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130
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, ..."
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Cited by 121 (11 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.
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 expec ..."
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Cited by 120 (7 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 Existence of Absolutely Continuous Local Martingale Measures", Annals of Applied Probability
, 1995
"... Abstract. We investigate the existence of an absolutely continuous martingale measure. For continuous processes we show that the absence of arbitrage for general admissible integrands implies the existence of an absolutely continuous (not necessarily equivalent) local martingale measure. We also re ..."
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Cited by 62 (1 self)
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Abstract. We investigate the existence of an absolutely continuous martingale measure. For continuous processes we show that the absence of arbitrage for general admissible integrands implies the existence of an absolutely continuous (not necessarily equivalent) local martingale measure. We also rephrase RadonNikodym theorems for predictable processes. 1.Introduction. In our paper Delbaen and Schachermayer (1994a) we showed that for locally bounded flnite dimensional stochastic price processes S, the existence of an equivalent (local) martingale measure { sometimes called risk neutral measure { is equivalent to a property called No Free Lunch with Vanishing Risk (NFLVR). We also proved that if the set of (local) martingale measures
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 55 (15 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 ..."
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Cited by 48 (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.
Lectures on Young Measure Theory and its Applications in Economics
 Rend. Istit. Mat. Univ. Trieste
, 1998
"... this paper we work with the following hypothesis: ..."
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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 ..."
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Cited by 27 (4 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...
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 25 (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...
Dynamically consistent nonlinear evaluations and expectations
 in arXiv:math.PR/0501415 v1 24
, 2005
"... Abstract. How an agent (or a firm, an investor, a financial market) evaluates a contingent claim, say a European type of derivatives X, with maturity t? In this paper we study a dynamic evaluation of this problem. We denote by {Ft} t≥0, the information acquired by this agent. The value X is known at ..."
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Cited by 19 (5 self)
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Abstract. How an agent (or a firm, an investor, a financial market) evaluates a contingent claim, say a European type of derivatives X, with maturity t? In this paper we study a dynamic evaluation of this problem. We denote by {Ft} t≥0, the information acquired by this agent. The value X is known at the maturity t means that X is an Ft–measurable random variable. We denote by Es,t[X] the evaluated value of X at the time s ≤ t. Es,t[X] is Fs–measurable since his evaluation is based on his information at the time s. Thus Es,t[·] is an operator that maps an Ft–measurable random variable to an Fs–measurable one. A system of operators {Es,t[·]} 0≤s≤t<∞ is called Ft–consistent evaluations if it satisfies the following conditions: (A1) Es,t[X] ≥ Es,t[Y], if X ≥ Y; (A2) Et,t[X] = X; (A3) Er,sEs,t[X] = Er,t[X], for r ≤ s ≤ t; (A4)
Vector Stochastic Integrals and the Fundamental Theorems of Asset
 Pricing”, Proceedings of the Steklov Mathematical Institute
, 2002
"... Abstract. This paper deals with the foundations of the stochastic mathematical finance, and it has three main purposes: 1. We present a selfcontained construction of the vector stochastic integral H • X with respect to a ddimensional semimartingale X = (X1t,..., Xdt). This notion is more general t ..."
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Cited by 19 (0 self)
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Abstract. This paper deals with the foundations of the stochastic mathematical finance, and it has three main purposes: 1. We present a selfcontained construction of the vector stochastic integral H • X with respect to a ddimensional semimartingale X = (X1t,..., Xdt). This notion is more general than the componentwise stochastic integral ∑d i=1H i •Xi. 2. We show that the vector stochastic integrals are important in the mathematical finance. To be more precise, the notion of the componentwise stochastic integral is insufficient in the First and the Second Fundamental Theorems of Asset Pricing. 3. We prove the Second Fundamental Theorem of Asset Pricing in the general setting, i.e. in the continuoustime case for a general multidimensional semimartingale. The proof is based on the martingale techniques and, in particular, on the properties of the vector stochastic integral.