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31.
We present a general framework for deriving continuous dependence estimates for, possibly polynomially growing, viscosity solutions of fully nonlinear degenerate parabolic integro-PDEs. We use this framework to provide explicit estimates for the continuous dependence on the coefficients and the “Lévy measure” in the Bellman/Isaacs integro-PDEs arising in stochastic control/differential games. Moreover, these explicit estimates are used to prove regularity results and rates of convergence for some singular perturbation problems. Finally, we illustrate our results on some integro-PDEs arising when attempting to price European/American options in an incomplete stock market driven by a geometric Lévy process. Many of the results obtained herein are new even in the convex case where stochastic control theory provides an alternative to our pure PDE methods. 相似文献
32.
The combined CFD-PBE (population balance models) are computationally intensive requiring efficient numerical methods for solving practical problems. In this paper, a high order method is presented based on the least-squares method (LSM) for the solution of a spatial-dependent population balance equation which includes advective processes. Numerical experiments are performed in order to study the behavior of the proposed method for one-dimensional cases using model problems with analytical solutions. 相似文献
33.
We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of
(0.1)
where φ is merely continuous and nondecreasing, and is the generator of a general symmetric Lévy process. This means that can have both local and nonlocal parts like, e.g., . New uniqueness results for bounded distributional solutions to this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for . Existence and a priori estimates are deduced from a numerical approximation, and energy-type estimates are also obtained. 相似文献
34.
We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (Lévy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments. 相似文献
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E. R. Jakobsen 《BIT Numerical Mathematics》2004,44(2):269-285
In this paper we provide estimates of the rates of convergence of monotone approximation schemes for non-convex equations in one space-dimension. The equations under consideration are the degenerate elliptic Isaacs equations with x-depending coefficients, and the results applies in particular to certain finite difference methods and control schemes based on the dynamic programming principle. Recently, Krylov, Barles, and Jakobsen obtained similar estimates for convex Hamilton–Jacobi–Bellman equations in arbitrary space-dimensions. Our results are only valid in one space-dimension, but they are the first results of this type for non-convex second-order equations. 相似文献
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39.
In solving a nonlinear equation by the use of a continuation method one of the crucial problems is the choice of the step sizes. We present a model for the total computational cost of a standard numerical continuation process and solve the problem of optimal step size control for this model. Using the theoretical results as a basis, we develop an adaptive step size algorithm for Newton's method. This procedure is computationally inexpensive and it gives quite satisfactory results compared to some other numerical experiments found in the literature. 相似文献
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