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Aleksandar Mijatović

Publications and source records attributed to Aleksandar Mijatović.

At least 55 records · Page 3Linked to original sources
arXiv2020

Reflecting random walks in curvilinear wedges

We study a random walk (Markov chain) in an unbounded planar domain whose boundary is described by two curves of the form $x_2 = a^+ x_1^{β^+}$ and $x_2 = -a^- x_1^{β^-}$, with $x_1 \geq 0$. In the interior of the domain, the random walk has zero drift and a given increment covariance matrix. From the vicinity of the upper and lower sections of the boundary, the walk drifts back into the interior at a given angle $α^+$ or $α^-$ to the relevant inwards-pointing normal vector. Here we focus on the case where $α^+$ and $α^-$ are equal but opposite, which includes the case of normal reflection. For $0 \leq β^+, β^- < 1$, we identify the phase transition between recurrence and transience, depending on the model parameters, and quantify recurrence via moments of passage times.

math.PR↗
arXiv2020

Stationary entrance Markov chains, inducing, and level-crossings of random walks

For a Markov chain $Y$ with values in a Polish space, consider the entrance Markov chain obtained by sampling $Y$ at the moments when it enters a fixed set $A$ from its complement $A^c$. Similarly, consider the exit Markov chain, obtained by sampling $Y$ at the exit times from $A^c$ to $A$. This paper provides a framework for analysing invariant measures of these two types of Markov chains in the case when the initial chain $Y$ has a known $σ$-finite invariant measure. Under certain recurrence-type assumptions ($Y$ can be transient), we give explicit formulas for invariant measures of these chains. Then we study their uniqueness and ergodicity assuming that $Y$ is topologically recurrent, irreducible, and weak Feller. Our approach is based on the technique of inducing from infinite ergodic theory. This also yields, in a natural way, the versions of the results above (provided in the paper) for the classical induced Markov chains. We give applications to random walks in $R^d$, which we regard as "stationary" Markov chains started under the Lebesgue measure. We are mostly interested in dimension one, where we study the Markov chain of overshoots above the zero level of a random walk that oscillates between $-\infty$ and $+\infty$. We show that this chain is ergodic, and use this result to prove a central limit theorem for the number of level crossings for random walks with zero mean and finite variance of increments.

math.PR↗
arXiv2019

Martingale approach to control for general jump processes

We provide verification theorems (at different levels of generality) for infinite horizon stochastic control problems in continuous time for semimartingales. The control framework is given as an abstract "martingale formulation", which encompasses a broad range of standard control problems. Under appropriate conditions we show that the set of admissible controls gives rise to a certain class of controlled special semimartingales. Our results generalise both the standard controlled Itô- and Lévy-diffusion settings as we allow ourselves to locally control not only the drift and diffusion coefficients, but also the jump intensity measure of the jumps. As an illustration, we present a few examples with explicit solutions.

math.PR↗
arXiv2019

$\varepsilon$-strong simulation of the convex minorants of stable processes and meanders

Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of Lévy processes, which includes subordinated stable and symmetric Lévy processes. We apply this characterisaiton to construct $\varepsilon$-strong simulation ($\varepsilon$SS) algorithms for the convex minorant of stable meanders, the finite dimensional distributions of stable meanders and the convex minorants of weakly stable processes. We prove that the running times of our $\varepsilon$SS algorithms have finite exponential moments. We implement the algorithms in Julia 1.0 (available on GitHub) and present numerical examples supporting our convergence results.

math.PR↗
arXiv2019

Invariance principles for local times in regenerative settings

Consider a stochastic process $\mathfrak{X}$, regenerative at a state $x$ which is instantaneous and regular. Let $L$ be a regenerative local time for $\mathfrak{X}$ at $x$. Suppose furthermore that $\mathfrak{X}$ can be approximated by discrete time regenerative processes $\mathfrak{X}^n$ for which $x$ is accesible. We give conditions on $\mathfrak{X}$ and $\mathfrak{X}^n$ so that the naturally defined local time of $\mathfrak{X}^n$ converges weakly to $L$. This limit theorem generalizes previous invariance principles that have appeared in the literature.

math.PR↗
arXiv2019

Non-asymptotic bounds for sampling algorithms without log-concavity

Discrete time analogues of ergodic stochastic differential equations (SDEs) are one of the most popular and flexible tools for sampling high-dimensional probability measures. Non-asymptotic analysis in the $L^2$ Wasserstein distance of sampling algorithms based on Euler discretisations of SDEs has been recently developed by several authors for log-concave probability distributions. In this work we replace the log-concavity assumption with a log-concavity at infinity condition. We provide novel $L^2$ convergence rates for Euler schemes, expressed explicitly in terms of problem parameters. From there we derive non-asymptotic bounds on the distance between the laws induced by Euler schemes and the invariant laws of SDEs, both for schemes with standard and with randomised (inaccurate) drifts. We also obtain bounds for the hierarchy of discretisation, which enables us to deploy a multi-level Monte Carlo estimator. Our proof relies on a novel construction of a coupling for the Markov chains that can be used to control both the $L^1$ and $L^2$ Wasserstein distances simultaneously. Finally, we provide a weak convergence analysis that covers both the standard and the randomised (inaccurate) drift case. In particular, we reveal that the variance of the randomised drift does not influence the rate of weak convergence of the Euler scheme to the SDE.

math.PR↗
arXiv2019

A note on the exact simulation of spherical Brownian motion

We describe an exact simulation algorithm for the increments of Brownian motion on a sphere of arbitrary dimension, based on the skew-product decomposition of the process with respect to the standard geodesic distance. The radial process is closely related to a Wright-Fisher diffusion, increments of which can be simulated exactly using the recent work of Jenkins & Spanò (2017) [JS17]. The rapid spinning phenomenon of the skew-product decomposition then yields the algorithm for the increments of the process on the sphere.

math.PR↗
arXiv2019

Asymptotic variance for Random walk Metropolis chains in high dimensions: logarithmic growth via the Poisson equation

There are two ways of speeding up MCMC algorithms: (1) construct more complex samplers that use gradient and higher order information about the target and (2) design a control variate to reduce the asymptotic variance. While the efficiency of (1) as a function of dimension has been studied extensively, this paper provides first rigorous results linking the growth of the asymptotic variance in (2) with dimension. Specifically, we construct a control variate for a $d$-dimensional Random walk Metropolis chain with an IID target using the solution of the Poisson equation for the scaling limit in the seminal paper "Weak convergence and optimal scaling of random walk Metropolis algorithms" of Gelman, Gilks and Roberts. We prove that the asymptotic variance of the corresponding estimator is bounded above by a multiple of $\log d/d$ over the spectral gap of the chain. The proof hinges on large deviations theory, optimal Young's inequality and Berry-Esseen type bounds. Extensions of the result to non-product targets are discussed.

math.PR↗
arXiv2019

Exact Simulation of the Extrema of Stable Processes

We exhibit an exact simulation algorithm for the supremum of a stable process over a finite time interval using dominated coupling from the past (DCFTP). We establish a novel perpetuity equation for the supremum (via the representation of the concave majorants of Lévy processes) and apply it to construct a Markov chain in the DCFTP algorithm. We prove that the number of steps taken backwards in time before the coalescence is detected is finite. We analyse numerically the performance of the algorithm (the code, written in Julia 1.0, is available on GitHub).

math.PR↗
arXiv2019

Stability of overshoots of zero mean random walks

We prove that for a random walk on the real line whose increments have zero mean and are either integer-valued or spread out (i.e. the distributions of the steps of the walk are eventually non-singular), the Markov chain of overshoots above a fixed level converges in total variation to its stationary distribution. We find the explicit form of this distribution heuristically and then prove its invariance using a time-reversal argument. If, in addition, the increments of the walk are in the domain of attraction of a non-one-sided $α$-stable law with index $α\in(1,2)$ (resp. have finite variance), we establish geometric (resp. uniform) ergodicity for the Markov chain of overshoots. All the convergence results above are also valid for the Markov chain obtained by sampling the walk at the entrance times into an interval.

math.PR↗
arXiv2018

Projections of spherical Brownian motion

We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R}^{n+\ell}$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R}^{n+\ell}$ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.

math.PR↗
arXiv2018

Invariance principle for non-homogeneous random walks

We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in $\mathbb{R}^d$, which may be recurrent in any dimension. The limit $\mathcal{X}$ is an elliptic martingale diffusion, which may be point-recurrent at the origin for any $d\geq2$. To characterise $\mathcal{X}$, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in $\mathbb{R}^d$ and use it to express a related spherical diffusion as a Brownian motion with drift. This representation allows us to establish the skew-product decomposition of the excursions of $\mathcal{X}$ and thus develop the excursion theory of $\mathcal{X}$ without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for $\mathcal{X}$ in $\mathbb{R}^d$, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of $\mathcal{X}$ is time-reversible. If so, the excursions of $\mathcal{X}$ in $\mathbb{R}^d$ generalise the classical Pitman-Yor splitting-at-the-maximum property of Bessel excursions.

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arXiv2017

A radial invariance principle for non-homogeneous random walks

Consider non-homogeneous zero-drift random walks in $\mathbb{R}^d$, $d \geq 2$, with the asymptotic increment covariance matrix $σ^2 (\mathbf{u})$ satisfying $\mathbf{u}^\top σ^2 (\mathbf{u}) \mathbf{u} = U$ and $\mathrm{tr}\ σ^2 (\mathbf{u}) = V$ in all in directions $\mathbf{u}\in\mathbb{S}^{d-1}$ for some positive constants $U<V$. In this paper we establish weak convergence of the radial component of the walk to a Bessel process with dimension $V/U$. This can be viewed as an extension of an invariance principle of Lamperti.

math.PR↗
arXiv2015

On the policy improvement algorithm in continuous time

We develop a general approach to the Policy Improvement Algorithm (PIA) for stochastic control problems for continuous-time processes. The main results assume only that the controls lie in a compact metric space and give general sufficient conditions for the PIA to be well-defined and converge in continuous time (i.e. without time discretisation). It emerges that the natural context for the PIA in continuous time is weak stochastic control. We give examples of control problems demonstrating the need for the weak formulation as well as diffusion-based classes of problems where the PIA in continuous time is applicable.

math.PR↗
arXiv2015

On the weak limit law of the maximal uniform k-spacing

This paper gives a simple proof of a limit theorem for the lenght of the largest interval straddling a fixed number of i.i.d. points uniformly disributed on a unit interval. The key step in our argument is a classical theorem of Watson (1954) on the maxima of m-dependent stationary stochastic sequences.

math.PR↗
arXiv2015

Anomalous recurrence properties of many-dimensional zero-drift random walks

Famously, a $d$-dimensional, spatially homogeneous random walk whose increments are non-degenerate, have finite second moments, and have zero mean is recurrent if $d \in \{1,2\}$ but transient if $d \geq 3$. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially non-homogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension $d \geq 2$, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk's current position; these \emph{elliptic random walks} generalize the classical homogeneous Pearson--Rayleigh walk (the spherical case). Our proof of the recurrence classification is based on fundamental work of Lamperti.

math.PR↗
arXiv2015

Markov chain approximations to scale functions of Lévy processes

We introduce a general algorithm for the computation of the scale functions of a spectrally negative Lévy process $X$, based on a natural weak approximation of $X$ via upwards skip-free continuous-time Markov chains with stationary independent increments. The algorithm consists of evaluating a finite linear recursion with its (nonnegative) coefficients given explicitly in terms of the Lévy triplet of $X$. Thus it is easy to implement and numerically stable. Our main result establishes sharp rates of convergence of this algorithm providing an explicit link between the semimartingale characteristics of $X$ and its scale functions, not unlike the one-dimensional Itô diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.

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