Results 1  10
of
11
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 98 (9 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
Multivariate COGARCH(1, 1) processes
, 2010
"... Multivariate COGARCH(1, 1) processes are introduced as a continuoustime models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate Lévy process and the latent timevarying covariance matrix is directly specified as a stochastic process in the positive semi ..."
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Cited by 5 (1 self)
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Multivariate COGARCH(1, 1) processes are introduced as a continuoustime models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate Lévy process and the latent timevarying covariance matrix is directly specified as a stochastic process in the positive semidefinite matrices. After defining the COGARCH(1, 1) process, we analyze its probabilistic properties. We show a sufficient condition for the existence of a stationary distribution for the stochastic covariance matrix process and present criteria ensuring the finiteness of moments. Under certain natural assumptions on the moments of the driving Lévy process, explicit expressions for the first and secondorder moments and (asymptotic) secondorder stationarity of the covariance matrix process are obtained. Furthermore, we study the stationarity and secondorder structure of the increments of the multivariate COGARCH(1, 1) process and their “squares”.
Free Jacobi processes
, 2006
"... Abstract. In this paper, we define and study free Jacobi processes of parameters λ> 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not depend o ..."
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Cited by 4 (1 self)
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Abstract. In this paper, we define and study free Jacobi processes of parameters λ> 0 and 0 < θ ≤ 1, as the limit of the complex version of the matrix Jacobi process already defined by Y. Doumerc. In the first part, we focus on the stationary case for which we compute the law (that does not depend on time) and derive, for λ ∈]0, 1] and 1/θ ≥ λ + 1 a free SDE analogous to the classical one. In the second part, we generalize this result under an additional condition. To proceed, we set a recurrence formula for the moments of the process using free stochastic calculus. This will also be used to compute the p. d. e. satisfied by the Cauchy transform of the free Jacobi’s law. 1.
Radial dunkl processes : Existence and uniqueness, hitting time, beta processes and random matrices. arXiv:0707.0367v1
"... Abstract. We begin with the study of some properties of the radial Dunkl process associated to a reduced root system R. It is shown that this diffusion is the unique strong solution for all t ≥ 0 of a SDE with singular drift. Then, we study T0, the first hitting time of the positive Weyl chamber: we ..."
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Cited by 4 (1 self)
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Abstract. We begin with the study of some properties of the radial Dunkl process associated to a reduced root system R. It is shown that this diffusion is the unique strong solution for all t ≥ 0 of a SDE with singular drift. Then, we study T0, the first hitting time of the positive Weyl chamber: we prove, via stochastic calculus, a result already obtained by Chybiryakov on the finiteness of T0. The second and new part deals with the law of T0 for which we compute the tail distribution, as well as some insight via stochastic calculus on how root systems are connected with eigenvalues of standard matrixvalued processes. This gives rise to the socalled βprocesses. The ultraspherical βJacobi case still involves a reduced root system while the general case is closely connected to a non reduced one. This process lives in a convex bounded domain known as principal Weyl alcove and the strong uniqueness result remains valid. The last part deals with the first hitting time of the alcove’s boundary and the semi group density which enables us to answer some open questions. 1. Preliminaries We begin by pointing out some facts on root systems and radial Dunkl processes. We refer to [38] for the Dunkl theory, to both [7] and [27] for a background on root systems and [11], [21] for facts on radial Dunkl processes. Let (V, <,>) be a finite real Euclidean space of dimension m. A reduced root system R is a finite set of non zero vectors spanning V such that: 1 R ∩ Rα = {α, −α} for all α ∈ R. 2 σα(R) = R where σα is the reflection with respect to the hyperplane Hα orthogonal to α: < α, x> σα(x) = x − 2 α, x ∈ V < α, α> A simple system ∆ is a basis of V which induces a total ordering in R. A root α is positive if it is a positive linear combination of elements of ∆. The set of positive roots is called a positive subsystem and is denoted by R+. Note that the choice of ∆ is not unique and that R+ is uniquely determined by ∆. The reflection group
AFFINE PROCESSES ON POSITIVE SEMIDEFINITE MATRICES
, 910
"... Abstract. This paper provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. These matrixvalued affine processes have arisen from a large and growing range of useful applications in finance, including multiasset o ..."
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Cited by 4 (1 self)
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Abstract. This paper provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. These matrixvalued affine processes have arisen from a large and growing range of useful applications in finance, including multiasset option pricing with stochastic volatility and correlation structures, and fixedincome models with stochastically correlated risk factors and default intensities.
ON STRONG SOLUTIONS FOR POSITIVE DEFINITE JUMP–DIFFUSIONS
"... Abstract. We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes. ..."
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Cited by 1 (0 self)
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Abstract. We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes. 1.
LAGUERRE PROCESS AND GENERALIZED HARTMANWATSON
, 2006
"... functions of matrix argument. Abstract.: In this paper, we study complex Wishart processes or the socalled Laguerre processes (Xt)t≥0. We are interested in the behaviour of the eigenvalue process, we derive some useful stochastic differential equations and compute both the infinitesimal generator a ..."
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functions of matrix argument. Abstract.: In this paper, we study complex Wishart processes or the socalled Laguerre processes (Xt)t≥0. We are interested in the behaviour of the eigenvalue process, we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semigroup. We also give absolutecontinuity relations between different indices. Finally, we compute the density function of the socalled generalized HartmanWatson law as well as the law of T0: = inf{t, det(Xt) = 0} when the size of the matrix is 2. 1.
The Laguerre process and generalized
, 708
"... In this paper, we study complex Wishart processes or the socalled Laguerre processes (Xt)t≥0. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semigroup. We also give absolute ..."
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In this paper, we study complex Wishart processes or the socalled Laguerre processes (Xt)t≥0. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semigroup. We also give absolutecontinuity relations between different indices. Finally, we compute the density function of the socalled generalized Hartman–Watson law as well as the law of T0: = inf{t,det(Xt) = 0} when the size of the matrix is 2. Keywords: generalized Hartman–Watson law; Gross–Richards formula; Laguerre process; special functions of matrix argument
LAGUERRE PROCESS AND GENERALIZED HARTMANWATSON
"... functions of matrix argument. Abstract.: In this paper, we study complex Wishart processes or the socalled Laguerre processes (Xt)t≥0. We are interested in the behaviour of the eigenvalue process, we derive some useful stochastic differential equations and compute both the infinitesimal generator a ..."
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functions of matrix argument. Abstract.: In this paper, we study complex Wishart processes or the socalled Laguerre processes (Xt)t≥0. We are interested in the behaviour of the eigenvalue process, we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semigroup. We also give absolutecontinuity relations between different indices. Finally, we compute the density function of the socalled generalized HartmanWatson law as well as the law of T0: = inf{t, det(Xt) = 0} when the size of the matrix is 2. hal00009439, version 3 28 Nov 2006 1.
Ecole Doctorale: MATHÉMATIQUES ET SCIENCES ET TECHNOLOGIES DE L’INFORMATION ET DE LA COMMUNICATION
, 2012
"... Thèse présentée pour obtenir le grade de ..."