arXiv subjects

Kristian Debrabant

Publications and source records attributed to Kristian Debrabant.

29 records · Page 2Linked to original sources

arXiv2014

Semi-Lagrangian schemes for linear and fully non-linear Hamilton-Jacobi-Bellman equations

We consider the numerical solution of Hamilton-Jacobi-Bellman equations arising in stochastic control theory. We introduce a class of monotone approximation schemes relying on monotone interpolation. These schemes converge under very weak assumptions, including the case of arbitrary degenerate diffusions. Besides providing a unifying framework that includes several known first order accurate schemes, stability and convergence results are given, along with two different robust error estimates. Finally, the method is applied to a super-replication problem from finance.

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arXiv2013

Families of efficient second order Runge-Kutta methods for the weak approximation of Itô stochastic differential equations

Recently, a new class of second order Runge-Kutta methods for Itô stochastic differential equations with a multidimensional Wiener process was introduced by Rößler. In contrast to second order methods earlier proposed by other authors, this class has the advantage that the number of function evaluations depends only linearly on the number of Wiener processes and not quadratically. In this paper, we give a full classification of the coefficients of all explicit methods with minimal stage number. Based on this classification, we calculate the coefficients of an extension with minimized error constant of the well-known RK32 method to the stochastic case. For three examples, this method is compared numerically with known order two methods and yields very promising results.

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arXiv2013

Classification of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations

In the present paper, a class of stochastic Runge-Kutta methods containing the second order stochastic Runge-Kutta scheme due to E. Platen for the weak approximation of Itô stochastic differential equation systems with a multi-dimensional Wiener process is considered. Order one and order two conditions for the coefficients of explicit stochastic Runge-Kutta methods are solved and the solution space of the possible coefficients is analyzed. A full classification of the coefficients for such stochastic Runge-Kutta schemes of order one and two with minimal stage numbers is calculated. Further, within the considered class of stochastic Runge-Kutta schemes coefficients for optimal schemes in the sense that additionally some higher order conditions are fulfilled are presented.

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arXiv2013

Continuous Weak Approximation for Stochastic Differential Equations

A convergence theorem for the continuous weak approximation of the solution of stochastic differential equations by general one step methods is proved, which is an extension of a theorem due to Milstein. As an application, uniform second order conditions for a class of continuous stochastic Runge-Kutta methods containing the continuous extension of the second order stochastic Runge-Kutta scheme due to Platen are derived. Further, some coefficients for optimal continuous schemes applicable to Itô stochastic differential equations with respect to a multi-dimensional Wiener process are presented.

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arXiv2013

Convergence of Runge-Kutta Methods Applied to Linear Partial Differential-Algebraic Equations

We apply Runge-Kutta methods to linear partial differential-algebraic equations of the form $Au_t(t,x) + B(u_{xx}(t,x)+ru_x(t,x))+Cu(t,x) = f(t,x)$, where $A,B,C\in\R^{n,n}$ and the matrix $A$ is singular. We prove that under certain conditions the temporal convergence order of the fully discrete scheme depends on the time index of the partial differential-algebraic equation. In particular, fractional orders of convergence in time are encountered. Furthermore we show that the fully discrete scheme suffers an order reduction caused by the boundary conditions. Numerical examples confirm the theoretical results.

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arXiv2013

On quasi-linear PDAEs with convection: applications, indices, numerical solution

For a class of partial differential algebraic equations (PDAEs) of quasi-linear type which include nonlinear terms of convection type a possibility to determine a time and spatial index is considered. As a typical example we investigate an application from plasma physics. Especially we discuss the numerical solution of initial boundary value problems by means of a corresponding finite difference splitting procedure which is a modification of a well known fractional step method coupled with a matrix factorization. The convergence of the numerical solution towards the exact solution of the corresponding initial boundary value problem is investigated. Some results of a numerical solution of the plasma PDAE are given.

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arXiv2012

Semi-Lagrangian schemes for linear and fully non-linear diffusion equations

For linear and fully non-linear diffusion equations of Bellman-Isaacs type, we introduce a class of approximation schemes based on differencing and interpolation. As opposed to classical numerical methods, these schemes work for general diffusions with coefficient matrices that may be non-diagonal dominant and arbitrarily degenerate. In general such schemes have to have a wide stencil. Besides providing a unifying framework for several known first order accurate schemes, our class of schemes includes new first and higher order versions. The methods are easy to implement and more efficient than some other known schemes. We prove consistency and stability of the methods, and for the monotone first order methods, we prove convergence in the general case and robust error estimates in the convex case. The methods are extensively tested.

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arXiv2011

A micro/macro algorithm to accelerate Monte Carlo simulation of stochastic differential equations

We present and analyze a micro/macro acceleration technique for the Monte Carlo simulation of stochastic differential equations (SDEs) in which there is a separation between the (fast) time-scale on which individual trajectories of the SDE need to be simulated and the (slow) time-scale on which we want to observe the (macroscopic) function of interest. The method performs short bursts of microscopic simulation using an ensemble of SDE realizations, after which the ensemble is restricted to a number of macroscopic state variables. The resulting macroscopic state is then extrapolated forward in time and the ensemble is projected onto the extrapolated macroscopic state. We provide a first analysis of its convergence in terms of extrapolation time step and number of macroscopic state variables. The effects of the different approximations on the resulting error are illustrated via numerical experiments.

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arXiv2010

Composition of stochastic B-series with applications to implicit Taylor methods

In this article, we construct a representation formula for stochastic B-series evaluated in a B-series. This formula is used to give for the first time the order conditions of implicit Taylor methods in terms of rooted trees. Finally, as an example we apply these order conditions to derive in a simple manner a family of strong order 1.5 Taylor methods applicable to Itô SDEs.

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arXiv2010

Stochastic B-series analysis of iterated Taylor methods

For stochastic implicit Taylor methods that use an iterative scheme to compute their numerical solution, stochastic B--series and corresponding growth functions are constructed. From these, convergence results based on the order of the underlying Taylor method, the choice of the iteration method, the predictor and the number of iterations, for Itô and Stratonovich SDEs, and for weak as well as strong convergence are derived. As special case, also the application of Taylor methods to ODEs is considered. The theory is supported by numerical experiments.

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arXiv2010

Runge-Kutta methods for third order weak approximation of SDEs with multidimensional additive noise

A new class of third order Runge-Kutta methods for stochastic differential equations with additive noise is introduced. In contrast to Platen's method, which to the knowledge of the author has been up to now the only known third order Runge-Kutta scheme for weak approximation, the new class of methods affords less random variable evaluations and is also applicable to SDEs with multidimensional noise. Order conditions up to order three are calculated and coefficients of a four stage third order method are given. This method has deterministic order four and minimized error constants, and needs in addition less function evaluations than the method of Platen. Applied to some examples, the new method is compared numerically with Platen's method and some well known second order methods and yields very promising results.

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