7,839 research outputs found
An optimal polynomial approximation of Brownian motion
In this paper, we will present a strong (or pathwise) approximation of
standard Brownian motion by a class of orthogonal polynomials. The coefficients
that are obtained from the expansion of Brownian motion in this polynomial
basis are independent Gaussian random variables. Therefore it is practical
(requires independent Gaussian coefficients) to generate an approximate
sample path of Brownian motion that respects integration of polynomials with
degree less than . Moreover, since these orthogonal polynomials appear
naturally as eigenfunctions of an integral operator defined by the Brownian
bridge covariance function, the proposed approximation is optimal in a certain
weighted sense. In addition, discretizing Brownian paths as
piecewise parabolas gives a locally higher order numerical method for
stochastic differential equations (SDEs) when compared to the standard
piecewise linear approach. We shall demonstrate these ideas by simulating
Inhomogeneous Geometric Brownian Motion (IGBM). This numerical example will
also illustrate the deficiencies of the piecewise parabola approximation when
compared to a new version of the asymptotically efficient log-ODE (or
Castell-Gaines) method.Comment: 27 pages, 8 figure
Algebraic Structures and Stochastic Differential Equations driven by Levy processes
We construct an efficient integrator for stochastic differential systems
driven by Levy processes. An efficient integrator is a strong approximation
that is more accurate than the corresponding stochastic Taylor approximation,
to all orders and independent of the governing vector fields. This holds
provided the driving processes possess moments of all orders and the vector
fields are sufficiently smooth. Moreover the efficient integrator in question
is optimal within a broad class of perturbations for half-integer global root
mean-square orders of convergence. We obtain these results using the
quasi-shuffle algebra of multiple iterated integrals of independent Levy
processes.Comment: 41 pages, 11 figure
Algebraic structure of stochastic expansions and efficient simulation
We investigate the algebraic structure underlying the stochastic Taylor
solution expansion for stochastic differential systems.Our motivation is to
construct efficient integrators. These are approximations that generate strong
numerical integration schemes that are more accurate than the corresponding
stochastic Taylor approximation, independent of the governing vector fields and
to all orders. The sinhlog integrator introduced by Malham & Wiese (2009) is
one example. Herein we: show that the natural context to study stochastic
integrators and their properties is the convolution shuffle algebra of
endomorphisms; establish a new whole class of efficient integrators; and then
prove that, within this class, the sinhlog integrator generates the optimal
efficient stochastic integrator at all orders.Comment: 19 page
Optimal approximation of anticipating SDEs
In this article, we analyse the optimal approximation of anticipating
stochastic differential equations, where the integral is interpreted in
Skorohod sense. We derive optimal rate of convergence for the mean squared
error at the terminal point and an asymptotically optimal scheme for a class of
linear anticipating SDEs. Although alternative proof techniques are needed, our
results can be seen as generalizations of the corresponding results for It\=o
SDEs. As a key tool we carry over optimal approximation from vectors of
correlated Wiener integrals to a general class of random vectors, which cover
the solutions of the Skorohod SDEs
Strong Approximations for Nonlinear Transformations of Integrated Time Series
In this paper we establish the strong approximations for the nonlinear transformations of integrated time series. Both the asymptotically homogeneous and integrable transformations are considered, and the explicit rates for the convergence to their limit distributions are obtained under mild regularity conditions that are satisfied by virtually all nonlinear models used in practical applications. The first order asymptotics are also derived under the conditions that are significantly weaker than those required by earlier works.
From rough path estimates to multilevel Monte Carlo
New classes of stochastic differential equations can now be studied using
rough path theory (e.g. Lyons et al. [LCL07] or Friz--Hairer [FH14]). In this
paper we investigate, from a numerical analysis point of view, stochastic
differential equations driven by Gaussian noise in the aforementioned sense.
Our focus lies on numerical implementations, and more specifically on the
saving possible via multilevel methods. Our analysis relies on a subtle
combination of pathwise estimates, Gaussian concentration, and multilevel
ideas. Numerical examples are given which both illustrate and confirm our
findings.Comment: 34 page
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