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Horatio Boedihardjo

Publications and source records attributed to Horatio Boedihardjo.

At least 19 recordsLinked to original sources
arXiv2026

Statistical Inference for Fractional Diffusions

This is a review of statistical inference methodology for stochastic differential equations driven by fractional Brownian motion, otherwise called fractional diffusions. The first section reviews the theory needed to rigorously define them. The second section reviews existing theory of statistical inference for fractional diffusions, identifies remaining challenges and introduces a novel approach. The final section discusses results for the case where fractional diffusions result as a homogenisation limit.

math.PR↗
arXiv2025

Cartan's Path Development, the Logarithmic Signature and a Conjecture of Lyons-Sidorova

The signature transform, which is defined in terms of iterated path integrals of all orders, provides a faithful representation of the group of tree-reduced geometric rough paths. While the signature coefficients are known to decay factorially fast, the coefficients of the logarithimic signature generically only possess geometric decay. It was conjectured by T. Lyons and N. Sidorova that the only tree-reduced paths with bounded variation (BV) whose logarithmic signature can have infinite radius of convergence are straight lines. This conjecture was confirmed in the same work for certain types of paths and the general BV case remains unsolved. The aim of the present article is to develop a deeper understanding towards the Lyons-Sidorova conjecture. We prove that, if the logarithmic signature has infinite radius of convergence, the signature coefficients must satisfy an infinite system of rigid algebraic identities defined in terms of iterated integrals along complex exponential one-forms. These iterated integral identities impose strong geometric constraints on the underlying path, and in some special situations, confirm the conjecture. As a non-trivial application of our integral identities, we prove a strengthened version of the conjecture, which asserts that if the logarithmic signature of a BV path has infinite radius of convergence over all sub-intervals of time, the underlying path must be a straight line. Our methodology relies on Cartan's path development onto the complex semisimple Lie algebras $\mathfrak{sl}_m(\mathbb{C})$. The special root patterns of $\mathfrak{sl}_m(\mathbb{C})$ allow one to project the infinite-dimensional Baker-Campbell-Hausdorff (BCH) formula in a very special finite dimensional manner to yield meaningful quantitative relations between BCH-type singularities and the vanishing of certain iterated path integrals.

math.PR↗
arXiv2025

Statistical Inference for the Rough Homogenization Limit of Multiscale Fractional Ornstein-Uhlenbeck Processes

We study the problem of parameter estimation for the homogenization limit of multiscale systems involving fractional dynamics. In the case of stochastic multiscale systems driven by Brownian motion, it has been shown that in order for the Maximum Likelihood Estimators of the parameters of the limiting dynamics to be consistent, data needs to be subsampled at an appropriate rate. We extend these results to a class of fractional multiscale systems, often described as scaled fractional kinetic Brownian motions. We provide convergence results for the MLE of the diffusion coefficient of the limiting dynamics, computed using multiscale data. This requires the development of a different methodology to that used in the standard Brownian motion case, which is based on controlling the spectral norm of the inverse covariance matrix of a discretized fractional Gaussian noise on an interval.

math.ST↗
arXiv2023

A representation for the Expected Signature of Brownian motion up to the first exit time of the planar unit disc

The signature of a sample path is a formal series of iterated integrals along the path. The expected signature of a stochastic process gives a summary of the process that is especially useful for studying stochastic differential equations driven by the process. Lyons-Ni derived a partial differential equation for the expected signature of Brownian motion, starting at a point z in a bounded domain, until it hits to boundary of the domain. We focus on the domain of planar unit disc centred at 0. Motivated by recently found explicit formulae for some terms in the expected signature of this process in terms of Bessel functions, we derive a tensor series representation for this expected signature, coming from from studying Lyons-Ni's PDE. Although the representation is rather involved, it simplifies significantly to give a formula for the polynomial leading order term in each tensor component of the expected signature.

math.PR↗
arXiv2022

SL_2(R)-developments and Signature Asymptotics for Planar Paths with Bounded Variation

The signature transform, defined by the formal tensor series of global iterated path integrals, is a homomorphism between the path space and the tensor algebra that has been studied in geometry, control theory, number theory as well as stochastic analysis. An elegant isometry conjecture states that the length of a bounded variation path $γ$ can be recovered from the asymptotics of its normalised signature: $\text{Length}(γ)=\lim_{n\rightarrow\infty}\big\Vert n!\int_{0<t_{1}<\cdots<t_{n}<T}dγ_{t_{1}}\otimes\cdots\otimes dγ_{t_{n}}\big\Vert^{\frac{1}{n}}$. This property depends on a key topological non-degeneracy notion known as tree-reducedness (namely, with no tree-like pieces). Existing arguments have relied crucially on $γ$ having a continuous derivative under the unit speed parametrisation. In this article, we prove the above isometry conjecture for planar paths by assuming only local bounds on the angle of $γ'$ (which ensures the absence of tree-like pieces). Our technique is based on lifting the path onto the special linear group ${\rm SL}_{2}(\mathbb{R})$ and analysing the behaviour of the associated angle dynamics at a microscopic level.

math.CA↗
arXiv2022

On the Lack of Gaussian Tail for Rough Line Integrals along Fractional Brownian Paths

We show that the tail probability of the rough line integral $\int_{0}^{1}ϕ(X_{t})dY_{t}$, where $(X,Y)$ is a 2D fractional Brownian motion with Hurst parameter $H\in(1/4,1/2)$ and $ϕ$ is a $C_{b}^{\infty}$-function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a $γ$-Weibull tail with any exponent $γ>2H+1$. In particular, this produces a simple class of examples of differential equations driven by fBM, whose solutions fail to have Gaussian tail even though the underlying vector fields are assumed to be of class $C_{b}^{\infty}$. This also demonstrates that the well-known upper tail estimate proved by Cass-Litterer-Lyons in 2013 is essentially sharp.

math.PR↗
arXiv2020

Lipschitz-stability of Controlled Rough Paths and Rough Differential Equations

We provide an account for the existence and uniqueness of solutions to rough differential equations under the framework of controlled rough paths. The case when the driving path is $β$-Hölder continuous, for $β>1/3$, is widely available in the literature. In its extension to the case when $β\leqslant1/3,$ a main challenge and missing ingredient is to show that controlled roughs paths are closed under composition with Lipschitz transformations. Establishing such a property precisely, which has a strong algebraic nature, is a main purpose of the present article.

math.CA↗
arXiv2020

The expected signature of Brownian motion stopped on the boundary of a circle has finite radius of convergence

The expected signature is an analogue of the Laplace transform for rough paths. Chevyrev and Lyons showed that, under certain moment conditions, the expected signature determines the laws of signatures. Lyons and Ni posed the question of whether the expected signature of Brownian motion up to the exit time of a domain satisfies Chevyrev and Lyons' moment condition. We provide the first example where the answer is negative.

math.PR↗
arXiv2019

Path Developments and Tail Asymptotics of Signature for Pure Rough Paths

Solutions to linear controlled differential equations can be expressed in terms of iterated path integrals of the driving path. This collection of iterated integrals encodes essentially all information about the driving path. While upper bounds for iterated path integrals are well known, lower bounds are much less understood, and it is known only relatively recently that some type of asymptotics for the $n$-th order iterated integral can be used to recover some intrinsic quantitative properties of the path, such as the length of $C^1$ paths. In the present paper, we investigate the simplest type of rough paths (the rough path analogue of line segments), and establish uniform upper and lower estimates for the tail asymptotics of iterated integrals in terms of the local variation of the path. Our methodology, which we believe is new for this problem, involves developing paths into complex semisimple Lie algebras and using the associated representation theory to study spectral properties of Lie polynomials under the Lie algebraic development.

math.CA↗
arXiv2018

A Non-vanishing Property for the Signature of a Path

We prove that a continuous path with finite length in a real Banach space cannot have infinitely many zero components in its signature unless it is tree-like. In particular, this allows us to strengthen a limit theorem for signature recently proved by Chang, Lyons and Ni. What lies at the heart of our proof is a complexification idea together with deep results from holomorphic polynomial approximations in the theory of several complex variables.

math.CA↗
arXiv2018

An isomorphism between branched and geometric rough paths

We exhibit an explicit natural isomorphism between spaces of branched and geometric rough paths. This provides a multi-level generalisation of the isomorphism of Lejay-Victoir (2006) as well as a canonical version of the Itô-Stratonovich correction formula of Hairer-Kelly (2015). Our construction is elementary and uses the property that the Grossman-Larson algebra is isomorphic to a tensor algebra. We apply this isomorphism to study signatures of branched rough paths. Namely, we show that the signature of a branched rough path is trivial if and only if the path is tree-like, and construct a non-commutative Fourier transform for probability measures on signatures of branched rough paths. We use the latter to provide sufficient conditions for a random signature to be determined by its expected value, thus giving an answer to the uniqueness moment problem for branched rough paths.

math.PR↗
arXiv2018

The Tail Asymptotics of the Brownian Signature

The signature of a path γis a sequence whose n-th term is the order-n iterated integrals of γ. It arises from solving multidimensional linear differential equations driven by γ. We are interested in relating the path properties of γwith its signature. If γis C^{1}, then an elegant formula of Hambly and Lyons relates the length of γto the tail asymptotics of the signature. We show an analogous formula for the multidimensional Brownian motion, with the quadratic variation playing a similar role to the length. In the proof, we study the hyperbolic development of Brownian motion and also obtain a new subadditive estimate for the asymptotic of signature, which may be of independent interest. As a corollary, we strengthen the existing uniqueness results for the signatures of Brownian motion.

math.PR↗
arXiv2017

Decay Rate of Iterated Integrals of Branched Rough Paths

Iterated integrals of paths arise frequently in the study of the Taylor's expansion for controlled differential equations. We will prove a factorial decay estimate, conjectured by M. Gubinelli, for the iterated integrals of non-geometric rough paths. We will explain, with a counter example, why the conventional approach of using the neoclassical inequality fails. Our proof involves a concavity estimate for sums over rooted trees and a non-trivial extension of T. Lyons' proof in 1994 for the factorial decay of iterated Young's integrals.

math.PR↗
arXiv2016

Factorial Decay of Iterated Rough Integrals

In this complementary note to [1] (arXiv:1501.05641), we provide an alternative proof for the factorial decay estimate of iterated integrals for geometric rough paths without using the neoclassical inequality. This note intends to aid the readers on the proof in [1] which works also for branched rough paths. Just as in [1], the proof here is an extension of Lyons 94' [4] from Young's integration to geometric rough paths.

math.CA↗
arXiv2016

A Quasi-sure Non-degeneracy Property for the Brownian Rough Path

In the present paper, we are going to show that outside a slim set in the sense of Malliavin (or quasi-surely), the signature path (which consists of iterated path integrals in every degree) of Brownian motion is non-self-intersecting. This property relates closely to a non-degeneracy property for the Brownian rough path arising naturally from the uniqueness of signature problem in rough path theory. As an important consequence we conclude that quasi-surely, the Brownian rough path does not have any tree-like pieces and every sample path of Brownian motion is uniquely determined by its signature up to reparametrization.

math.PR↗
arXiv2015

The Signature of a Rough Path: Uniqueness

In the context of controlled differential equations, the signature is the exponential function on paths. B. Hambly and T. Lyons proved that the signature of a bounded variation path is trivial if and only if the path is tree-like. We extend Hambly-Lyons' result and their notion of tree-like paths to the setting of weakly geometric rough paths in a Banach space. At the heart of our approach is a new definition for reduced path and a lemma identifying the reduced path group with the space of signatures.

math.CA↗