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65
An empirical investigation of continuoustime equity return models
 Journal of Finance
, 2002
"... This paper extends the class of stochastic volatility diffusions for asset returns to encompass Poisson jumps of timevarying intensity. We find that any reasonably descriptive continuoustime model for equityindex returns must allow for discrete jumps as well as stochastic volatility with a pronou ..."
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Cited by 240 (13 self)
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This paper extends the class of stochastic volatility diffusions for asset returns to encompass Poisson jumps of timevarying intensity. We find that any reasonably descriptive continuoustime model for equityindex returns must allow for discrete jumps as well as stochastic volatility with a pronounced negative relationship between return and volatility innovations. We also find that the dominant empirical characteristics of the return process appear to be priced by the option market. Our analysis indicates a general correspondence between the evidence extracted from daily equityindex returns and the stylized features of the corresponding options market prices. MUCH ASSET AND DERIVATIVE PRICING THEORY is based on diffusion models for primary securities. However, prescriptions for practical applications derived from these models typically produce disappointing results. A possible explanation could be that analytic formulas for pricing and hedging are available for only a limited set of continuoustime representations for asset returns
An algorithmic introduction to numerical simulation of stochastic differential equations
 SIAM Review
, 2001
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Meansquare and asymptotic stability of the stochastic theta method
 SIAM J. Numer. Anal
"... Abstract. Stability analysis of numerical methods for ordinary differential equations (ODEs) is motivated by the question “for what choices of stepsize does the numerical method reproduce the characteristics of the test equation? ” We study a linear test equation with a multiplicative noise term, an ..."
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Cited by 64 (10 self)
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Abstract. Stability analysis of numerical methods for ordinary differential equations (ODEs) is motivated by the question “for what choices of stepsize does the numerical method reproduce the characteristics of the test equation? ” We study a linear test equation with a multiplicative noise term, and consider meansquare and asymptotic stability of a stochastic version of the theta method. We extend some meansquare stability results in [Saito and Mitsui, SIAM. J. Numer. Anal., 33 (1996), pp. 2254–2267]. In particular, we show that an extension of the deterministic Astability property holds. We also plot meansquare stability regions for the case where the test equation has real parameters. For asymptotic stability, we show that the issue reduces to finding the expected value of a parametrized random variable. We combine analytical and numerical techniques to get insights into the stability properties. For a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region. This allows us to prove a number of results. The technique introduced is widely applicable, and we use it to show that a fully implicit method suggested by [Kloeden and Platen, Numerical Solution of Stochastic Differential Equations, SpringerVerlag, 1992] has an asymptotic stability extension of the deterministic Astability property. We also use the approach to explain some numerical results
A selective overview of nonparametric methods in financial econometrics
 Statist. Sci
, 2005
"... Abstract. This paper gives a brief overview of the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inference for instantaneous returns and volatility functions of timehomogeneous and timedependent diffusion processes, and estimation ..."
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Cited by 52 (8 self)
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Abstract. This paper gives a brief overview of the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inference for instantaneous returns and volatility functions of timehomogeneous and timedependent diffusion processes, and estimation of transition densities and state price densities. We first briefly describe the problems and then outline the main techniques and main results. Some useful probabilistic aspects of diffusion processes are also briefly summarized to facilitate our presentation and applications.
Numerical Methods for Strong Solutions of Stochastic Differential Equations: an Overview
, 2003
"... This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations (SDEs). We give a brief survey of the area focusing on a number of application areas where approximations to strong solution ..."
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Cited by 46 (2 self)
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This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations (SDEs). We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications (section 1), and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes (section 2). In section 3 we present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods which preserve the underlying structure of the problem. In sections 4 and 5 we present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, in section 6 we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable stepsize implementations based on various types of control.
The Magnus expansion and some of its applications
, 2008
"... Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an ..."
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Cited by 29 (3 self)
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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as TimeDependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial resummation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related nonperturbative
Weak Discrete Time Approximation of Stochastic Differential Equations with Time Delay
 MATH. COMPUT. SIMULATION
, 2001
"... The paper considers the derivation of weak discrete time approximations for solutions of stochastic differential equations with time delay. These are suitable for Monte Carlo simulation and allow the computation of expectations for functionals of stochastic delay equations. The suggested approximati ..."
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Cited by 24 (0 self)
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The paper considers the derivation of weak discrete time approximations for solutions of stochastic differential equations with time delay. These are suitable for Monte Carlo simulation and allow the computation of expectations for functionals of stochastic delay equations. The suggested approximations converge in a weak sense.
Numerical analysis of micromacro simulations of polymeric fluid flows: a simple case
 Math. Models and Methods in Applied Sciences
, 2001
"... We present in this article the numerical analysis of a simple micromacro simulation of a polymeric fluid flow, namely the shear flow for the Hookean dumbbells model. Although restricted to this academic case (which is however used in practice as a test problem for new numerical strategies to be app ..."
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Cited by 23 (4 self)
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We present in this article the numerical analysis of a simple micromacro simulation of a polymeric fluid flow, namely the shear flow for the Hookean dumbbells model. Although restricted to this academic case (which is however used in practice as a test problem for new numerical strategies to be applied to more sophisticated cases), our study can be considered as a first step towards that of more complicated models. Our main result states the convergence of the fully discretized scheme (finite element in space, finite difference in time, plus Monte Carlo realizations) towards the coupled solution of a partial differential equation / stochastic differential equation system.
Astability and stochastic meansquare stability
 BIT
"... This note extends and interprets a result of Saito and Mitsui [SIAM J. Numer. Anal., 33 (1996), pp. 2254–2267] for a method of Milstein. The result concerns meansquare stability on a stochastic differential equation test problem with multiplicative noise. The numerical method reduces to the Theta M ..."
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Cited by 22 (3 self)
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This note extends and interprets a result of Saito and Mitsui [SIAM J. Numer. Anal., 33 (1996), pp. 2254–2267] for a method of Milstein. The result concerns meansquare stability on a stochastic differential equation test problem with multiplicative noise. The numerical method reduces to the Theta Method on deterministic problems. Saito and Mitsui showed that the deterministic Astability property of the Theta Method does not carry through to the meansquare context in general, and gave a condition under which unconditional stability holds. The main purpose of this note is to emphasize that the approach of Saito and Mitsui makes it possible to quantify precisely the point where unconditional stability is lost in terms of the ratio of the drift (deterministic) and diffusion (stochastic) coefficients. This leads to a concept akin to deterministic A(α)stability that may be useful in the stability analysis of more general methods. It is also shown that meansquare Astability is recovered if the Theta Method parameter is increased beyond its normal range to the value 3/2.