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Christian Litterer

Publications and source records attributed to Christian Litterer.

10 recordsLinked to original sources
arXiv2024

High-degree cubature on Wiener space through unshuffle expansions

Utilising classical results on the structure of Hopf algebras, we develop a novel approach for the construction of cubature formulae on Wiener space based on unshuffle expansions. We demonstrate the effectiveness of this approach by constructing the first explicit degree-7 cubature formula on $d$-dimensional Wiener space with drift, in the sense of Lyons and Victoir. The support of our degree-7 formula is significantly smaller than that of currently implemented or proposed constructions.

math.PR↗
arXiv2021

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 $1 < p$-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 $p$.

math.CA↗
arXiv2015

Smoothness of the density for solutions to Gaussian rough differential equations

We consider stochastic differential equations of the form $dY_t=V(Y_t)\,dX_t+V_0(Y_t)\,dt$ driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields $V_0$ and $V=(V_1,\ldots,V_d)$ satisfy Hörmander's bracket condition, we demonstrate that $Y_t$ admits a smooth density for any $t\in(0,T]$, provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter $H>1/4$, the Ornstein-Uhlenbeck process and the Brownian bridge returning after time $T$.

math.PR↗
arXiv2014

Constrained Rough Paths

We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an efficient and intrinsic theory of rough differential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold valued rough paths. Finally, this framework is used to show there is a one to one correspondence between rough paths on a d-dimensional manifold and rough paths on d-dimensional Euclidean space. This last result is a rough path analogue of Cartan's development map and its stochastic version which was developed by Eeels and Elworthy and Malliavin.

math.PR↗
arXiv2013

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 [7],[9],[10],[11], 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 in the spirit of classical work by Ocone [14]. The estimates we derive have sharp small time asymptotics.

math.PR↗
arXiv2013

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.

math.PR↗
arXiv2011

A Chen-Fliess approximation for diffusion functionals

We show that an interesting class of functionals of stochastic differential equations can be approximated by a Chen-Fliess series of iterated stochastic integrals and give a L^{2} error estimate, thus generalizing the standard stochastic Taylor expansion. The coefficients in this series are given a very intuitive meaning by using functional derivatives, recently introduced by B. Dupire.

math.PR↗
arXiv2011

On the error estimate for cubature on Wiener space

It was pointed out in Crisan, Ghazali [2] that the error estimate for the cubature on Wiener space algorithm developed in Lyons, Victoir [11] requires an additional assumption on the drift. In this note we demonstrate that it is straightforward to adopt the analysis of Kusuoka [7] to obtain a general estimate without an additional assumptions on the drift. In the process we slightly sharpen the bounds derived in [7].

math.PR↗
arXiv2011

Rough Paths on Manifolds

We develop a fundamental framework for and extend the theory of rough paths to Lipschitz-gamma manifolds.

math.CA↗