arXiv subjects

Han Cheng Lie

Publications and source records attributed to Han Cheng Lie.

25 records · Page 2Linked to original sources

arXiv2019

Fréchet differentiable drift dependence of Perron--Frobenius and Koopman operators for non-deterministic dynamics

We consider Perron-Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations, and establish their Fréchet differentiability with respect to the drift. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron-Frobenius and Koopman operators.

math.PR↗

arXiv2019

On a degeneracy ratio for bounded convex polytopes

We consider a quantity that measures the roundness of a bounded, convex $d$-polytope in $\mathbb{R}^d$. We majorise this quantity in terms of the smallest singular value of the matrix of outer unit normals to the facets of the polytope.

math.OC↗

arXiv2018

Implicit Probabilistic Integrators for ODEs

We introduce a family of implicit probabilistic integrators for initial value problems (IVPs), taking as a starting point the multistep Adams-Moulton method. The implicit construction allows for dynamic feedback from the forthcoming time-step, in contrast to previous probabilistic integrators, all of which are based on explicit methods. We begin with a concise survey of the rapidly-expanding field of probabilistic ODE solvers. We then introduce our method, which builds on and adapts the work of Conrad et al. (2016) and Teymur et al. (2016), and provide a rigorous proof of its well-definedness and convergence. We discuss the problem of the calibration of such integrators and suggest one approach. We give an illustrative example highlighting the effect of the use of probabilistic integrators - including our new method - in the setting of parameter inference within an inverse problem.

stat.ME↗

arXiv2018

Equivalence of weak and strong modes of measures on topological vector spaces

A strong mode of a probability measure on a normed space $X$ can be defined as a point $u$ such that the mass of the ball centred at $u$ uniformly dominates the mass of all other balls in the small-radius limit. Helin and Burger weakened this definition by considering only pairwise comparisons with balls whose centres differ by vectors in a dense, proper linear subspace $E$ of $X$, and posed the question of when these two types of modes coincide. We show that, in a more general setting of metrisable vector spaces equipped with measures that are finite on bounded sets, the density of $E$ and a uniformity condition suffice for the equivalence of these two types of modes. We accomplish this by introducing a new, intermediate type of mode. We also show that these modes can be inequivalent if the uniformity condition fails. Our results shed light on the relationships between among various notions of maximum a posteriori estimator in non-parametric Bayesian inference.

math.FA↗

arXiv2018

Quasi-invariance of countable products of Cauchy measures under non-unitary dilations

Consider an infinite sequence $(U_n)_{n\in\mathbb{N}}$ of independent Cauchy random variables, defined by a sequence $(δ_n)_{n\in\mathbb{N}}$ of location parameters and a sequence $(γ_n)_{n\in\mathbb{N}}$ of scale parameters. Let $(W_n)_{n\in\mathbb{N}}$ be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence $(σ_nγ_n)_{n\in\mathbb{N}}$ of scale parameters, with $σ_n\neq 0$ for all $n\in\mathbb{N}$. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of $(U_n)_{n\in\mathbb{N}}$ and $(W_n)_{n\in\mathbb{N}}$ are equivalent if and only if the sequence $(\vert σ_n\vert-1)_{n\in\mathbb{N}}$ is square-summable.

math.PR↗

arXiv2016

Cameron-Martin theorems for sequences of symmetric Cauchy-distributed random variables

Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences.

math.PR↗

arXiv2016

Convexity of a stochastic control functional related to importance sampling of Itô diffusions

We consider the problem of rare event importance sampling, where the random variable of interest is a path functional of an Itô diffusion computed up to the first exit from a $d$-dimensional bounded domain. Dupuis and Wang (\textit{Ann. Appl. Probab.}, 15 (2005), pp. 1-38) studied the importance sampling problem by formulating it as a stochastic optimal control problem, where the value function is related to the conditional cumulant generating function of the random variable. In this paper, we show that the sufficient conditions for the value function to be twice-differentiable and $α$-uniformly Hölder continuous on the closure of the domain are also sufficient conditions for positive definiteness of the second variation of the control functional on the space of differentiable, $α$-uniformly Hölder continuous $\mathbb{R}^d$-valued feedback controls. We derive an expression for the second variation using Kazamaki's sufficient condition for $L^q$-boundedness of exponential martingales, and using Fredholm theory to prove the finiteness of the moment generating function of the first exit time over any bounded interval containing the origin. The strict convexity result suggests that one may be able to solve the corresponding Hamilton-Jacobi-Bellman boundary value problem in a dimension-robust way, by combining convex optimisation and Monte Carlo methods. We apply the result to analyse a gradient descent algorithm proposed by Hartmann and Schütte (\textit{J. Stat. Mech. Theor. Exp.} (2012), P11004) for efficient rare event simulation.

math.OC↗