then applied to an approximation problem arising in Mathematical Finance. 1.
Modelling of Default Risk: An Overview
, 1999
"... The aim of these notes is to provide a relatively concise but still selfcontained overview of mathematical notions and results which underpin the valuation of defaultable claims. Though the default risk modelling was extensively studied in numerous recent papers, it seems nonetheless that some of ..."
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Cited by 15 (7 self)
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The aim of these notes is to provide a relatively concise but still selfcontained overview of mathematical notions and results which underpin the valuation of defaultable claims. Though the default risk modelling was extensively studied in numerous recent papers, it seems nonetheless that some of these papers lack a sound theoretical background. Our goal is to furnish results which cover both the classic valueofthefirm (or structural) approach, as well as the more recent intensitybased methodology. For a more detailed account of mathematical results
Lectures on Young Measure Theory and its Applications in Economics
 Rend. Istit. Mat. Univ. Trieste
, 1998
"... this paper we work with the following hypothesis: ..."
Optional Decomposition and Lagrange Multipliers
, 1997
"... Let Q be the set of equivalent martingale measures for a given process S, and let X be a process which is a local supermartingale with respect to any measure in Q. The optional decomposition theorem for X states that there exists a predictable integrand # such that the di#erence X# · S is a ..."
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Cited by 13 (1 self)
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Let Q be the set of equivalent martingale measures for a given process S, and let X be a process which is a local supermartingale with respect to any measure in Q. The optional decomposition theorem for X states that there exists a predictable integrand # such that the di#erence X# · S is a decreasing process. In this paper we give a new proof which uses techniques from stochastic calculus rather than functional analysis, and which removes any boundedness assumption. Key words: optional decomposition, semimartingale, equivalent martingale measure, Hellinger process, Lagrange multiplier JEL Classification: G10, G12 AMS Classification: 60H05, 90A09 # The paper will appear in Finance and Stochastics. Support of the Deutsche Forschungsgemeinschaft (SFB 373 at Humboldt University) and of Volkswagenstiftung is gratefully acknowledged. <E901> 1 Introduction Let S be an R d valued rightcontinuous semimartingale given on a stochastic basis (#, F , F = (F t ), P ) with the us...
A nonSkorohod topology on the Skorohod space
 Electron. J. Probab
, 1997
"... : A new topology (called S) is defined on the space ID of functions x : [0; 1] ! IR 1 which are rightcontinuous and admit limits from the left at each t ? 0. Although S cannot be metricized, it is quite natural and shares many useful properties with the traditional Skorohod's topologies J 1 and M ..."
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Cited by 12 (2 self)
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: A new topology (called S) is defined on the space ID of functions x : [0; 1] ! IR 1 which are rightcontinuous and admit limits from the left at each t ? 0. Although S cannot be metricized, it is quite natural and shares many useful properties with the traditional Skorohod's topologies J 1 and M 1 . In particular, on the space P(ID) of laws of stochastic processes with trajectories in ID the topology S induces a sequential topology for which both the direct and the converse Prohorov's theorems are valid, the a.s. Skorohod representation for subsequences exists and finite dimensional convergence outside a countable set holds. Keywords: Skorohod space, Skorohod representation, convergence in distribution, sequential spaces, semimartingales. AMS subject classification: 60F17, 60B05, 60G17, 54D55. Research supported by Komitet Bada'n Naukowych under Grant No 2 1108 91 01. Submitted to EJP on April 1, 1996. Final version accepted on July 4, 1997. A nonSkorohod topology on the Skoroh...
Hitting probabilities for systems of nonlinear stochastic heat equations with additive noise
, 2007
"... ..."
New Fundamentals of Young Measure Convergence
 in Calculus of Variations and Optimal Control
, 2000
"... New fundamentals of Young measure convergence ..."