SIQING GAN

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Oleg Yu Vorobyev

Siberian Federal University

David Seamon

Kansas State University

Diego Gabriel Krivochen

Basil Lourié

Institute for Philosophy and Law of the Siberian Branch of the Russian Academy of Sciences

Kerstin P. Hofmann

German Archaeological Institute

Jack Sarfatti

Cornell University

Alan Garfinkel

California State University, Bakersfield

Mohammed Najm Abdullah Al Salam

University of Technology/Iraq

Ashok Paranjothi

Sri Sairam Institute Of Technology

Sonja Smets

University of Amsterdam

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Papers by SIQING GAN

Analytical Approximation for the Distorted Expectations

SSRN Electronic Journal, 2015

Given the density function or the characteristic function of a random variable, we propose an ana... more

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Weak error estimate of a fully discrete method for stochastic Cahn-Hilliard equation with additive noise

Cornell University - arXiv, Nov 15, 2021

We prove a weak error estimate of a fully discrete scheme for stochastic Cahn-Hilliard equation w... more We prove a weak error estimate of a fully discrete scheme for stochastic Cahn-Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen-Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. Besides, the weak analysis is also troublesome caused by the lack of the associated Kolmogorov equation. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence, shown to be higher than the strong convergence rates, are revealed in the stochastic Cahn-Hilliard equation setting for the first time. Numerical examples are finally performed to confirm the theoretical results.

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Split-step theta Milstein methods for SDEs with non-globally Lipschitz diffusion coefficients

Applied Numerical Mathematics

Symmetric-Adjoint and Symplectic-Adjoint Runge-Kutta Methods and Their Applications

Numerical Mathematics: Theory, Methods and Applications, 2022

Symmetric and symplectic methods are classical notions in the theory of numerical methods for sol... more Symmetric and symplectic methods are classical notions in the theory of numerical methods for solving ordinary differential equations. They can generate numerical flows that respectively preserve the symmetry and symplecticity of the continuous flows in the phase space. This article is mainly concerned with the symmetric-adjoint and symplectic-adjoint Runge-Kutta methods as well as their applications. It is a continuation and an extension of the study in [14], where the authors introduced the notion of symplectic-adjoint method of a Runge-Kutta method and provided a simple way to construct symplectic partitioned Runge-Kutta methods via the symplectic-adjoint method. In this paper, we provide a more comprehensive and systematic study on the properties of the symmetric-adjoint and symplecticadjoint Runge-Kutta methods. These properties reveal some intrinsic connections among some classical Runge-Kutta methods. Moreover, those properties can be used to significantly simplify the order conditions and hence can be applied to the construction of high-order Runge-Kutta methods. As a specific and illustrating application, we construct a novel class of explicit Runge-Kutta methods of stage 6 and order 5. Finally, with the help of symplectic-adjoint method, we thereby obtain a new simple proof of the nonexistence of explicit Runge-Kutta method with stage 5 and order 5.

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First order strong approximation of Ait-Sahalia-type interest rate model with Poisson jumps

ArXiv, 2021

For Ait-Sahalia-type interest rate model with Poisson jumps, we are interested in strong converge... more For Ait-Sahalia-type interest rate model with Poisson jumps, we are interested in strong convergence of a novel time-stepping method, called transformed jump-adapted backward Euler method (TJABEM). Under certain hypothesis, the considered model takes values in positive domain (0,∞). It is shown that the TJABEM can preserve the domain of the underlying problem. Furthermore, for the above model with non-globally Lipschitz drift and diffusion coefficients, the strong convergence rate of order one of the TJABEM is recovered with respect to a Lp-error criterion. Finally, numerical experiments are given to illustrate the theoretical results. Mathematics Subject Classification: 60H35, 60H15, 65C30.

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Weak convergence rates for a full implicit scheme of stochastic Cahn-Hilliard equation with additive noise

ArXiv, 2021

The aim of this study is the weak convergence rate of a temporal and spatial discretization schem... more The aim of this study is the weak convergence rate of a temporal and spatial discretization scheme for stochastic Cahn–Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler scheme is used in time. The presence of the unbounded operator in front of the nonlinear term and the lack of the associated Kolmogorov equations make the error analysis much more challenging and demanding. To overcome these difficulties, we further exploit a novel approach proposed in [7] and combine it with Malliavin calculus to obtain an improved weak rate of convergence, in comparison with the corresponding strong convergence rates. The techniques used here are quite general and hence have the potential to be applied to other non-Markovian equations. As a byproduct the rate of the strong error can also be easily obtained.

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First order strong convergence of an explicit scheme for the stochastic SIS epidemic model

Journal of Computational and Applied Mathematics, 2021

$Research paper thumbnail of A fractional trapezoidal rule type difference scheme for fractional order integro-differential equation$

A fractional trapezoidal rule type difference scheme for fractional order integro-differential equation

A fractional trapezoidal rule type difference scheme for fractional order integro-differential eq... more A fractional trapezoidal rule type difference scheme for fractional order integro-differential equation is considered. The equation is discretized in time by means of a method based on the trapezoidal rule: while the time derivative is approximated by the standard trapezoidal rule, the integral term is discretized by means of a fractional quadrature rule constructed again from the trapezoidal rule. The solvability, stability and L2-norm convergence are proved. The convergence order is second order both in temporal and spatial directions. Furthermore, a spatial compact scheme, based on the fractional trapezoidal rule type difference scheme, is also proposed and the similar results are derived. The convergence order is second for time and fourth for space. Preliminary numerical experiment confirms our theoretical results.

Dissipativity of Linear 0-Methods for Integro-Differential Equations

This paper is concerned with the analytic and numerical dissipativity of nonlinear integro-differ... more This paper is concerned with the analytic and numerical dissipativity of nonlinear integro-differential equations (IDEs). A dissipativity criteria for IDEs is given. It is shown that any A-stable linear 0-method for the systems is dissipative. Numerical examples are given to confirm the theoretical results. © 2006 Elsevier Ltd. All rights reserved.

A full-discrete exponential Euler approximation of the invariant measure for parabolic stochastic partial differential equations

Applied Numerical Mathematics, 2020

We discrete the ergodic semilinear stochastic partial differential equations in space dimension d... more We discrete the ergodic semilinear stochastic partial differential equations in space dimension d ≤ 3 with additive noise, spatially by a spectral Galerkin method and temporally by an exponential Euler scheme. It is shown that both the spatial semi-discretization and the spatio-temporal full discretization are ergodic. Further, convergence orders of the numerical invariant measures, depending on the regularity of noise, are recovered based on an easy time-independent weak error analysis without relying on Malliavin calculus. To be precise, the convergence order is 1 − ǫ in space and 1 2 − ǫ in time for the space-time white noise case and 2 − ǫ in space and 1 − ǫ in time for the trace class noise case in space dimension d = 1, with arbitrarily small ǫ > 0. Numerical results are finally reported to confirm these theoretical findings.

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Stability of the drift-implicit and double-implicit Milstein schemes for nonlinear SDEs

Applied Mathematics and Computation, 2018

This paper examines the stability of numerical solutions of nonlinear stochastic differential equ... more This paper examines the stability of numerical solutions of nonlinear stochastic differential equations (SDEs) with non-global Lipschitz continuous coefficients. Two implicit Milstein schemes, called drift-implicit Milstein scheme and double-implicit Milstein scheme, are considered to simulate the underlying SDEs. It is proved that the schemes can preserve the stability and contractivity in mean square of the underlying systems.

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High order method for Black–Scholes PDE

Computers & Mathematics with Applications, 2018

In this paper, the Black-Scholes PDE is solved numerically by using the high order numerical meth... more In this paper, the Black-Scholes PDE is solved numerically by using the high order numerical method. Fourth-order central scheme and fourth-order compact scheme in space are performed, respectively. The comparison of these two kinds of difference schemes shows that under the same computational accuracy, the compact scheme has simpler stencil, less computation and higher efficiency. The fourth-order backward differentiation formula (BDF4) in time is then applied. However, the overall convergence order of the scheme is less than O(h 4 + δ 4). The reason is, in option pricing, terminal conditions (also called pay-off function) is not able to be differentiated at the strike price and this problem will spread to the initial time, causing a second-order convergence solution. To tackle this problem, in this paper, the grid refinement method is performed, as a result, the overall rate of convergence could revert to fourth-order. The numerical experiments show that the method in this paper has high precision and high efficiency, thus it can be used as a practical guide for option pricing in financial markets.

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Stability of numerical solutions to the delay logistic equations with piecewise continuous arguments

It is shown that spurious solutions are obtained when Runge-Kutta method is directly applied to t... more It is shown that spurious solutions are obtained when Runge-Kutta method is directly applied to the delay logistic equations with piecewise continuous arguments.Runge-Kutta method which does not admit spurious solutions is constructed.The convergent order of this method is investigated.It is shown that this method is locally asymptotically stable and globally asymptotically stable under certain conditions.

Errors of linear multistep methods for singularly perturbed Volterra delay-integro-differential equations

Mathematics and Computers in Simulation, 2009

This paper is concerned with the error behaviour of linear multistep methods applied to singularl... more This paper is concerned with the error behaviour of linear multistep methods applied to singularly perturbed Volterra delayintegro-differential equations. We derive global error estimates of A(α)-stable linear multistep methods with convergent quadrature rule. Numerical experiments confirm our theoretical analysis.

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Weak convergence analysis of the linear implicit Euler method for semilinear stochastic partial differential equations with additive noise

Journal of Mathematical Analysis and Applications, 2013

ABSTRACT In this paper, we analyze the weak error of a semi-discretization in time by the linear ... more ABSTRACT In this paper, we analyze the weak error of a semi-discretization in time by the linear implicit Euler method for semilinear stochastic partial differential equations (SPDEs) with additive noise. The main result reveals how the weak order depends on the regularity of noise and that the order of weak convergence is twice that of strong convergence. In particular, the linear implicit Euler method for SPDEs driven by trace class noise achieves an almost optimal order 1−ϵ1−ϵ for arbitrarily small ϵ&gt;0ϵ&gt;0.

The split-step backward Euler method for linear stochastic delay differential equations

Journal of Computational and Applied Mathematics, 2009

In this paper, the numerical approximation of solutions of linear stochastic delay differential e... more In this paper, the numerical approximation of solutions of linear stochastic delay differential equations (SDDEs) in the Itô sense is considered. We construct split-step backward Euler (SSBE) method for solving linear SDDEs and develop the fundamental numerical analysis concerning its strong convergence and mean-square stability. It is proved that the SSBE method is convergent with strong order γ = 1 2 in the mean-square sense. The conditions under which the SSBE method is mean-square stable (MS-stable) and general mean-square stable (GMS-stable) are obtained. Some illustrative numerical examples are presented to demonstrate the order of strong convergence and the meansquare stability of the SSBE method.

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Delay-dependent stability analysis of numerical methods for stochastic delay differential equations

Journal of Computational and Applied Mathematics, 2012

This paper is concerned with the numerical solution of stochastic delay differential equations. T... more This paper is concerned with the numerical solution of stochastic delay differential equations. The focus is on the delay-dependent stability of numerical methods for a linear scalar test equation with real coefficients. By using the so-called root locus technique, the full asymptotic stability region in mean square of stochastic theta methods is obtained, which is characterized by a sufficient and necessary condition in terms of the drift and diffusion coefficients as well as time stepsize and method parameter theta. Then, this condition is compared with the analytical stability condition. It is proved that the Backward Euler method completely preserves the asymptotic mean square stability of the underlying system and the Euler-Maruyama method preserves the instability of the system. Our investigation also shows that not all theta methods with θ ≥ 1 2 preserve this delay-dependent stability. Some numerical examples are presented to confirm the theoretical results.

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The improved split-step backward Euler method for stochastic differential delay equations

International Journal of Computer Mathematics, 2011

A new, improved split-step backward Euler (SSBE) method is introduced and analyzed for stochastic... more A new, improved split-step backward Euler (SSBE) method is introduced and analyzed for stochastic differential delay equations(SDDEs) with generic variable delay. The method is proved to be convergent in mean-square sense under conditions (Assumption 3.1) that the diffusion coefficient g(x, y) is globally Lipschitz in both x and y, but the drift coefficient f (x, y) satisfies one-sided Lipschitz condition in x and globally Lipschitz in y. Further, exponential mean-square stability of the proposed method is investigated for SDDEs that have a negative one-sided Lipschitz constant. Our results show that the method has the unconditional stability property in the sense that it can well reproduce stability of underlying system, without any restrictions on stepsize h. Numerical experiments and comparisons with existing methods for SDDEs illustrate the computational efficiency of our method.

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Convergence and stability of the balanced methods for stochastic differential equations with jumps

International Journal of Computer Mathematics, 2011

This paper deals with the balanced methods which are implicit methods for stochastic differential... more This paper deals with the balanced methods which are implicit methods for stochastic differential equations with Poisson-driven jumps. It is shown that the balanced methods give a strong convergence rate of at least 1/2 and can preserve the linear mean-square stability with the sufficiently small stepsize. Weak variants are also considered and their mean-square stability analysed. Some numerical experiments are given to demonstrate the conclusions.

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Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion

Discrete Dynamics in Nature and Society, 2011

A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differen... more A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations NSDDEs driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the mean-square convergence theory is illustrated by the stochastic θ-methods and the balanced implicit methods. It is derived from Theorem 3.1 that the order of the mean-square convergence of both of them for NSDDEs with jumps is 1/2. Numerical experiments illustrate the theoretical results. It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.

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Analytical Approximation for the Distorted Expectations

SSRN Electronic Journal, 2015

Given the density function or the characteristic function of a random variable, we propose an ana... more

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Weak error estimate of a fully discrete method for stochastic Cahn-Hilliard equation with additive noise

Cornell University - arXiv, Nov 15, 2021

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Split-step theta Milstein methods for SDEs with non-globally Lipschitz diffusion coefficients

Applied Numerical Mathematics

Symmetric-Adjoint and Symplectic-Adjoint Runge-Kutta Methods and Their Applications

Numerical Mathematics: Theory, Methods and Applications, 2022