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198
Exponential functional of a new family of Lévy processes and selfsimilar continuous state branching processes with immigration
 Bull. Sci. Math
"... Abstract. We first introduce and derive some basic properties of a twoparameters (α, γ) family of onesided Lévy processes, with 1 < α < 2 and γ> −α. Their Laplace exponents are given in terms of the Pochhammer symbol as follows ψ(γ)(λ) = c ((λ+ γ)α − (γ)α) , λ ≥ 0, where c is a positive ..."
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Cited by 27 (1 self)
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Abstract. We first introduce and derive some basic properties of a twoparameters (α, γ) family of onesided Lévy processes, with 1 < α < 2 and γ> −α. Their Laplace exponents are given in terms of the Pochhammer symbol as follows ψ(γ)(λ) = c ((λ+ γ)α − (γ)α) , λ ≥ 0, where c is a positive constant, (λ)α = Γ(λ+α) Γ(λ) stands for the Pochhammer symbol and Γ for the gamma function. These are a generalization of the Brownian motion, since in the limit case α → 2, we end up to the Laplace exponent of a Brownian motion with drift γ+ 1 2. Then, we proceed by computing the density of the law of the exponential functional associated to some elements of this family (and their dual) and some transformations of these elements. More precisely, we shall consider the Lévy processes which admit the following Laplace exponent, for any δ> α−1 α ψ(0,δ)(λ) = ψ(0)(λ) − αδ λ+ α − 1ψ (0)(λ), λ ≥ 0. These densities are expressed in terms of the Wright hypergeometric functions. By means of probabilistic arguments, we derive some interesting properties enjoyed by these functions. On the way, we also characterize explicitly the semigroup of the family of selfsimilar continuous state branching processes with immigration. Résumé. Nous introduisons et étudions quelques propriétés élémentaires d’une famille de processus de Lévy complètement asymmétriques. Leurs lois sont caractérisées par leurs exposants de Laplace qui s’expriment en termes du symbole de Pochhammer. Ensuite, nous calculons la loi de la fonctionnelle exponentielle associée a ̀ certains éléments de cette famille et d’une tranformation de ces éléments. Ces lois s’avèrent absolument continues et leurs densités s’expriment en termes des fonctions hypergéométriques de Wright. En utilisant des arguments probabilistes, nous déduisons que ces fonctions possèdent des propriétés analytiques intéressantes. Lors du déroulement de la preuve, nous caractérisons également le semigroupe des processus autosimilaires de branchement avec immigration. Key words and phrases. Lévy processes, exponential functional, selfsimilar processes, continuous state branching processes with immigration, Wright hypergeometric functions
Note of option pricing for constant elasticity of variance model
 AsiaPacific Financial Markets
"... Abstract: We study the arbitrage free option pricing problem for constant elasticity of variance (CEV) model. To treat the stochastic aspect of the CEV model, we direct attention to the relationship between the CEV model and squared Bessel processes. Then we show the existence of a unique equivalent ..."
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Cited by 24 (0 self)
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Abstract: We study the arbitrage free option pricing problem for constant elasticity of variance (CEV) model. To treat the stochastic aspect of the CEV model, we direct attention to the relationship between the CEV model and squared Bessel processes. Then we show the existence of a unique equivalent martingale measure and derive the Cox’s arbitrage free option pricing formula through the properties of squared Bessel processes. Finally we show that the CEV model admits arbitrage opportunities when it is conditioned to be strictly positive.
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 23 (1 self)
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this paper we encounter a number of examples of strictly local martingales,
THE SPECTRAL DECOMPOSITION OF THE OPTION VALUE
, 2004
"... This paper develops a spectral expansion approach to the valuation of contingent claims when the underlying state variable follows a onedimensional diffusion with the infinitesimal variance a 2 (x), drift b(x) and instantaneous discount (killing) rate r(x). The Spectral Theorem for selfadjoint ope ..."
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Cited by 23 (10 self)
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This paper develops a spectral expansion approach to the valuation of contingent claims when the underlying state variable follows a onedimensional diffusion with the infinitesimal variance a 2 (x), drift b(x) and instantaneous discount (killing) rate r(x). The Spectral Theorem for selfadjoint operators in Hilbert space yields the spectral decomposition of the contingent claim value function. Based on the Sturm–Liouville (SL) theory, we classify Feller’s natural boundaries into two further subcategories: nonoscillatory and oscillatory/nonoscillatory with cutoff Λ ≥ 0 (this classification is based on the oscillation of solutions of the associated SL equation) and establish additional assumptions (satisfied in nearly all financial applications) that allow us to completely characterize the qualitative nature of the spectrum from the behavior of a, b and r near the boundaries, classify all diffusions satisfying these assumptions into the three spectral categories, and present simplified forms of the spectral expansion for each category. To obtain explicit expressions, we observe that the Liouville transformation reduces the SL equation to the onedimensional Schrödinger equation with a potential function constructed from a, b and r. If analytical solutions are available for the Schrödinger equation, inverting the Liouville transformation yields analytical solutions for the original SL equation, and the spectral representation for the diffusion process can be constructed explicitly. This produces an explicit spectral decomposition of the contingent claim value function.
The Wiener Disorder Problem with Finite Horizon
"... The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ’disorder ’ when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is fin ..."
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Cited by 20 (12 self)
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The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ’disorder ’ when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is finite. The method of proof is based on reducing the initial problem to a parabolic freeboundary problem where the continuation region is determined by a continuous curved boundary. By means of the changeofvariable formula containing the local time of a diffusion process on curves we show that the optimal boundary can be characterized as a unique solution of the nonlinear integral equation. 1.
Bessel processes, the integral of geometric Brownian motion, and Asian options
 Theor. Probab. Appl
, 2004
"... Schröder Abstract. This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been stu ..."
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Cited by 15 (0 self)
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Schröder Abstract. This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the HartmanWatson theory of [Y80]. Consequences of this approach for valuing Asian options proper have been spelled out in [GY] whose Laplace transform results were in fact regarded as a noted advance. Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper. One of them in particular is of a principal nature and originates with the HartmanWatson approach itself: this approach is in general applicable without modifications only if it does not involve Bessel processes of negative indices. The main mathematical contribution of this paper is the developement of three principal ways to overcome these restrictions, in particular by merging stochastics and complex analysis in what seems a novel way, and the discussion of their consequences for the valuation of Asian options proper.
A TRANSFORMATION FOR LÉVY PROCESSES WITH ONESIDED JUMPS AND APPLICATIONS
"... Abstract. The aim of this work is to extend and study a family of transformations between Laplace exponents of Lévy processes which have been introduced recently in a variety of different contexts, [27, 29, 21, 17], as well as in older work of Urbanik [35]. We show how some specific instances of thi ..."
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Cited by 15 (7 self)
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Abstract. The aim of this work is to extend and study a family of transformations between Laplace exponents of Lévy processes which have been introduced recently in a variety of different contexts, [27, 29, 21, 17], as well as in older work of Urbanik [35]. We show how some specific instances of this mapping prove to be useful for a variety of applications. 1.