53 research outputs found
High order recombination and an application to cubature on Wiener space
Particle methods are widely used because they can provide accurate
descriptions of evolving measures. Recently it has become clear that by
stepping outside the Monte Carlo paradigm these methods can be of higher order
with effective and transparent error bounds. A weakness of particle methods
(particularly in the higher order case) is the tendency for the number of
particles to explode if the process is iterated and accuracy preserved. In this
paper we identify a new approach that allows dynamic recombination in such
methods and retains the high order accuracy by simplifying the support of the
intermediate measures used in the iteration. We describe an algorithm that can
be used to simplify the support of a discrete measure and give an application
to the cubature on Wiener space method developed by Lyons and Victoir [Proc. R.
Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004) 169-198].Comment: Published in at http://dx.doi.org/10.1214/11-AAP786 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Kusuoka-Stroock gradient bounds for the solution of the filtering equation
© 2014 Elsevier Inc.We obtain sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock [13,14,16,17], and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. It uses a pathwise representation of the perturbed semigroup following Ocone [22]. The estimates we derive have sharp small time asymptotics
A combinatorial approach to geometric rough paths and their controlled paths
We develop the structure theory for transformations of weakly geometric rough paths of bounded -variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We derive an explicit combinatorial expression for the rough path lift of a controlled path, and use it to obtain fundamental identities such as the associativity of the rough integral, the adjunction between pushforwards and pullbacks, and a change of variables formula for rough differential equations (RDEs). As applications we define rough paths, rough integration and RDEs on manifolds, extending the results of [CDL15] to the case of arbitrary
Kusuoka–Stroock gradient bounds for the solution of the filtering equation
We obtain sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock [13], [14], [16], [17], and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. It uses a pathwise representation of the perturbed semigroup following Ocone [22]. The estimates we derive have sharp small time asymptotics
Integrability and tail estimates for Gaussian rough differential equations
We derive explicit tail-estimates for the Jacobian of the solution flow for stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter H>1/4. We remark on the relevance of such estimates to a number of significant open problems
Complexity matters: The effect of non-native algal species on habitat architecture, biodiversity and predator-prey interactions
Reconstitution of abnormalities in the malignant hyperthermia-susceptible pig ryanodine receptor
Skeletal muscle junctional membrane protein content in pigs with different ryanodine receptor genotypes
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