A Natural Approach to the Numerical Integration of Riccati Differential Equations

Schiff, Jeremy; Shnider, Steven

doi:10.1137/s0036142996307946

Cited by 63 publications

(86 citation statements)

References 15 publications

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“…Riccati system. Our first application is for stochastic Riccati differential systems-some classes of which can be reformulated as linear systems (see Freiling [18] and Schiff and Shnider [50]). Such systems arise in stochastic linear-quadratic optimal control problems, for example, mean-variance hedging in finance (see Bobrovnytska and Schweizer [5] and Kohlmann and Tang [34])-though often these are backward problems (which we intend to investigate in a separate study).…”

Section: Uniformly Accurate Magnus Integratorsmentioning

confidence: 99%

Efficient Strong Integrators for Linear Stochastic Systems

Lord¹,

Malham²,

Wiese³

2008

SIAM J. Numer. Anal.

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Abstract. We present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of Wiener processes. These schemes are based on the Neumann (stochastic Taylor) and Magnus expansions. Firstly, we consider the case when the governing linear diffusion vector fields commute with each other, but not with the linear drift vector field. We prove that numerical methods based on the Magnus expansion are more accurate in the mean-square sense than corresponding stochastic Taylor integration schemes. Secondly, we derive the maximal rate of convergence for arbitrary multi-dimensional stochastic integrals approximated by their conditional expectations. Consequently, for general nonlinear stochastic differential equations with non-commuting vector fields, we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order. Thirdly, we consider the consequences in two numerical studies, one of which is an application arising in stochastic linear-quadratic optimal control.

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Section: Uniformly Accurate Magnus Integratorsmentioning

confidence: 99%

Efficient Strong Integrators for Linear Stochastic Systems

Lord¹,

Malham²,

Wiese³

2008

SIAM J. Numer. Anal.

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“…For a detailed expression of the scheme at order 4, see [13]. Inserting the definition of the admittance matrix into equation (3.12) shows that it can then be computed with the scheme 14) where the matrices E j are the elements of the matrix exponential e Ω(ỹ n ) identified by…”

Section: (A) Numerical Integrationmentioning

confidence: 99%

Multimodal method and conformal mapping for the scattering by a rough surface

Favraud

Pagneux

2015

Proc. R. Soc. A.

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The scattering of a wave from a periodic rough surface in two dimensions is considered. The proposed method is based on the use of a conformal mapping and of the multimodal admittance method. The use of the conformal mappings induces a spatially varying refractive index and transforms the boundary condition on the rough surface into a boundary condition on a flat surface. Then, the multimodal admittance method reduces this problem to a Riccati equation for the modal admittance matrix, which is solved numerically with a MagnusMöbius scheme. The method is shown to converge exponentially with the number of Fourier modes. The method also allows to find geometries having trapped modes, or quasi-trapped modes, at a given frequency, by looking at singularities, or quasisingularities, of this Riccati equation. Besides, a simple perturbation expansion, based on a small roughness approximation, is developed.

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“…where the block matrices α, β, γ and δ are sizes k × k, k × (n − k), (n − k) × k and (n − k) × (n − k), respectively (see Schiff and Shnider [43]; Munthe-Kaas [40]). We choose the action of GL(n) on Gr(k, n) to be the generalized Möbius transformation Λ y0 (S) = (αy 0 + β)(γy 0 + δ) −1 .…”

Section: Dual Of the Euclidean Algebra Se(3)mentioning

confidence: 99%

“…Stochastic Lie group integrators in the form of Magnus integrators for linear stochastic differential equations were investigated by Burrage and Burrage [5]. They were also used in the guise of Möbius schemes (see Schiff and Shnider [43]) to solve stochastic Riccati equations by Lord, Malham and Wiese [31] where they outperformed direct stochastic Taylor methods. Further applications where they might be applied include: backward stochastic Riccati equations arising in optimal stochastic linear-quadratic control (Kohlmann and Tang [28]); jump diffusion processes on matrix Lie groups for Bayesian inference (Srivastava, Miller and Grenander [44]); fractional Brownian motions on Lie groups (Baudoin and Coutin [3]) and stochastic dynamics triggered by DNA damage (Chickarmane, Ray, Sauro and Nadim [10]).…”

mentioning

confidence: 99%

Stochastic Lie Group Integrators

Malham¹,

Wiese²

2008

SIAM J. Sci. Comput.

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Abstract. We present Lie group integrators for nonlinear stochastic differential equations with non-commutative vector fields whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action, and subsequently pull back the flow to the corresponding Lie algebra via the exponential map. We construct an approximation to the stochastic flow in the Lie algebra via closed operations and then push back to the Lie group and then to the manifold, thus ensuring our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas methods after their deterministic counterparts. We also present stochastic Lie group integration schemes based on Castell-Gaines methods. These involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential Lie series. They become stochastic Lie group integrator schemes if we use Munthe-Kaas methods as the underlying ordinary differential integrator. Further, we show that some Castell-Gaines methods are uniformly more accurate than the corresponding stochastic Taylor schemes. Lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes.

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A Natural Approach to the Numerical Integration of Riccati Differential Equations

Cited by 63 publications

References 15 publications

Efficient Strong Integrators for Linear Stochastic Systems

Efficient Strong Integrators for Linear Stochastic Systems

Multimodal method and conformal mapping for the scattering by a rough surface

Stochastic Lie Group Integrators

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