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600
NonUniform Random Variate Generation
, 1986
"... Abstract. This is a survey of the main methods in nonuniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various ..."
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Cited by 632 (21 self)
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Abstract. This is a survey of the main methods in nonuniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods.
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 NonGaussian models that are so ..."
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Cited by 100 (7 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 NonGaussian models that are solutions to OU (OrnsteinUhlenbeck) 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
TimeChanged Lévy Processes and Option Pricing
, 2002
"... As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return ..."
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Cited by 90 (13 self)
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As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that timechanged Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.
Stochastic Partial Differential Equations. Birkhauser
 Stoch. Dyn
, 1996
"... In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a dparameter (pure jump) Lévy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Lévy white noise for any dimension d. The soluti ..."
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Cited by 85 (2 self)
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In this paper we develop a white noise framework for the study of stochastic partial differential equations driven by a dparameter (pure jump) Lévy white noise. As an example we use this theory to solve the stochastic Poisson equation with respect to Lévy white noise for any dimension d. The solution is a stochastic distribution process given explicitly. We also show that if d≤3, then this solution can be represented as a classical random field in L 2 (µ), where µ is the probability law of the Lévy process. The starting point of our theory is a chaos expansion in terms of generalized Charlier polynomials. Based on this expansion we define Kondratiev spaces and the Lévy Hermite transform. 1. Introduction. White
Stochastic models for generic images
 Quart. Appl. Math
"... images has been used for at least 20 years, going back, for instance, to the work of Grenander [Gr] and Cooper [Co]. To apply these techniques, one needs, of course, a probabilistic model for some class of images or some class of structures present in images. Many models of this type have been intro ..."
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Cited by 76 (4 self)
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images has been used for at least 20 years, going back, for instance, to the work of Grenander [Gr] and Cooper [Co]. To apply these techniques, one needs, of course, a probabilistic model for some class of images or some class of structures present in images. Many models of this type have been introduced. There are stochastic models for image
Affine processes and applications in finance
 Annals of Applied Probability
, 2003
"... Abstract. We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and OrnsteinUhlenbeck type processes. We show, and provide foundations for, a wide range of financial ap ..."
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Cited by 38 (5 self)
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Abstract. We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and OrnsteinUhlenbeck type processes. We show, and provide foundations for, a wide range of financial applications for regular affine processes.
Model specification and risk premia: Evidence from futures options
 Journal of Finance
, 2007
"... This paper examines specification issues and estimates diffusive and jump risk premia using S&P futures option prices from 1987 to 2003. We first develop a test to detect the presence of jumps in volatility, and find strong evidence supporting their presence. Based on the crosssectional fit of opti ..."
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Cited by 38 (4 self)
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This paper examines specification issues and estimates diffusive and jump risk premia using S&P futures option prices from 1987 to 2003. We first develop a test to detect the presence of jumps in volatility, and find strong evidence supporting their presence. Based on the crosssectional fit of option prices, we find strong evidence for jumps in prices and modest evidence for jumps in volatility. We are not able to identify a statistically significant diffusive volatility risk premium. We do find modest but statistically and economically significant jump risk premia.
Ruin probabilities and overshoots for general Lévy insurance risk processes
 ANN. APPL. PROBAB
, 2004
"... We formulate the insurance risk process in a general Lévy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to − ∞ a.s. and the positive tail of the Lévy measure, or of the l ..."
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Cited by 37 (20 self)
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We formulate the insurance risk process in a general Lévy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to − ∞ a.s. and the positive tail of the Lévy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Klüppelberg [Stochastic Process. Appl. 64 (1996) 103–125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207–226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewaltype properties of Lévy processes.
Random Walks with Strongly Inhomogeneous Rates and Singular Diffusions: Convergence, Localization and Aging in One Dimension
, 2000
"... Let = ( i : i 2 Z) denote i.i.d. positive random variables with common distribution F and (conditional on ) let X = (X t : t 0; X 0 = 0), be a continuoustime simple symmetric random walk on Z with inhomogeneous rates ( \Gamma1 i : i 2 Z). When F is in the domain of attraction of a stable law o ..."
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Cited by 36 (4 self)
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Let = ( i : i 2 Z) denote i.i.d. positive random variables with common distribution F and (conditional on ) let X = (X t : t 0; X 0 = 0), be a continuoustime simple symmetric random walk on Z with inhomogeneous rates ( \Gamma1 i : i 2 Z). When F is in the domain of attraction of a stable law of exponent ff ! 1 (so that E( i ) = 1 and X is subdiffusive), we prove that (X; ), suitably rescaled (in space and time), converges to a natural (singular) diffusion Z = (Z t : t 0; Z 0 = 0) with a random (discrete) speed measure ae. The convergence is such that the "amount of localization", E P i2Z [P(X t = ij )] 2 converges as t ! 1 to E P z2R [P(Z s = zjae)] 2 ? 0, which is independent of s ? 0 because of scaling/selfsimilarity properties of (Z; ae). The scaling properties of (Z; ae) are also closely related to the "aging" of (X; ). Our main technical result is a general convergence criterion for localization and aging functionals of diffusions/walks Y (ffl) with (nonrando...