H. Schurz - Academia.edu (original) (raw)

Papers by H. Schurz

Research paper thumbnail of Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise

Stochastic Analysis and Applications, 1996

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Research paper thumbnail of Numerical regularization for SDEs: Construction of nonnegative solutions

In the numerical solution of stochastic differential equations (SDEs) such appearances as sudden,... more In the numerical solution of stochastic differential equations (SDEs) such appearances as sudden, large fluctuations (explosions), negative paths or unbounded solutions are sometimes observed in contrast to the qualitative behaviour of the exact solution. To overcome this dilemma we construct regular (bounded) numerical solutions through implicit techniques without discretizing the state space. For discussion and classification, the notation of life time of numerical solutions is introduced. Thereby the task consists in construction of numerical solutions with lengthened life time up to eternal one. During the exposition we outline the role of implicitness for this "process of numerical regularization". Boundedness(Nonnegativity) of some implicit numerical solutions can be proved at least for a class of linearly bounded models. Balanced implicit methods (BIMs) turn out to be very efficient for this purpose. Furthermore, the local property of conditional positivity of numer...

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Research paper thumbnail of The numerical solution of stochastic differential equations

The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1977

A method is proposed for the numerical solution of Itô stochastic differential equations by means... more A method is proposed for the numerical solution of Itô stochastic differential equations by means of a second-order Runge–Kutta iterative scheme rather than the less efficient Euler iterative scheme. It requires the Runge–Kutta iterative scheme to be applied to a different stochastic differential equation obtained by subtraction of a correction term from the given one.

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Research paper thumbnail of Noise and Fluctuations in Econophysics and Finance (Proceedings Volume)

Credit risk models like Moody's KMV are now well established in the market and give bond man... more Credit risk models like Moody's KMV are now well established in the market and give bond managers reliable default probabilities for individual firms. Until now it has been hard to relate those probabilities to the actual credit spreads observed on the market for corporate bonds. Inspired by the existence of scaling laws in financial markets by Dacorogna et al. 2001 and DiMatteo et al. 2005 deviating from the Gaussian behavior, we develop a model that quantitatively links those default probabilities to credit spreads (market prices). The ...

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Research paper thumbnail of Modelling, analysis and simulation of stochastic innovation diffusion

The well-known BASS model for description of diffusion of innovations has been extensively invest... more The well-known BASS model for description of diffusion of innovations has been extensively investigated within deterministic framework. One of the basic processes in modelling of these diffusions concerns with the propagation through word of mouth which is inherently nonlinear. As a more realistic modelling, the diffusion of an innovation in the presence of uncertainty is generally formulated in terms of nonlinear stochastic differential equations (SDEs). At first we discuss well-posedness, regularity (boundedness) and uniqueness of solutions of these SDEs. However, an explicit expression for analytical solution itself is not available. Accordingly one has to resort to numerical solution of SDEs for studying various aspects like the time-development of growth patterns, exit frequencies, mean passage times and impact of advertising policies. In this respect we present some basic aspects of numerical analysis of these random extensions of the BASS model, e.g. numerical regularity and ...

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Research paper thumbnail of Mean square stability for discrete linear stochastic systems

Several results concerning asymptotical mean square stability of the null solution of specific li... more Several results concerning asymptotical mean square stability of the null solution of specific linear stochastic systems are presented and proven. It is shown that the mean square stability of the implicit Euler method, taken from the monography of Kloeden and Platen (1992) and applied to linear stochastic differential equations, is necessary for the mean square stability of the corresponding implicit Milstein method (using the same implicitness parameter). Furthermore, a sufficient condition for the mean square stability of the implicit Euler method can be varified for autonomous systems. Additionally, the principle of 'monotonous inclusion' of the sequel of mean square stability domains holds for linear systems. The paper generalizes the results due to Schurz (1993) where one-dimensional linear complex systems with respect to asymptotical p-th mean stability have been investigated. Finally, a simple example confirms these assertions. The results can also be used to deduce ...

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Research paper thumbnail of Institut fiir Angewandt~ Analysis und Stochastik Balanced implicit methods for stiff stochastic systems: An introduction and numerical experiments

The paper introduces implicitness in stochastic terms of numerical methods for solving of stiff s... more The paper introduces implicitness in stochastic terms of numerical methods for solving of stiff stochastic differential equations and especially a class of fully implicit methods, the balanced methods. Their order of strong convergence is proved. Systematic numerical experiments compare the numerical behaviour of these schemes with that of different other schemes. A wide class of model equations are also provided as one by-product in order to test numerical methods in the case of stochastic stiffness in the given system.

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Research paper thumbnail of Waveform relaxation methods for stochastic differential equations

An operator equation X = Π X + G in a Banach space 퓔 of 퓕<sub>t</sub>-adapted random ... more An operator equation X = Π X + G in a Banach space 퓔 of 퓕<sub>t</sub>-adapted random elements describing an initial- or boundary value problem of a system of stochastic differential equations (SDEs) is considered. Our basic assumption is that the underlying system consists of weakly coupled subsystems. The proof of the convergence of corresponding waveform relaxation methods depends on the property that the spectral radius of an associated matrix is less than one. The entries of this matrix depend on the Lipschitz-constants of a decomposition of Π. In proving an existence result for the operator equation we show how the entries of the matrix depend on the right hand side of the stochastic differential equations. We derive conditions for the convergence under "classical" vector-valued Lipschitz-continuity of an appropriate splitting of the system of stochastic ODEs. A generalization of these key results under one-sided Lipschitz continuous and anticoercive drift...

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Research paper thumbnail of Moment Evolution of the Outflow-Rate from Nonlinear Conceptual Reservoirs

Water Science and Technology Library

The temporal evolution of moments of outflow-rate is investigated in a stochastically perturbed n... more The temporal evolution of moments of outflow-rate is investigated in a stochastically perturbed nonlinear reservoir due to precipitation. The detailed stochastic behaviour of outflow is obtained from the numerical solution of a nonlinear stochastic differential equation with multiplicative noise. The time-development of first two moments is studied for various choices of parameters. Using Stratonovich interpretation, it turns out that the mean outflow-rate is above that given by the deterministic solution. Based on the set of 9000 simulation runs, 90 % confidence intervals for the mean evolution of outflow-rate are computed. The effect of stochastic perturbations with finite correlation time is investigated.

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Research paper thumbnail of A Theorem on Asymptotic Stability of Solutions of Nonlinear Stochastic Difference Equations with Volterra Type Noise

A general theorem on global a.s. asymptotic stability of solutions to some nonlinear stochastic d... more A general theorem on global a.s. asymptotic stability of solutions to some nonlinear stochastic difference equations in IR 1 with in-the-arithmetic-mean-sense monotone terms as main part of its drift and Volterra-type dependence of its diffusion terms is presented as a certain application of convergence theorems for semimartingale inequalities to the decomposition of appropriate functionals.

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Research paper thumbnail of On global asymptotic stability of solutions of differential equations

Transactions of the American Mathematical Society, 1962

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Research paper thumbnail of Numerical Analysis of Stochastic Differential Equations with Explosions

Stochastic Analysis and Applications, 2005

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Research paper thumbnail of Boundary Preserving Semianalytic Numerical Algorithms for Stochastic Differential Equations

SIAM Journal on Scientific Computing, 2007

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Research paper thumbnail of Undamped nonlinear beam excited by additive -regular noise

Journal of Computational and Applied Mathematics, 2011

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Research paper thumbnail of Almost sure asymptotic stability of drift-implicit -methods for bilinear ordinary stochastic differential equations in

Journal of Computational and Applied Mathematics, 2005

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Research paper thumbnail of Krée, P.; Wedig, W. (eds.): Probabilistic Methods in Applied Physics. Berlin etc., Springer-Verlag 1995. IX, 393 pp., DM 116.–. ISBN 3-540-60214-3 (Lecture Notes in Physics 451)

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1997

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Research paper thumbnail of Existence and uniqueness of solutions of a semilinear differential equation

Indiana University Mathematics Journal, 2003

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Research paper thumbnail of Global asymptotic stability of solutions of cubic stochastic difference equations

Advances in Difference Equations, 2004

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Research paper thumbnail of On eeects of discretization on estimators of drift parameters for diiusion processes

In this paper statistical properties of estimators of drift parameters for diiusion processes are... more In this paper statistical properties of estimators of drift parameters for diiusion processes are studied by m o d e r n n umerical methods for stochastic diier-ential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diiusions. A review is given of the necessary theory for parameter estimation for diiusion processes and for simulation of diiusion processes. Three examples are studied.

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Research paper thumbnail of The invariance of asymptotic laws of stochastic systems under discretization

The stochastic trapezoidal rule provides the only discretization scheme from the family of implic... more The stochastic trapezoidal rule provides the only discretization scheme from the family of implicit Euler methods (see 11]) which possesses the same asymptotic (stationary) law as underlying linear continuous time stochastic systems with white or coloured noise. This identity is shown for systems with multiplicative (para-metric) and additive noise using xed p oint principles and the theo r y o f p ositive operators. The key result is useful for adequate implementation of stochastic algorithms applied to numerical solution of autonomous stochastic diierential equations. In particular it has practical importance when accurate long time integration is required such as in the process of estimation of Lyapunov exponents or stationary measures for oscillators in Mechanical Engineering.

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Research paper thumbnail of Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise

Stochastic Analysis and Applications, 1996

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Research paper thumbnail of Numerical regularization for SDEs: Construction of nonnegative solutions

In the numerical solution of stochastic differential equations (SDEs) such appearances as sudden,... more In the numerical solution of stochastic differential equations (SDEs) such appearances as sudden, large fluctuations (explosions), negative paths or unbounded solutions are sometimes observed in contrast to the qualitative behaviour of the exact solution. To overcome this dilemma we construct regular (bounded) numerical solutions through implicit techniques without discretizing the state space. For discussion and classification, the notation of life time of numerical solutions is introduced. Thereby the task consists in construction of numerical solutions with lengthened life time up to eternal one. During the exposition we outline the role of implicitness for this "process of numerical regularization". Boundedness(Nonnegativity) of some implicit numerical solutions can be proved at least for a class of linearly bounded models. Balanced implicit methods (BIMs) turn out to be very efficient for this purpose. Furthermore, the local property of conditional positivity of numer...

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Research paper thumbnail of The numerical solution of stochastic differential equations

The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 1977

A method is proposed for the numerical solution of Itô stochastic differential equations by means... more A method is proposed for the numerical solution of Itô stochastic differential equations by means of a second-order Runge–Kutta iterative scheme rather than the less efficient Euler iterative scheme. It requires the Runge–Kutta iterative scheme to be applied to a different stochastic differential equation obtained by subtraction of a correction term from the given one.

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Research paper thumbnail of Noise and Fluctuations in Econophysics and Finance (Proceedings Volume)

Credit risk models like Moody's KMV are now well established in the market and give bond man... more Credit risk models like Moody's KMV are now well established in the market and give bond managers reliable default probabilities for individual firms. Until now it has been hard to relate those probabilities to the actual credit spreads observed on the market for corporate bonds. Inspired by the existence of scaling laws in financial markets by Dacorogna et al. 2001 and DiMatteo et al. 2005 deviating from the Gaussian behavior, we develop a model that quantitatively links those default probabilities to credit spreads (market prices). The ...

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Research paper thumbnail of Modelling, analysis and simulation of stochastic innovation diffusion

The well-known BASS model for description of diffusion of innovations has been extensively invest... more The well-known BASS model for description of diffusion of innovations has been extensively investigated within deterministic framework. One of the basic processes in modelling of these diffusions concerns with the propagation through word of mouth which is inherently nonlinear. As a more realistic modelling, the diffusion of an innovation in the presence of uncertainty is generally formulated in terms of nonlinear stochastic differential equations (SDEs). At first we discuss well-posedness, regularity (boundedness) and uniqueness of solutions of these SDEs. However, an explicit expression for analytical solution itself is not available. Accordingly one has to resort to numerical solution of SDEs for studying various aspects like the time-development of growth patterns, exit frequencies, mean passage times and impact of advertising policies. In this respect we present some basic aspects of numerical analysis of these random extensions of the BASS model, e.g. numerical regularity and ...

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Research paper thumbnail of Mean square stability for discrete linear stochastic systems

Several results concerning asymptotical mean square stability of the null solution of specific li... more Several results concerning asymptotical mean square stability of the null solution of specific linear stochastic systems are presented and proven. It is shown that the mean square stability of the implicit Euler method, taken from the monography of Kloeden and Platen (1992) and applied to linear stochastic differential equations, is necessary for the mean square stability of the corresponding implicit Milstein method (using the same implicitness parameter). Furthermore, a sufficient condition for the mean square stability of the implicit Euler method can be varified for autonomous systems. Additionally, the principle of 'monotonous inclusion' of the sequel of mean square stability domains holds for linear systems. The paper generalizes the results due to Schurz (1993) where one-dimensional linear complex systems with respect to asymptotical p-th mean stability have been investigated. Finally, a simple example confirms these assertions. The results can also be used to deduce ...

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Research paper thumbnail of Institut fiir Angewandt~ Analysis und Stochastik Balanced implicit methods for stiff stochastic systems: An introduction and numerical experiments

The paper introduces implicitness in stochastic terms of numerical methods for solving of stiff s... more The paper introduces implicitness in stochastic terms of numerical methods for solving of stiff stochastic differential equations and especially a class of fully implicit methods, the balanced methods. Their order of strong convergence is proved. Systematic numerical experiments compare the numerical behaviour of these schemes with that of different other schemes. A wide class of model equations are also provided as one by-product in order to test numerical methods in the case of stochastic stiffness in the given system.

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Research paper thumbnail of Waveform relaxation methods for stochastic differential equations

An operator equation X = Π X + G in a Banach space 퓔 of 퓕<sub>t</sub>-adapted random ... more An operator equation X = Π X + G in a Banach space 퓔 of 퓕<sub>t</sub>-adapted random elements describing an initial- or boundary value problem of a system of stochastic differential equations (SDEs) is considered. Our basic assumption is that the underlying system consists of weakly coupled subsystems. The proof of the convergence of corresponding waveform relaxation methods depends on the property that the spectral radius of an associated matrix is less than one. The entries of this matrix depend on the Lipschitz-constants of a decomposition of Π. In proving an existence result for the operator equation we show how the entries of the matrix depend on the right hand side of the stochastic differential equations. We derive conditions for the convergence under "classical" vector-valued Lipschitz-continuity of an appropriate splitting of the system of stochastic ODEs. A generalization of these key results under one-sided Lipschitz continuous and anticoercive drift...

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Research paper thumbnail of Moment Evolution of the Outflow-Rate from Nonlinear Conceptual Reservoirs

Water Science and Technology Library

The temporal evolution of moments of outflow-rate is investigated in a stochastically perturbed n... more The temporal evolution of moments of outflow-rate is investigated in a stochastically perturbed nonlinear reservoir due to precipitation. The detailed stochastic behaviour of outflow is obtained from the numerical solution of a nonlinear stochastic differential equation with multiplicative noise. The time-development of first two moments is studied for various choices of parameters. Using Stratonovich interpretation, it turns out that the mean outflow-rate is above that given by the deterministic solution. Based on the set of 9000 simulation runs, 90 % confidence intervals for the mean evolution of outflow-rate are computed. The effect of stochastic perturbations with finite correlation time is investigated.

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Research paper thumbnail of A Theorem on Asymptotic Stability of Solutions of Nonlinear Stochastic Difference Equations with Volterra Type Noise

A general theorem on global a.s. asymptotic stability of solutions to some nonlinear stochastic d... more A general theorem on global a.s. asymptotic stability of solutions to some nonlinear stochastic difference equations in IR 1 with in-the-arithmetic-mean-sense monotone terms as main part of its drift and Volterra-type dependence of its diffusion terms is presented as a certain application of convergence theorems for semimartingale inequalities to the decomposition of appropriate functionals.

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Research paper thumbnail of On global asymptotic stability of solutions of differential equations

Transactions of the American Mathematical Society, 1962

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Numerical Analysis of Stochastic Differential Equations with Explosions

Stochastic Analysis and Applications, 2005

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Boundary Preserving Semianalytic Numerical Algorithms for Stochastic Differential Equations

SIAM Journal on Scientific Computing, 2007

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Undamped nonlinear beam excited by additive -regular noise

Journal of Computational and Applied Mathematics, 2011

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Almost sure asymptotic stability of drift-implicit -methods for bilinear ordinary stochastic differential equations in

Journal of Computational and Applied Mathematics, 2005

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Krée, P.; Wedig, W. (eds.): Probabilistic Methods in Applied Physics. Berlin etc., Springer-Verlag 1995. IX, 393 pp., DM 116.–. ISBN 3-540-60214-3 (Lecture Notes in Physics 451)

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1997

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Existence and uniqueness of solutions of a semilinear differential equation

Indiana University Mathematics Journal, 2003

Bookmarks Related papers MentionsView impact

Research paper thumbnail of Global asymptotic stability of solutions of cubic stochastic difference equations

Advances in Difference Equations, 2004

Bookmarks Related papers MentionsView impact

Research paper thumbnail of On eeects of discretization on estimators of drift parameters for diiusion processes

In this paper statistical properties of estimators of drift parameters for diiusion processes are... more In this paper statistical properties of estimators of drift parameters for diiusion processes are studied by m o d e r n n umerical methods for stochastic diier-ential equations. This is a particularly useful method for discrete time samples, where estimators can be constructed by making discrete time approximations to the stochastic integrals appearing in the maximum likelihood estimators for continuously observed diiusions. A review is given of the necessary theory for parameter estimation for diiusion processes and for simulation of diiusion processes. Three examples are studied.

Bookmarks Related papers MentionsView impact

Research paper thumbnail of The invariance of asymptotic laws of stochastic systems under discretization

The stochastic trapezoidal rule provides the only discretization scheme from the family of implic... more The stochastic trapezoidal rule provides the only discretization scheme from the family of implicit Euler methods (see 11]) which possesses the same asymptotic (stationary) law as underlying linear continuous time stochastic systems with white or coloured noise. This identity is shown for systems with multiplicative (para-metric) and additive noise using xed p oint principles and the theo r y o f p ositive operators. The key result is useful for adequate implementation of stochastic algorithms applied to numerical solution of autonomous stochastic diierential equations. In particular it has practical importance when accurate long time integration is required such as in the process of estimation of Lyapunov exponents or stationary measures for oscillators in Mechanical Engineering.

Bookmarks Related papers MentionsView impact