H. Schurz - Academia.edu (original) (raw)
Papers by H. Schurz
Stochastic Analysis and Applications, 1996
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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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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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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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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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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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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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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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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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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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Transactions of the American Mathematical Society, 1962
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Stochastic Analysis and Applications, 2005
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SIAM Journal on Scientific Computing, 2007
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Journal of Computational and Applied Mathematics, 2011
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Journal of Computational and Applied Mathematics, 2005
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ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1997
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Indiana University Mathematics Journal, 2003
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Advances in Difference Equations, 2004
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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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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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Stochastic Analysis and Applications, 1996
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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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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.
Bookmarks Related papers MentionsView impact
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 ...
Bookmarks Related papers MentionsView impact
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 ...
Bookmarks Related papers MentionsView impact
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 ...
Bookmarks Related papers MentionsView impact
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.
Bookmarks Related papers MentionsView impact
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...
Bookmarks Related papers MentionsView impact
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.
Bookmarks Related papers MentionsView impact
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.
Bookmarks Related papers MentionsView impact
Transactions of the American Mathematical Society, 1962
Bookmarks Related papers MentionsView impact
Stochastic Analysis and Applications, 2005
Bookmarks Related papers MentionsView impact
SIAM Journal on Scientific Computing, 2007
Bookmarks Related papers MentionsView impact
Journal of Computational and Applied Mathematics, 2011
Bookmarks Related papers MentionsView impact
Journal of Computational and Applied Mathematics, 2005
Bookmarks Related papers MentionsView impact
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1997
Bookmarks Related papers MentionsView impact
Indiana University Mathematics Journal, 2003
Bookmarks Related papers MentionsView impact
Advances in Difference Equations, 2004
Bookmarks Related papers MentionsView impact
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
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