Linear Multistep Schemes for BSDEs

Chassagneux, Jean-François

doi:10.1137/120902951

“…Higher-order schemes are developed in [5,4]. In our case a Fourier-based method results in a very efficient numerical scheme.…”

Section: Global Error -Time-discretization Scheme Fbsdementioning

confidence: 99%

Numerical Fourier Method and Second-Order Taylor Scheme for Backward SDEs in Finance

Ruijter¹,

Oosterlee

²

2014

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We develop a Fourier method to solve quite general backward stochastic differential equations (BSDEs) with second-order accuracy. The underlying forward stochastic differential equation (FSDE) is approximated by different Taylor schemes, such as the Euler, Milstein, and Order 2.0 weak Taylor schemes, or by exact simulation. A θ -time-discretization of the time-integrands leads to an induction scheme with conditional expectations. The computation of the conditional expectations appearing relies on the availability of the characteristic function for these schemes. We will use the characteristic function of the discrete forward process. The expected values are approximated by Fourier cosine series expansions. Numerical experiments show rapid convergence of our efficient probabilistic numerical method. Second-order accuracy is observed and also proved. We apply the method to, among others, option pricing problems under the Constant Elasticity of Variance and Cox-Ingersoll-Ross processes.

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“…Similarly, we get Higher-order schemes are developed in [5,4]. In our case a Fourier-based method results in a very efficient numerical scheme.…”

Section: Global Error -Time-discretization Scheme Fbsdementioning

confidence: 93%

Numerical Fourier method and second-order Taylor scheme for backward SDEs in finance

Ruijter

¹

,

Oosterlee

²

2016

Applied Numerical Mathematics

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We develop a Fourier method to solve quite general backward stochastic differential equations (BSDEs) with second-order accuracy. The underlying forward stochastic differential equation (FSDE) is approximated by different Taylor schemes, such as the Euler, Milstein, and Order 2.0 weak Taylor schemes, or by exact simulation. A θ -time-discretization of the time-integrands leads to an induction scheme with conditional expectations. The computation of the conditional expectations appearing relies on the availability of the characteristic function for these schemes. We will use the characteristic function of the discrete forward process. The expected values are approximated by Fourier cosine series expansions. Numerical experiments show rapid convergence of our efficient probabilistic numerical method. Second-order accuracy is observed and also proved. We apply the method to, among others, option pricing problems under the Constant Elasticity of Variance and Cox-Ingersoll-Ross processes.

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“…Since then, many extensions have been considered: high order schemes e.g. [8,10], schemes for reflected BSDEs [1,9], for fully coupled BSDEs [20,2], for quadratic BSDEs [12] or McKean-Vlasov BSDEs [13,11]. It is also important to mention that, quite recently, very promising probabilistic forward methods have been introduced to approximate (2) [5] or directly the non-linear parabolic PDE (3) [24].…”

Section: Introductionmentioning

confidence: 99%

“…Essentially, the algorithm estimates the value of the field u on the points on the support of the cubature approximating law, thus giving an approximation scheme for the solution of the BSDE (2). By its nature, this algorithm can be easily used to implement second order discretization schemes as in [8,17], and applied in the context of McKean-Vlasov BSDEs as in [13]. The cubature algorithm has been studied under a set of assumptions that guarantee sufficient regularity for the field (for example, smooth coefficients for the forward equation and the generator of the backward equation, Lipschitz regularity on the boundary condition plus a structural condition of the type UFG, see Section 1.1 below).…”

Section: Introductionmentioning

confidence: 99%

Cubature method to solve BSDEs: Error expansion and complexity control

Chassagneux

¹

,

Trillos

²

2020

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We obtain an explicit error expansion for the solution of Backward Stochastic Differential Equations (BSDEs) using the cubature on Wiener spaces method. The result is proved under a mild strengthening of the assumptions needed for the application of the cubature method. The explicit expansion can then be used to construct implementable higher order approximations via Richardson-Romberg extrapolation. To allow for an effective efficiency improvement of the interpolated algorithm, we introduce an additional projection on finite grids through interpolation operators. We study the resulting complexity reduction in the case of the linear interpolation.

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Linear Multistep Schemes for BSDEs

Cited by 56 publications

References 22 publications

Numerical Fourier Method and Second-Order Taylor Scheme for Backward SDEs in Finance

Numerical Fourier Method and Second-Order Taylor Scheme for Backward SDEs in Finance

Numerical Fourier method and second-order Taylor scheme for backward SDEs in finance

Cubature method to solve BSDEs: Error expansion and complexity control

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