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833
Brownian Excursions, Critical Random Graphs and the Multiplicative Coalescent
, 1996
"... Let (B t (s); 0 s ! 1) be reflecting inhomogeneous Brownian motion with drift t \Gamma s at time s, started with B t (0) = 0. Consider the random graph G(n; n \Gamma1 +tn \Gamma4=3 ), whose largest components have size of order n 2=3 . Normalizing by n \Gamma2=3 , the asymptotic joint d ..."
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Cited by 86 (10 self)
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Let (B t (s); 0 s ! 1) be reflecting inhomogeneous Brownian motion with drift t \Gamma s at time s, started with B t (0) = 0. Consider the random graph G(n; n \Gamma1 +tn \Gamma4=3 ), whose largest components have size of order n 2=3 . Normalizing by n \Gamma2=3 , the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of B t (Corollary 2). The dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process. The states of this process are vectors x of nonnegative real cluster sizes (x i ), and clusters with sizes x i and x j merge at rate x i x j . The multiplicative coalescent is shown to be a Feller process on l 2 . The random graph limit specifies the standard multiplicative coalescent, which starts from infinitesimally small clusters at time \Gamma1: the existence of such a process is not obvious. AMS 1991 subject classifications. 60C05, 60J50, Key words and phras...
The Dynamics of Stochastic Volatility: Evidence from Underlying and Option Markets
, 2000
"... This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultane ..."
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Cited by 72 (1 self)
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This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultaneously. I conclude that the square root stochastic variance model of Heston (1993) and others is incapable of generating realistic returns behavior and find that the data are more accurately represented by a stochastic variance model in the CEV class or a model that allows the price and variance processes to have a timevarying correlation. Specifically, I find that as the level of market variance increases, the volatility of market variance increases rapidly and the correlation between the price and variance processes becomes substantially more negative. The heightened heteroskedasticity in market variance that results generates realistic crash probabilities and dynamics and causes returns to display values of skewness and kurtosis much more consistent with their sample values. While the model dramatically improves the fit of options prices relative to the square root process, it falls short of explaining the implied volatility smile for shortdated options.
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...
Exact and computationally efficient likelihoodbased estimation for discretely observed diffusion processes
 Journal of the Royal Statistical Society, Series B: Statistical Methodology
, 2006
"... The objective of this paper is to present a novel methodology for likelihoodbased inference for discretely observed diffusions. We propose Monte Carlo methods, which build on recent advances on the exact simulation of diffusions, for performing maximum likelihood and Bayesian estimation. ..."
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Cited by 58 (12 self)
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The objective of this paper is to present a novel methodology for likelihoodbased inference for discretely observed diffusions. We propose Monte Carlo methods, which build on recent advances on the exact simulation of diffusions, for performing maximum likelihood and Bayesian estimation.
Probability laws related to the Jacobi theta and Riemann zeta functions, and the Brownian excursions
 Bulletin (New series) of the American Mathematical Society
"... Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional ..."
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Cited by 57 (11 self)
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Abstract. This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to onedimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws. Contents
Branched rough paths
, 2006
"... Abstract. We formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints. This allows us to define a good notion of integral with respect to irregular paths with Hölder exponent greater than 1/3 (e.g. samp ..."
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Cited by 53 (9 self)
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Abstract. We formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints. This allows us to define a good notion of integral with respect to irregular paths with Hölder exponent greater than 1/3 (e.g. samples of Brownian motion) and study the problem of the existence, uniqueness and continuity of solution of differential equations driven by such paths. We recover Young’s theory of integration and the main results of Lyons’ theory of rough paths in Hölder topology. 1.
MICROSTRUCTURE NOISE, REALIZED VARIANCE, AND OPTIMAL SAMPLING
, 2005
"... Observed asset prices are known to deviate from their efficient values due to market microstructure frictions. This paper studies the effects of market microstructure noise on nonparametric estimates of the efficient price integrated variance. Specifically, we consider both asymptotic and finite sam ..."
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Cited by 49 (5 self)
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Observed asset prices are known to deviate from their efficient values due to market microstructure frictions. This paper studies the effects of market microstructure noise on nonparametric estimates of the efficient price integrated variance. Specifically, we consider both asymptotic and finite sample effects of general market microstructure noise on realized variance estimates. The finite sample results culminate in a variance/bias tradeoff that serves as a basis for an optimal sampling theory. Our theory also considers the effects of prefiltering the data and proposes a novel biascorrection. We show that this theory is easily implementable in practise requiring only the calculation of sample moments of the observable highfrequency return data.
An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions
, 1996
"... this paper, we denote by #Z t # t#0 a normalized fractional Brownian motion #FBM# with selfsimilarity parameter H #0; 1#, characterized by the following properties: #i# Z t has stationary increments; #ii# Z 0 =0, and EZ t =0for all t; #iii# EZ jtj for all t; #iv# Z t is Gaussian; #v# Z t h ..."
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Cited by 48 (9 self)
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this paper, we denote by #Z t # t#0 a normalized fractional Brownian motion #FBM# with selfsimilarity parameter H #0; 1#, characterized by the following properties: #i# Z t has stationary increments; #ii# Z 0 =0, and EZ t =0for all t; #iii# EZ jtj for all t; #iv# Z t is Gaussian; #v# Z t has continuous sample paths. We assume that Z is de#ned on a probability and denote by #F t # t#0 the #ltration #history# generated by Z. This process was originally de#ned and studied by Kolmogorov #1940# within a Hilbert space framework. In another pioneering paper on FBM, Mandelbrot and Van Ness #1968# de#ned the process more constructively as the integral Z s = c H dW u + ; #1.1# where W t is the standard Brownian motion. The normalization EZ 1 = 1 is achieved with the choice c H = 2H,# ,H# ,#H + #,#2 where B denotes the beta function B##; ##= x# dx = ,###,### ,## + ## : Note that the choice H =1=2 gives a standard Brownian motion, and another special case H = 1 gives a deterministic process with linear paths. In all cases, FBMs are selfsimilar processes. The sample path assumption #v# above can be strengthened to H#lder continuity #this follows from Kolmogorov's criterion for the continuity of sample paths, see Revuz and Yor 1991#: Theorem 1.1 The sample paths of a continuous fractional Brownian motion with parameter H are, outside a negligible event, H#lder continuous with every exponent # #H. When H = 2f1=2; 1g, the FBM is neither a Markov process nor a semimartingale #see, e.g., Liptser and Shiryaev 1986, Example 2 of Section 4.9.13#. It is, however, a process with a simple structure, and several interesting objects related to it have explicite expressions. For example, a formula for the conditional expectation E...