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93
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...
Pricing of American PathDependent Contingent Claims
, 1994
"... We consider the problem of pricing pathdependent contingent claims. Classically, this problem can be cast into the BlackScholes valuation framework through inclusion of the pathdependent variables into the state space. This leads to solving a degenerate advectiondiffusion Partial Differential Eq ..."
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Cited by 39 (1 self)
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We consider the problem of pricing pathdependent contingent claims. Classically, this problem can be cast into the BlackScholes valuation framework through inclusion of the pathdependent variables into the state space. This leads to solving a degenerate advectiondiffusion Partial Differential Equation (PDE). Standard Finite Difference (FD) methods are known to be inadequate for solving such degenerate PDE. Hence, pathdependent European claims are typically priced through MonteCarlo simulation. To date, there is no numerical method for pricing pathdependent American claims. We first establish necessary and sufficient conditions amenable to a Lie algebraic characterization, under which degenerate diffusions can be reduced to lowerdimensional nondegenerate diffusions on a submanifold of the underlying asset space. We apply these results to pathdependent options. Then, we describe a new numerical technique, called Forward Shooting Grid (FSG) method, that efficiently copes with de...
Exponential functionals of Lévy processes
 Probabilty Surveys
, 2005
"... Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0 ..."
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Cited by 34 (4 self)
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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
A Survey and Some Generalizations of Bessel Processes
 Bernoulli
, 1999
"... Bessel processes play an important role in financial mathematics because of their strong relation to financial processes like geometric Brownian motion or CIR processes. We are interested in the first time Bessel processes and more generally, radial OrnsteinUhlenbeck processes hit a given barrier. ..."
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Cited by 27 (1 self)
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Bessel processes play an important role in financial mathematics because of their strong relation to financial processes like geometric Brownian motion or CIR processes. We are interested in the first time Bessel processes and more generally, radial OrnsteinUhlenbeck processes hit a given barrier. We give explicit expressions of the Laplace transforms of first hitting times by (squared) radial OrnsteinUhlenbeck processes, i. e., CIR processes. As a natural extension we study squared Bessel processes and squared OrnsteinUhlenbeck processes with negative dimensions or negative starting points and derive their properties. Keywords: First hitting times; CIR processes; Bessel processes; radial Ornstein Uhlenbeck processes; Bessel processes with negative dimensions 1 Introduction Bessel processes have come to play a distinguished role in financial mathematics for at least two reasons, which have a lot to do with the models being usually considered. One of these models is the CoxI...
Arbitrage Possibilities In Bessel Processes And Their Relations To Local Martingales
 Probab. Theory Related Fields
, 1994
"... . 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 i ..."
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Cited by 24 (1 self)
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. 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 supermartingale. Introduction. In our paper DelbaenSchachermayer [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 DelbaenSchachermayer...
Levy Integrals and the Stationarity of generalised OrnsteinUhlenbeck processes
"... The generalised OrnsteinUhlenbeck process constructed from a bivariate Lévy process (ξt, ηt)t≥0 is defined as Vt = e −ξt ( ∫ t ..."
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Cited by 21 (9 self)
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The generalised OrnsteinUhlenbeck process constructed from a bivariate Lévy process (ξt, ηt)t≥0 is defined as Vt = e −ξt ( ∫ t
On Continuity Properties of the Law of Integrals of Lévy Processes
, 2008
"... Let (ξ,η) be a bivariate Lévy process such that the integral ∫ ∞ 0 e−ξt − dηt converges almost surely. We characterise, in terms of their Lévy measures, those Lévy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the f ..."
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Cited by 14 (4 self)
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Let (ξ,η) be a bivariate Lévy process such that the integral ∫ ∞ 0 e−ξt − dηt converges almost surely. We characterise, in terms of their Lévy measures, those Lévy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the form I: = ∫ ∞ 0 g(ξt)dt, where g is a deterministic function. We give sufficient conditions ensuring that I has no atoms, and under further conditions derive that I has a Lebesgue density. The results are also extended to certain integrals of the form ∫ ∞ 0 g(ξt)dYt, where Y is an almost surely strictly increasing stochastic process, independent of ξ.
The importance of strictly local martingales; applications to radial OrnsteinUhlenbeck processes
, 1998
"... this paper we encounter a number of examples of strictly local martingales, ..."
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Cited by 12 (1 self)
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this paper we encounter a number of examples of strictly local martingales,
Equivalent and absolutely continuous measure changes for jumpdiffusion processes” to appear in the Annals of Applied Probability
"... We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jumpdiffusion processes that can explode and be killed by a potential. ..."
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Cited by 11 (2 self)
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We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jumpdiffusion processes that can explode and be killed by a potential.
A Refined Binomial Lattice for Pricing American Asian Options
, 1998
"... . We present simple and fast algorithms for computing very tight upper and lower bounds on the prices of American Asian options in the binomial model. We introduce a new refined version of the CoxRossRubinstein [4] binomial lattice of stock prices. Each node in the lattice is partitioned into "node ..."
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Cited by 11 (0 self)
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. We present simple and fast algorithms for computing very tight upper and lower bounds on the prices of American Asian options in the binomial model. We introduce a new refined version of the CoxRossRubinstein [4] binomial lattice of stock prices. Each node in the lattice is partitioned into "nodelets", each of which represents all paths arriving at the node with a specific geometric stock price average. The upper bound uses an interpolation idea similar to the HullWhite [5] method. From the backwardrecursive upperbound computation, we estimate a good exercise rule that is consistent with the refined lattice. This exercise rule is used to obtain a lower bound on the option price using a modification of a conditionalexpectation based idea from RogersShi [11] and ChalasaniJhaVarikooty [3]. Our algorithms run in time proportional to the number of nodelets in the refined lattice, which is smaller than n 4 =20 for n periods. Keywords: American Options, Asian Options, Pathdepend...