Numerical Analysis of Stochastic Schemes in Geophysics

Ewald, Brian D.; Témam, Roger

doi:10.1137/s0036142902418333

Cited by 23 publications

(18 citation statements)

References 10 publications

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“…C. The stochastic Adam-Bashforth (SAB) scheme (cf. [16]). It is easily seen from (3.13) that the Platen-Wagner derivatives of coefficient function b but still achieves the 3 2 -order convergence rate.…”

Section: Numerical Schemes For the Sde (24)mentioning

confidence: 99%

“…To compare with the existing results, we shall use an example proposed in Milstein and Tretyakov [31], and we test several methods for the forward SDEs. These include the standard Euler scheme, the Milstein scheme, the Platen-Wagner scheme, and the stochastic Adam-Bashforth (SAB) scheme proposed in [16]. By the nature of these schemes, we expect the Euler scheme to be 1/2-order convergent, the Milstein first-order scheme to be first-order convergent, and the Platen-Wagner and SAB schemes to be 3/2-order convergent.…”

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confidence: 99%

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On Numerical Approximations of Forward-Backward Stochastic Differential Equations

Ma¹,

Shen²,

Zhao³

2008

SIAM J. Numer. Anal.

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Abstract.A numerical method for a class of forward-backward stochastic differential equations (FBSDEs) is proposed and analyzed. The method is designed around the four step scheme [J. Douglas, Jr., J. Ma, and P. Protter, Ann. Appl. Probab., 6 (1996), pp. 940-968] but with a Hermite-spectral method to approximate the solution to the decoupling quasi-linear PDE on the whole space. A rigorous synthetic error analysis is carried out for a fully discretized scheme, namely a first-order scheme in time and a Hermite-spectral scheme in space, of the FBSDEs. Equally important, a systematical numerical comparison is made between several schemes for the resulting decoupled forward SDE, including a stochastic version of the Adams-Bashforth scheme. It is shown that the stochastic version of the Adams-Bashforth scheme coupled with the Hermite-spectral method leads to a convergence rate of 3 2 (in time) which is better than those in previously published work for the FBSDEs. , finding an efficient numerical scheme for both BSDEs and FBSDEs has also become an independent but integral part of the theory. Tremendous efforts have been made during the past decade to circumvent the fundamental difficulties caused by the combination of the "backward" nature of the SDEs and the associated decoupling techniques for FBSDEs. In the "pure backward" (or "decoupled" forwardbackward) case, various methods have been proposed. These include the PDE method in the Markovian case (e.g., Chevance [8] (Lemor, Gobet, and Warin [23] and [24]).In the case of coupled FBSDEs, however, the results are very limited, largely due to the lack of a solution method itself in such cases. It has been well understood

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Section: Numerical Schemes For the Sde (24)mentioning

confidence: 99%

mentioning

confidence: 99%

On Numerical Approximations of Forward-Backward Stochastic Differential Equations

Ma¹,

Shen²,

Zhao³

2008

SIAM J. Numer. Anal.

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“…We now recall the well-known strong convergence results for the explicit and implicit Euler schemes and the so-called Heun scheme; we then present the convergence results for the stochastic Adams-Bashforth scheme following [ET2].…”

Section: Strong Convergencementioning

confidence: 99%

“…By inspection of the stochastic Taylor formula at order 2, a stochastic version of this scheme has been proposed for equation (1) in [ET2]. Namely, for the equation…”

Section: Implicit Euler Scheme the Scheme Readsmentioning

confidence: 99%