The Rate of Convergence of Finite-Difference Approximations for Parabolic Bellman Equations with Lipschitz Coefficients in Cylindrical Domains

Dong, Hongjie; Крылов, Н. В.

doi:10.1007/s00245-007-0879-4

Cited by 45 publications

(97 citation statements)

References 21 publications

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“…The following corollary is one of the results of [3] proved there by using the theory of diffusion processes. We obtain it immediately from case (iii) with k = 1.…”

Section: Formulation Of the Main Results For Parabolic Equationsmentioning

confidence: 76%

“…Actually, in [3] a full discretization in time and space is considered for parabolic equations, so that, formally, Corollary 2.4 does not yield the corresponding result of [3]. On the other hand, a similar corollary can be derived from Theorem 3.5 below which treats elliptic equations and it does imply the corresponding result of [3].…”

Section: Corollary 24 Let Conditions (S) and (216) Be Satisfied Lmentioning

confidence: 99%

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Accelerated finite difference schemes for second order degenerate elliptic and parabolic problems in the whole space

Gyöngy

Крылов

2011

Math. Comp.

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“…The following corollary is one of the results of [3] proved there by using the theory of diffusion processes. We obtain it immediately from case (iii) with k = 1.…”

Section: Formulation Of the Main Results For Parabolic Equationsmentioning

confidence: 76%

Section: Corollary 24 Let Conditions (S) and (216) Be Satisfied Lmentioning

confidence: 99%

Accelerated finite difference schemes for second order degenerate elliptic and parabolic problems in the whole space

Gyöngy

Крылов

2011

Math. Comp.

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“…If u (1) and u (2) are solutions to of (4.25) on [0, τ ] for a stopping time τ , then using Itô's formula for y t = |u…”

Section: Gyöngymentioning

confidence: 99%

“…The rate of convergence of various finite difference schemes for elliptic and parabolic PDEs have been studied extensively in the literature when the equations are non degenerate, but there are only a few publications dedicated to the numerical analysis of finite difference schemes for degenerate equations. Sharp rate of convergence in sup norm are obtained in [2] for fully discretized degenerate elliptic and parabolic PDEs. The finite difference schemes investigated in [2] are monotone schemes.…”

Section: Introductionmentioning

confidence: 99%

“…Sharp rate of convergence in sup norm are obtained in [2] for fully discretized degenerate elliptic and parabolic PDEs. The finite difference schemes investigated in [2] are monotone schemes. In [6] for a large class of monotone finite difference schemes (in the spatial variables) power series expansions are obtained and Richardson extrapolation is used to get accelerated schemes for degenerate elliptic and parabolic PDEs.…”

Section: Introductionmentioning

confidence: 99%

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On stochastic finite difference schemes

Gyöngy

2014

Stoch PDE: Anal Comp

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Abstract. Finite difference schemes in the spatial variable for degenerate stochastic parabolic PDEs are investigated. Sharp results on the rate of Lp and almost sure convergence of the finite difference approximations are presented and results on Richardson extrapolation are established for stochastic parabolic schemes under smoothness assumptions. IntroductionWe consider finite difference schemes to stochastic partial differential equations (SPDEs). The stochastic PDEs we are interested in are linear second order stochastic parabolic equations in the whole R d in the spatial variable. They may degenerate and become first order stochastic or deterministic PDEs. The finite difference schemes which we investigate are spatial discretizations of such SPDEs. One can view them as (possibly degenerate) infinite systems of stochastic differential equations, whose components describe the time evolution of approximate values at the grid points of the solutions to SPDEs. Adapting the approach of [13] we view stochastic finite difference schemes, like in [7] and [3], as stochastic equations for random fields on the whole R d not only on grids.Our aim is to investigate the rate of convergence in the supremum norm of the finite difference approximations. We show that under the stochastic parabolicity condition, if the coefficients and the data are sufficiently smooth, then the solutions to the finite difference schemes admit power series expansions in terms of h, the mesh-size of the grid. The coefficients in these power series are random fields, independent of h, and for any p > 0 the p-th moments of the sup norm of the remainder term is estimated by a power of h. This is Theorem 2.2. Hence for h → 0 we get the convergence (and the sharp rate) of the solutions of the finite difference schemes to a random field which is the solution to the corresponding SPDE. Moreover, by Richardson extrapolation we get that the rate of convergence can be accelerated to any high order if one takes appropriate mixtures of approximations corresponding to different grid sizes. In Theorem 2.4 we obtain convergence estimates for any (high) p-th moments of the sup norm of the approximation error. Hence in 2000 Mathematics Subject Classification. 65M15, 35J70, 35K65.

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Monotone Schemes

Jakobsen¹

2010

Encyclopedia of Quantitative Finance

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The Rate of Convergence of Finite-Difference Approximations for Parabolic Bellman Equations with Lipschitz Coefficients in Cylindrical Domains

Cited by 45 publications

References 21 publications

Accelerated finite difference schemes for second order degenerate elliptic and parabolic problems in the whole space

Accelerated finite difference schemes for second order degenerate elliptic and parabolic problems in the whole space

On stochastic finite difference schemes

Monotone Schemes

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