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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 29 (11 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.
Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison
 ANNALS OF APPLIED PROBABILITY
, 2008
"... It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the socalled Mokobodski cond ..."
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Cited by 23 (10 self)
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It is now established that under quite general circumstances, including in models with jumps, the existence of a solution to a reflected BSDE is guaranteed under mild conditions, whereas the existence of a solution to a doubly reflected BSDE is essentially equivalent to the socalled Mokobodski condition. As for uniqueness of solutions, this holds under mild integrability conditions. However, for practical purposes, existence and uniqueness are not enough. In order to further develop these results in Markovian setups, one also needs a (simply or doubly) reflected BSDE to be well posed, in the sense that the solution satisfies suitable bound and error estimates, and one further needs a suitable comparison theorem. In this paper, we derive such estimates and comparison results. In the last section, applicability of the results is illustrated with a pricing problem in finance.
A fourier transform method for nonparametric estimation of multivariate volatility
 Annals of Statistics
"... We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semimartingales, which is based on Fourier analysis. The covolatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the pr ..."
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Cited by 18 (2 self)
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We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semimartingales, which is based on Fourier analysis. The covolatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the prices process and the Fourier transform of the covolatility process. A nonparametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes. The asymptotic properties of the random estimator are studied: namely, consistency in probability uniformly in time and convergence in law to a mixture of Gaussian distributions. 1. Introduction. The
MULTIVARIATE SUPOU PROCESSES
"... called supOU processes, provide a class of continuous time processes capable of exhibiting long memory behavior. This paper introduces multivariate supOU processes and gives conditions for their existence and finiteness of moments. Moreover, the secondorder moment structure is explicitly calculated ..."
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Cited by 17 (7 self)
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called supOU processes, provide a class of continuous time processes capable of exhibiting long memory behavior. This paper introduces multivariate supOU processes and gives conditions for their existence and finiteness of moments. Moreover, the secondorder moment structure is explicitly calculated, and examples exhibit the possibility of longrange dependence. Our supOU processes are defined via homogeneous and factorizable Lévy bases. We show that the behavior of supOU processes is particularly nice when the mean reversion parameter is restricted to normal matrices and especially to strictly negative definite ones. For finite variation Lévy bases we are able to give conditions for supOU processes to have locally bounded càdlàg paths of finite variation and to show an analogue of the stochastic differential equation of OUtype processes, which has been suggested in [2] in the univariate case. Finally, as an important special case, we introduce positive semidefinite supOU processes, and we discuss the relevance of multivariate supOU processes in applications.
Do price and volatility jump together
, 2009
"... We consider a process Xt, which is observed on a finite time interval [0,T], at discrete times 0,∆n,2∆n,.... This process is an Itô semimartingale with stochastic volatility σ 2 t. Assuming that X has jumps on [0,T], we derive tests to decide whether the volatility process has jumps occurring simult ..."
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Cited by 17 (5 self)
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We consider a process Xt, which is observed on a finite time interval [0,T], at discrete times 0,∆n,2∆n,.... This process is an Itô semimartingale with stochastic volatility σ 2 t. Assuming that X has jumps on [0,T], we derive tests to decide whether the volatility process has jumps occurring simultaneously with the jumps of Xt. There are two different families of tests for the two possible null hypotheses (common jumps or disjoint jumps). They have a prescribed asymptotic level as the mesh ∆n goes to 0. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use on S&P 500 index data. 1. Introduction. Financial
Scaling limits via excursion theory: Interplay between CrumpModeJagers branching processes and ProcessorSharing queues
 Ann. Appl. Probab
"... We study the convergence of the M/G/1 processorsharing, queue length process in the heavy traffic regime, in the finite variance case. To do so, we combine results pertaining to Lévy processes, branching processes and queuing theory. These results yield the convergence of long excursions of the que ..."
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Cited by 13 (9 self)
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We study the convergence of the M/G/1 processorsharing, queue length process in the heavy traffic regime, in the finite variance case. To do so, we combine results pertaining to Lévy processes, branching processes and queuing theory. These results yield the convergence of long excursions of the queue length processes, toward excursions obtained from those of some reflected Brownian motion with drift, after taking the image of their local time process by the Lamperti transformation. We also show, via excursion theoretic arguments, that this entails the convergence of the entire processes to some (other) reflected Brownian motion with drift. Along the way, we prove various invariance principles for homogeneous, binary Crump–Mode–Jagers processes. In the last section we discuss potential implications of the state space collapse property, well known in the queuing literature, to branching processes. 1. Introduction. The
Consistent families of Brownian motions and stochastic flows of kernels. ArXiv: math.PR/0611292
"... Consider the following mechanism for the random evolution of a distribution of mass on the integer lattice Z. At unit rate, independently for each site, the mass at the site is split into two parts by choosing a random proportion distributed according to some specified probability measure on [0,1] a ..."
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Cited by 13 (1 self)
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Consider the following mechanism for the random evolution of a distribution of mass on the integer lattice Z. At unit rate, independently for each site, the mass at the site is split into two parts by choosing a random proportion distributed according to some specified probability measure on [0,1] and dividing the mass in that proportion. One part then moves to each of the two adjacent sites. This paper considers a continuous analogue of this evolution, which may be described by means of a stochastic flow of kernels, the theory of which was developed by Le Jan and Raimond. One of their results is that such a flow is characterized by specifying its N point motions, which form a consistent family of Brownian motions. This means for each dimension N we have a diffusion in R N, whose N coordinates are all Brownian motions. Any M coordinates taken from the Ndimensional process are distributed as the Mdimensional process in the family. Moreover, in this setting, the only interactions between coordinates are local: when coordinates differ in value they evolve independently of each other. In this paper we explain how such multidimensional diffusions may be constructed and characterized via martingale problems.
Quarticity and other functionals of volatility: efficient estimation
, 2012
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Countable state shifts and uniqueness of gmeasures
"... Abstract. In this paper we present a new approach to studying gmeasures which is based upon local absolute continuity. We extend the result in [11] that square summability of variations of gfunctions ensures uniqueness of gmeasures. The first extension is to the case of countably many symbols. Th ..."
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Cited by 10 (2 self)
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Abstract. In this paper we present a new approach to studying gmeasures which is based upon local absolute continuity. We extend the result in [11] that square summability of variations of gfunctions ensures uniqueness of gmeasures. The first extension is to the case of countably many symbols. The second extension is to some cases where g ≥ 0, relaxing the earlier requirement in [11] that inf g> 0. 1.
Strict local martingales and bubbles
"... This paper deals with asset price bubbles modeled by strict local martingales. With any strict local martingale, one can associate a new measure, which is studied in detail in the first part of the paper. In the second part, we determine the “default term ” apparent in riskneutral option prices if ..."
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Cited by 8 (0 self)
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This paper deals with asset price bubbles modeled by strict local martingales. With any strict local martingale, one can associate a new measure, which is studied in detail in the first part of the paper. In the second part, we determine the “default term ” apparent in riskneutral option prices if the underlying stock exhibits a bubble modeled by a strict local martingale. Results for certain path dependent options and last passage time formulas are given. 1. Introduction. The