Results 1 - 10
of
30
Stochastic Volatility for Lévy Processes
, 2001
"... Three processes re°ecting persistence of volatility are initially formulated by evaluating three L¶evy processes at a time change given by the integral of a mean reverting square root process. The model for the mean reverting time change is then generalized to include Non-Gaussian models that are so ..."
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Cited by 209 (12 self)
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Three processes re°ecting persistence of volatility are initially formulated by evaluating three L¶evy processes at a time change given by the integral of a mean reverting square root process. The model for the mean reverting time change is then generalized to include Non-Gaussian models that are solutions to OU (Ornstein-Uhlenbeck) equations driven by one sided discontinuous L¶evy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general, mean corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean corrected exponential model is not a martingale in the ¯ltration in which it is originally de¯ned. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered ¯ltrations consistent with the one dimensional marginal distributions of the level of the process at each future date. 1
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 Ornstein--Uhlenbeck processes hit a given barrier. ..."
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Cited by 66 (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 Ornstein--Uhlenbeck processes hit a given barrier. We give explicit expressions of the Laplace transforms of first hitting times by (squared) radial Ornstein--Uhlenbeck processes, i. e., CIR processes. As a natural extension we study squared Bessel processes and squared Ornstein--Uhlenbeck 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 Cox--I...
Self-similar processes with independent increments associated with Lévy and Bessel processes
, 2001
"... Wolfe [28] and Sato [24] gave two different representations of a random variable X 1 with a self-decomposable distribution in terms of processes with independent increments. This paper shows how either of these representations follows easily from the other, and makes these representations more e ..."
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Cited by 26 (8 self)
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Wolfe [28] and Sato [24] gave two different representations of a random variable X 1 with a self-decomposable distribution in terms of processes with independent increments. This paper shows how either of these representations follows easily from the other, and makes these representations more explicit when X 1 is either a first or last passage time for a Bessel process.
Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times
- ELECTRON. J. PROBAB
, 1999
"... For a random process X consider the random vector defined by the values of X at times 0 <U n,1 < ... < U n,n < 1 and the minimal values of X on each of the intervals between consecutive pairs of these times, where the U n,i are the order statistics of n independent uniform (0, 1) variabl ..."
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Cited by 24 (4 self)
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For a random process X consider the random vector defined by the values of X at times 0 <U n,1 < ... < U n,n < 1 and the minimal values of X on each of the intervals between consecutive pairs of these times, where the U n,i are the order statistics of n independent uniform (0, 1) variables, independent of X . The joint law of this random vector is explicitly described when X is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander are deduced by appropriate conditioning. These descriptions yield numerous new identities involving the laws of these processes, and simplified proofs of various known results, including Aldous's characterization of the random tree constructed by sampling the excursion at n independent uniform times, Vervaat's transformation of Brownian bridge into Brownian excursion, and Denisov's decomposition of the Brownian motion at the time of its minimum into two independent Brownian meanders. Other consequences of the sampling formulae a...
The importance of strictly local martingales; applications to radial Ornstein-Uhlenbeck processes
, 1998
"... this paper we encounter a number of examples of strictly local martingales, ..."
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Cited by 24 (1 self)
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this paper we encounter a number of examples of strictly local martingales,
Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes
, 2011
"... We study the distribution and various properties of exponential functionals of hypergeometric Lévy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic series expansions of its probability density function. As applic ..."
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Cited by 21 (9 self)
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We study the distribution and various properties of exponential functionals of hypergeometric Lévy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic series expansions of its probability density function. As applications we present a new proof of some of the results on the density of the supremum of a stable process, which were recently obtained in [25] and [23]. We also derive some new results related to (i) the entrance law of the stable process conditioned to stay positive, (ii) the entrance law of the excursion measure of the stable process reflected at its past infimum and (iii) the entrance law and the last passage time of the radial part of n-dimensional symmetric stable process.
The law of the maximum of a Bessel bridge
- Electronic J. Probability
, 1998
"... Let M ffi be the maximum of a standard Bessel bridge of dimension ffi . A series formula for P (M ffi a) due to Gikhman and Kiefer for ffi = 1; 2; : : : is shown to be valid for all real ffi ? 0. Various other characterizations of the distribution of M ffi are given, including formulae for its Mel ..."
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Cited by 18 (6 self)
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Let M ffi be the maximum of a standard Bessel bridge of dimension ffi . A series formula for P (M ffi a) due to Gikhman and Kiefer for ffi = 1; 2; : : : is shown to be valid for all real ffi ? 0. Various other characterizations of the distribution of M ffi are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of M ffi as is described both as ffi !1 and as ffi # 0. Keywords: Brownian bridge, Brownian excursion, Brownian scaling, local time, Bessel process, zeros of Bessel functions, Riemann zeta function Contents 1 Introduction 3 2 The maximum of a diffusion bridge 8 3 The Gikhman-Kiefer Formula 9 4 The law of T ffi and the agreement formula 11 5 The first passage transform and its derivatives 13 6 Moments 16 7 Dimensions one and three 20 8 Limits as ffi !1 22 9 Limits as ffi # 0 24 10 Relation to last exit times 27 11 A series involving the zeros of J 30 A Some Useful Formulae 33 A.1 Bessel Functions : : : : : : : : : : :...
Paths in Weyl chambers and random matrices
"... Baryshnikov [3] and Gravner, Tracy & Widom [14] have shown that the largest eigenvalue of a random matrix of the G.U.E. of order d has the same distribution as 1#t1 ##td-1#0 +W d (t d-1 )] , where W = (W 1 , , W d ) is a d-dimensional Brownian motion. We provide a generalization o ..."
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Cited by 13 (0 self)
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Baryshnikov [3] and Gravner, Tracy & Widom [14] have shown that the largest eigenvalue of a random matrix of the G.U.E. of order d has the same distribution as 1#t1 ##td-1#0 +W d (t d-1 )] , where W = (W 1 , , W d ) is a d-dimensional Brownian motion. We provide a generalization of this formula to all the eigenvalues and give a geometric interpretation. For any Weyl chamber a + of an Euclidean finite-dimensional space a, we define a natural continuous path transformation which associates to a path w in a a path w in a+ . This transformation occurs in the description of the asymptotic behaviour of some deterministic dynamical systems on the symmetric space G/K where G is the complex group with chamber a + . When a = R , a + = , x d ); x 1 > x 2 > > x d if W is the Euclidean Brownian motion on a then W is the process of the eigenvalues of the Dyson Brownian motion on the set of Hermitian matrices and (T W )(1) is distributed as the eigenvalues of the G.U.E.
On The Laws Of Homogeneous Functionals Of The Brownian Bridge.
- Studia Sci. Math. Hungar
, 1998
"... . We develop a general and elementary method, which allows in particular to compute the distributions of a large number of interesting homogeneous functionals of the standard Brownian bridge. 1. Introduction. Let (B t ; t 0) be a Brownian motion starting from 0, and f : R \Gamma! R a locally boun ..."
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Cited by 11 (5 self)
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. We develop a general and elementary method, which allows in particular to compute the distributions of a large number of interesting homogeneous functionals of the standard Brownian bridge. 1. Introduction. Let (B t ; t 0) be a Brownian motion starting from 0, and f : R \Gamma! R a locally bounded Borel function. It is well-known that the computation of the law of R S 0 f(B u ) du is more involved when S is a fixed time, t say, than when S is equal to either T a = infft; B t = ag, or ø l = infft; L t ? lg, where (L t ; t 0) denotes the local time of B at 0. An obvious "reason" for this is that the value of Brownian motion B at time t is not fixed, whereas B Ta = a and B ø l = 0. A classical manner to overcome the difficulty for time t is to replace t by S , an independent exponential time of parameter , and use Feynman-Kac formula, which allows to compute: E exp ` \Gamma¯ Z S 0 f(B u ) du ' = Z 1 0 e \Gammat E exp ` \Gamma¯ Z t 0 f(B u ) du ' dt i...
