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Numerical Integration of Stochastic Differential Equations
, 1995
"... Abstract. We propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients. Following this concept, we discard the approximate trajectories which leave a sufficiently ..."
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Cited by 191 (12 self)
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Abstract. We propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients. Following this concept, we discard the approximate trajectories which leave a sufficiently large sphere. We prove that accuracy of any method of weak order p is estimated by ε + O(hp), where ε can be made arbitrarily small with increasing radius of the sphere. The results obtained are supported by numerical experiments. Key words. SDEs with nonglobally Lipschitz coefficients, numerical integration of SDEs in the weak sense
An algorithmic introduction to numerical simulation of stochastic differential equations
 SIAM Review
, 2001
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HighOrder Collocation Methods for Differential Equations with Random Inputs
 SIAM Journal on Scientific Computing
"... Abstract. Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling met ..."
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Cited by 180 (9 self)
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Abstract. Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling methods. However, when the governing equations take complicated forms, numerical implementations of stochastic Galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations. On the other hand, the traditional sampling methods, e.g., Monte Carlo methods, are straightforward to implement, but they do not offer convergence as fast as stochastic Galerkin methods. In this paper, a highorder stochastic collocation approach is proposed. Similar to stochastic Galerkin methods, the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence. However, the numerical implementation of stochastic collocation is trivial, as it requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods. The computational cost of the collocation methods depends on the choice of the collocation points, and we present several feasible constructions. One particular choice, based on sparse grids, depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs. Numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods. Key words. collocation methods, stochastic inputs, differential equations, uncertainty quantification
The Relative Contribution of Jumps to Total Price Variance
 Journal of Financial Econometrics
, 2005
"... We examine tests for jumps based on recent asymptotic results; we interpret the tests as Hausmantype tests. Monte Carlo evidence suggests that the daily ratio zstatistic has appropriate size, good power, and good jump detection capabilities revealed by the confusion matrix comprised of jump class ..."
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Cited by 159 (5 self)
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We examine tests for jumps based on recent asymptotic results; we interpret the tests as Hausmantype tests. Monte Carlo evidence suggests that the daily ratio zstatistic has appropriate size, good power, and good jump detection capabilities revealed by the confusion matrix comprised of jump classification probabilities. We identify a pitfall in applying the asymptotic approximation over an entire sample. Theoretical and Monte Carlo analysis indicates that microstructure noise biases the tests against detecting jumps, and that a simple lagging strategy corrects the bias. Empirical work documents evidence for jumps that account for 7 % of stock market price variance.
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 149 (3 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.
Estimating Functions for Discretely Observed Diffusions: A Review
 EDS, `SELECTED PROCEEDINGS OF THE SYMPOSIUM ON ESTIMATING FUNCTIONS
, 1997
"... Several estimating functions for discretely observed diffusion processes are reviewed. First we discuss simple explicit estimating functions based on Gaussian approximations to the transition density. The corresponding estimators often have considerable bias, a problem that can be avoided by usi ..."
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Cited by 142 (14 self)
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Several estimating functions for discretely observed diffusion processes are reviewed. First we discuss simple explicit estimating functions based on Gaussian approximations to the transition density. The corresponding estimators often have considerable bias, a problem that can be avoided by using martingale estimating functions. These, on the other hand, are rarely explicit and therefore often requires a considerable computational effort. We review results on how to choose an optimal martingale estimating function and on asymptotic properties of the estimators. Martingale estimating functions based on polynomials of the increments of the observed process or on eigenfunctions for the generator of the diffusion model are considered in more detail.
A multilevel Monte Carlo algorithm for Lévy driven stochastic differential equations
, 2009
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Numerical Techniques for Maximum Likelihood Estimation of ContinuousTime Diffusion Processes
, 2002
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A stochastic mesh method for pricing highdimensional American options
 Journal of Computational Finance
, 1997
"... Highdimensional problems frequently arise in the pricing of derivative securities – for example, in pricing options on multiple underlying assets and in pricing term structure derivatives. American versions of these options, ie, where the owner has the right to exercise early, are particularly chal ..."
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Cited by 137 (8 self)
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Highdimensional problems frequently arise in the pricing of derivative securities – for example, in pricing options on multiple underlying assets and in pricing term structure derivatives. American versions of these options, ie, where the owner has the right to exercise early, are particularly challenging to price. We introduce a stochastic mesh method for pricing highdimensional American options when there is a finite, but possibly large, number of exercise dates. The algorithm provides point estimates and confidence intervals; we provide conditions under which these estimates converge to the correct values as the computational effort increases. Numerical results illustrate the performance of the method. 1