A Lie Algebraic Approach to Numerical Integration of Stochastic Differential Equations

Misawa, Tetsuya

doi:10.1137/s106482750037024x

Cited by 39 publications

(36 citation statements)

References 18 publications

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“…We also note that the texts [6,9,15] include examples of numerical simulations on SDEs of the form (1) and [25] derives a method that applies to a subclass of (1).…”

Section: The Mean-reverting Square Root Processmentioning

confidence: 99%

Convergence of Monte Carlo simulations involving the mean-reverting square root process

Higham¹,

Mao²

2005

JCF

144

124

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The mean-reverting square root process is a stochastic differential equation (SDE) that has found considerable use as a model for volatility, interest rate, and other financial quantities. The equation has no general, explicit, solution, although its transition density can be characterized. For valuing path-dependent options under this model, it is typically quicker and simpler to simulate the SDE directly than to compute with the exact transition density. Because the diffusion coefficient does not satisfy a global Lipschitz condition, there is currently a lack of theory to justify such simulations. We begin by showing that a natural Euler-Maruyama discretization provides qualitatively correct approximations to the first and second moments. We then derive explicitly computable bounds on the strong (pathwise) error over finite time intervals. These bounds imply strong convergence in the limit of the timestep tending to zero. The strong convergence result can be used to justify the method within Monte Carlo simulations that compute the expected payoff of financial products. We spell this out for a bond with interest rate given by the mean-reverting square root process, and for an up-and-out barrier option with asset price governed by the mean-reverting square root process. We also prove convergence for European and up-andout barrier options under Heston's stochastic volatility model-here the mean-reverting square root process feeds into the asset price dynamics as the squared volatility.

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“…We also note that the texts [6,9,15] include examples of numerical simulations on SDEs of the form (1) and [25] derives a method that applies to a subclass of (1).…”

Section: The Mean-reverting Square Root Processmentioning

confidence: 99%

Convergence of Monte Carlo simulations involving the mean-reverting square root process

Higham¹,

Mao²

2005

JCF

144

124

View full text Add to dashboard Cite

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“…At the linear level, the Neumann, stochastic Taylor and Runge-Kutta type methods are equivalent. In the stochastic context, Magnus integrators have been considered by Castell and Gaines [13], Burrage [7], Burrage and Burrage [8] and Misawa [46].…”

mentioning

confidence: 99%

Efficient Strong Integrators for Linear Stochastic Systems

Lord¹,

Malham²,

Wiese³

2008

SIAM J. Numer. Anal.

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Abstract. We present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of Wiener processes. These schemes are based on the Neumann (stochastic Taylor) and Magnus expansions. Firstly, we consider the case when the governing linear diffusion vector fields commute with each other, but not with the linear drift vector field. We prove that numerical methods based on the Magnus expansion are more accurate in the mean-square sense than corresponding stochastic Taylor integration schemes. Secondly, we derive the maximal rate of convergence for arbitrary multi-dimensional stochastic integrals approximated by their conditional expectations. Consequently, for general nonlinear stochastic differential equations with non-commuting vector fields, we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order. Thirdly, we consider the consequences in two numerical studies, one of which is an application arising in stochastic linear-quadratic optimal control.

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“…It is not trivial to establish regularity of the generator in (1), because the diffusion is not uniformly elliptic nor are the coefficients smooth. Further discussion of splitting methods for SDEs includes [17,3,14].…”

mentioning

confidence: 99%

Splitting for Dissipative Particle Dynamics

Shardlow¹

2003

SIAM J. Sci. Comput.

138

176

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Abstract. We study numerical methods for dissipative particle dynamics, a system of stochastic differential equations for simulating particles interacting pairwise according to a soft potential at constant temperature where the total momentum is conserved. We introduce splitting methods and examine the behavior of these methods experimentally. The performance of the methods, particularly temperature control, is compared to the modified velocity Verlet method used in many previous papers.

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A Lie Algebraic Approach to Numerical Integration of Stochastic Differential Equations

Cited by 39 publications

References 18 publications

Convergence of Monte Carlo simulations involving the mean-reverting square root process

Convergence of Monte Carlo simulations involving the mean-reverting square root process

Efficient Strong Integrators for Linear Stochastic Systems

Splitting for Dissipative Particle Dynamics

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