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Multidimensional random walks conditioned to stay ordered via generalized ladder height functionsMay 14 2019Random walks conditioned to stay positive are a prominent topic in fluctuation theory. One way to construct them is as a random walk conditioned to stay positive up to time $n$, and let $n$ tend to infinity. A second method is conditioning instead to ... More

Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measuresMay 01 2019The box-ball system (BBS) is a simple model of soliton interaction introduced by Takahashi and Satsuma in the 1990s. Recent work of the authors, together with Tsuyoshi Kato and Satoshi Tsujimoto, derived various families of invariant measures for the ... More

Quantitative homogenization in a balanced random environmentMar 28 2019We consider discrete non-divergence form difference operators in an i.i.d. random environment and the corresponding process--the random walk in a balanced random environment in $\mathbb{Z}^d$. We first quantify the ergodicity of the environment viewed ... More

Revising SA-CCRFeb 22 2019Apr 08 2019From SA-CCR to RSA-CCR: making SA-CCR self-consistent and appropriately risk-sensitive by cashflow decomposition in a 3-Factor Gaussian Market Model

From SA-CCR to RSA-CCR: making SA-CCR self-consistent and appropriately risk-sensitive by cashflow decomposition in a 3-Factor Gaussian Market ModelFeb 22 2019SA-CCR has major issues including: lack of self-consistency for linear trades; lack of appropriate risk sensitivity (zero positions can have material add-ons; moneyness is ignored); dependence on economically-equivalent confirmations. We show that SA-CCR ... More

Embedding quadratization gadgets on Chimera and Pegasus graphsJan 23 2019We group all known quadratizations of cubic and quartic terms in binary optimization problems into six and seven unique graphs respectively. We then perform a minor embedding of these graphs onto the well-known Chimera graph, and the brand new Pegasus ... More

Pegasus: The second connectivity graph for large-scale quantum annealing hardwareJan 22 2019Pegasus is a graph which offers substantially increased connectivity between the qubits of quantum annealing hardware compared to the graph Chimera. It is the first fundamental change in the connectivity graph of quantum annealers built by D-Wave since ... More

Quadratization in discrete optimization and quantum mechanicsJan 14 2019A book about turning high-degree optimization problems into quadratic optimization problems that maintain the same global minimum (ground state). This book explores quadratizations for pseudo-Boolean optimization, perturbative gadgets used in QMA completeness ... More

Transition probability estimates for subordinate random walksDec 09 2018Let $S_n$ be the simple random walk on the integer lattice $\mathbb{Z}^d$. For a Bernstein function $\phi$ we consider a random walk $S^\phi_n$ which is subordinated to $S_n$. Under a certain assumption on the behaviour of $\phi$ at zero we establish ... More

Random walks on graphs: new bounds on hitting, meeting, coalescing and returningJul 18 2018We prove new results on lazy random walks on finite graphs. To start, we obtain new estimates on return probabilities $P^t(x,x)$ and the maximum expected hitting time $t_{\rm hit}$, both in terms of the relaxation time. We also prove a discrete-time version ... More

Recurrence and winding number for a revolving random walkJul 10 2018Nov 26 2018We consider simple random walks on two partially directed square lattices. One common feature of these walks is that they are bound to revolve clockwise; however they exhibit different recurrence/transience behaviors. Our main result is indeed a proof ... More

Excessive Backlog Probabilities of Two Parallel QueuesJun 02 2018Let $X$ be the constrained random walk on ${\mathbb Z}_+^2$ with increments $(1,0)$, $(-1,0)$, $(0,1)$ and $(0,-1)$; $X$ represents, at arrivals and service completions, the lengths of two queues working in parallel whose service and interarrival times ... More

Quantum Fast-Forwarding: Markov Chains and Graph Property TestingApr 06 2018Jan 23 2019We introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. More specifically, with $P$ the Markov chain transition matrix and $D ... More

On the range of a two-dimensional conditioned simple random walkApr 01 2018May 14 2019We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient ... More

On the range of a two-dimensional conditioned simple random walkApr 01 2018We consider the two-dimensional simple random walk conditioned on never hitting the origin. This process is a Markov chain, namely it is the Doob $h$-transform of the simple random walk with respect to the potential kernel. It is known to be transient ... More

On the Mabinogion urn modelFeb 20 2018In this paper we discuss the Mabinogion urn model introduced by D. Williams in Probability with Martingales (1991). Therein he describes an optimal control problem where the objective is to maximize the expected final number of objects of one kind in ... More

Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operatorJan 30 2018We study the convergence of the graph Laplacian of a random geometric graph generated by an i.i.d. sample from a $m$-dimensional submanifold $M$ in $R^d$ as the sample size $n$ increases and the neighborhood size $h$ tends to zero. We show that eigenvalues ... More

Approximation of Excessive Backlog Probabilities of Two Tandem QueuesJan 15 2018Let $X$ be the constrained random walk on ${\mathbb Z}_+^2$ taking the steps $(1,0)$, $(-1,1)$ and $(0,-1)$ with probabilities $\lambda < (\mu_1\neq \mu_2)$; in particular, $X$ is assumed stable. Let $\tau_n$ be the first time $X$ hits $\partial A_n = ... More

Simulation of Quantum Walks and Fast Mixing with Classical ProcessesDec 05 2017We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains, that is, classical Markov chains with added memory. We show that these can simulate quantum walks, allowing us to answer ... More

Bounding the convergence time of local probabilistic evolutionNov 16 2017Isoperimetric inequalities form a very intuitive yet powerful characterization of the connectedness of a state space, that has proven successful in obtaining convergence bounds. Since the seventies they form an essential tool in differential geometry, ... More

Correlated continuous time random walks and fractional Pearson diffusionsAug 23 2017Continuous time random walks have random waiting times between particle jumps. We define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs). The jumps in these CTRWs are obtained from Markov chains ... More

Invariant Measures, Hausdorff Dimension and Dimension Drop of some Harmonic Measures on Galton-Watson TreesAug 23 2017May 04 2018We consider infinite Galton-Watson trees without leaves together with i.i.d.~random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic assumptions, ... More

On the centre of mass of a random walkAug 15 2017Dec 14 2018For a random walk $S_n$ on $\mathbb{R}^d$ we study the asymptotic behaviour of the associated centre of mass process $G_n = n^{-1} \sum_{i=1}^n S_i$. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the ... More

On the number of points skipped by a transient (1,2) random walk on the lineJul 20 2017Consider a transient near-critical (1,2) random walk on the positive half line. We give a criteria for the finiteness of the number of the skipped points (the points never visited) by the random walk. This result generalizes (partially) the criteria for ... More

Banks as Tanks: A Continuous-Time Model of Financial ClearingMay 16 2017Jun 26 2017We present a simple model of clearing in financial networks in continuous time. In the model, firms (banks) are represented as reservoirs (tanks) with liquid (money) flowing in and out. This approach provides a simple recursive solution to a classical ... More

Wolf Barth (1942--2016)Apr 24 2017In this article we describe the life and work of Wolf Barth who died on 30th December 2016. Wolf Barth's contributions to algebraic variety span a wide range of subjects. His achievements range from what is now called the Barth-Lefschetz theorems to his ... More

An Arcsine Law for Markov Random WalksMar 01 2017Mar 08 2018The classic arcsine law for the number $N_{n}^{>}:=n^{-1}\sum_{k=1}^{n}\mathbf{1}_{\{S_{k}>0\}}$ of positive terms, as $n\to\infty$, in an ordinary random walk $(S_{n})_{n\ge 0}$ is extended to the case when this random walk is governed by a positive ... More

Divisible sandpile on Sierpinski gasket graphsFeb 27 2017The divisible sandpile model is a growth model on graphs that was introduced by Levine and Peres as a tool to study internal diffusion limited aggregation. In this work we investigate the shape of the divisible sandpile model on the graphical Sierpinski ... More

Internal DLA on Sierpinski gasket graphsFeb 13 2017Aug 17 2017Internal diffusion-limited aggregation (IDLA) is a stochastic growth model on a graph $G$ which describes the formation of a random set of vertices growing from the origin (some fixed vertex) of $G$. Particles start at the origin and perform simple random ... More

Internal DLA on Sierpinski gasket graphsFeb 13 2017Feb 27 2019Internal diffusion-limited aggregation (IDLA) is a stochastic growth model on a graph $G$ which describes the formation of a random set of vertices growing from the origin (some fixed vertex) of $G$. Particles start at the origin and perform simple random ... More

A remark on a central limit theorem for non-symmetric random walks on crystal latticesJan 30 2017Apr 16 2017Recently, Ishiwata, Kawabi and Kotani [2] proved two kinds of central limit theorems for non-symmetric random walks on crystal lattices from the view point of discrete geometric analysis. In the present paper, we obtain yet another kind of the central ... More

Construction of a new class of quantum Markov fieldsDec 16 2016In the present paper, we propose a new construction of quantum Markov fields on arbitrary connected, infinite, locally finite graphs. The construction is based on a specific tessellation on the considered graph, that allows us to express the Markov property ... More

Average Entropy of the Ranges for Simple Random Walks on Discrete GroupsDec 03 2016Feb 20 2017Inspired by Benjamini et al (Ann. Inst. H. Poincar\'{e} Probab. Stat. 2010) and Windisch (Electron. J. Probab. 2010), we consider the entropy of the random walk range formed by a simple random walk on a discrete group. It is shown in this setting the ... More

The social network model on infinite graphsOct 13 2016Given an infinite connected regular graph $G=(V,E)$, place at each vertex Pois($\lambda$) walkers performing independent lazy simple random walks on $G$ simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. ... More

The social network model on infinite graphsOct 13 2016Sep 01 2018Given an infinite connected regular graph $G=(V,E)$, place at each vertex Pois($\lambda$) walkers performing independent lazy simple random walks on $G$ simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. ... More

Heavy-tailed random walks on complexes of half-linesOct 04 2016We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is governed ... More

XVA at the Exercise BoundaryOct 02 2016XVA is a material component of a trade valuation and hence it must impact the decision to exercise options within a given netting set. This is true for both unsecured trades and secured / cleared trades where KVA and MVA play a material role even if CVA ... More

Limit theorems for local and occupation times of random walks and Brownian motion on a spiderSep 27 2016May 10 2017A simple random walk and a Brownian motion are considered on a spider that is a collection of half lines (we call them legs) joined in the origin. We give a strong approximation of these two objects and their local times. For fixed number of legs we establish ... More

Limit theorems for local and occupation times of random walks and Brownian motion on a spiderSep 27 2016A simple random walk and a Brownian motion are considered on a spider that is a collection of half lines (we call them legs) joined in the origin. We give a strong approximation of these two objects and their local times. For fixed number of legs we establish ... More

Option-Based Pricing of Wrong Way Risk for CVASep 03 2016Oct 02 2016The two main issues for managing wrong way risk (WWR) for the credit valuation adjustment (CVA, i.e. WW-CVA) are calibration and hedging. Hence we start from a novel model-free worst-case approach based on static hedging of counterparty exposure with ... More

On the equivalence of various definitions of mixed Poisson processesJul 19 2016Under mild assumptions the equivalence of the mixed Poisson process with mixing parameter a real-valued random variable to the one with mixing distribution as well as to the mixed Poisson process in the sense of Huang is obtained, and a characterization ... More

General Edgeworth expansions with applications to profiles of random treesJun 13 2016Oct 05 2017We prove an asymptotic Edgeworth expansion for the profiles of certain random trees including binary search trees, random recursive trees and plane-oriented random trees, as the size of the tree goes to infinity. All these models can be seen as special ... More

General Edgeworth expansions with applications to profiles of random treesJun 13 2016Sep 14 2016We prove an asymptotic Edgeworth expansion for the profiles of certain random trees including binary search trees, random recursive trees and plane-oriented random trees, as the size of the tree goes to infinity. All these models can be seen as special ... More

Local Central Limit Theorem for a Random Walk Perturbed in One PointMay 20 2016Jun 05 2016We consider a symmetric random walk on the $\nu$-dimensional lattice, whose exit probability from the origin is perturbed. We prove the local central limit theorem for this process, finding a short-range correction to diffusive behaviour in any dimension ... More

Phase transitions for Quantum Markov Chains associated with Ising type models on a Cayley treeMay 15 2016The main aim of the present paper is to prove the existence of a phase transition in quantum Markov chain (QMC) scheme for the Ising type models on a Cayley tree. Note that this kind of models do not have one-dimensional analogous, i.e. the considered ... More

Joint Statistics of Random Walk on $Z^1$ and Accumulation of VisitsApr 28 2016We obtain the joint distribution $P_N (X, K|Z)$ of the location $X$ of a one-dimensional symmetric next neighbor random walk on the integer lattice, and the number of times the walk has visited a specified site $Z$. This distribution has a simple form ... More

About the distance between random walkers on some graphsApr 27 2016Jul 26 2016We consider two or more simple symmetric walks on some graphs, e.g. the real line, the plane or the two dimensional comb lattice, and investigate the properties of the distance among the walkers.

On Maximal Displacement of Bridges in the Random Conductance modelMar 21 2016Dec 18 2016We study a discrete time random walk in an environment of i.i.d. non-negative conductances in $\mathbb{Z}^d$. We consider the maximum displacements for bridges, i.e. we condition the random walk on returning to the origin, and we prove first a normal ... More

On Maximal Displacement of Bridges in the Random Conductance modelMar 21 2016We study a discrete time random walk in an environment of i.i.d. non-negative conductances in $\mathbb{Z}^d$. We consider the maximum displacements for bridges, i.e. we condition the random walk on returning to the origin, and we prove first a normal ... More

Enhancing the filtered derived categoryFeb 04 2016May 01 2018The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods of homotopical ... More

Enhancing the filtered derived categoryFeb 04 2016Jun 02 2016The filtered derived category of an abelian category has played a useful role in subjects including geometric representation theory, mixed Hodge modules, and the theory of motives. We develop a natural generalization using current methods of homotopical ... More

Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume samplingJan 13 2016Jan 20 2016We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral ... More

Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume samplingJan 13 2016May 02 2017We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral ... More

Crushing runtimes in adiabatic quantum computation with Energy Landscape Manipulation (ELM): Application to Quantum FactoringOct 26 2015We introduce two methods for speeding up adiabatic quantum computations by increasing the energy between the ground and first excited states. Our methods are even more general. They can be used to shift a Hamiltonian's density of states away from the ... More

Long time asymptotics of non-symmetric random walks on crystal latticesOct 17 2015In the present paper, we study long time asymptotics of non-symmetric random walks on crystal lattices from a view point of discrete geometric analysis due to Kotani and Sunada [11, 23]. We observe that the Euclidean metric associated with the standard ... More

Edge- and vertex-reinforced random walks with super-linear reinforcement on infinite graphsSep 02 2015Jun 02 2016In this paper we introduce a new simple but powerful general technique for the study of edge- and vertex-reinforced processes with super-linear reinforcement, based on the use of order statistics for the number of edge, respectively of vertex, traversals. ... More

Reducing multi-qubit interactions in adiabatic quantum computation without adding auxiliary qubits. Part 2: The "split-reduc" method and its application to quantum determination of Ramsey numbersAug 28 2015Quantum annealing has recently been used to determine the Ramsey numbers R(m,2) for 3 < m < 9 and R(3,3) [Bian et al. (2013) PRL 111, 130505]. This was greatly celebrated as the largest experimental implementation of an adiabatic evolution algorithm to ... More

Random walk on sparse random digraphsAug 26 2015Jan 22 2018A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of ... More

Random walk on sparse random digraphsAug 26 2015Sep 28 2015A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the context of ... More

Reducing multi-qubit interactions in adiabatic quantum computation without adding auxiliary qubits. Part 1: The "deduc-reduc" method and its application to quantum factorization of numbersAug 19 2015Oct 01 2015Adiabatic quantum computing has recently been used to factor 56153 [Dattani & Bryans, arXiv:1411.6758] at room temperature, which is orders of magnitude larger than any number attempted yet using Shor's algorithm (circuit-based quantum computation). However, ... More

Non-Universality of Nodal Length Distribution for Arithmetic Random WavesAug 03 2015Jun 30 2016"Arithmetic random waves" are the Gaussian Laplace eigenfunctions on the two-dimensional torus (Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013)). In this paper we find that their nodal length converges to a non-universal (non-Gaussian) ... More

Exit Probabilities and Balayage of Constrained Random WalksJun 29 2015Jul 27 2015Let $X$ be the constrained random walk on ${\mathbb Z}_+^d$ representing the queue lengths of a stable Jackson network and $x$ its initial position. Let $\tau_n$ be the first time the sum of the components of $X$ equals $n$. $p_n \doteq P_x(\tau_n < \tau_0)$ ... More

Anomalous recurrence properties of many-dimensional zero-drift random walksJun 29 2015Famously, 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 ... More

Logarithmic Coefficients and Multifractality of Whole-Plane SLEApr 21 2015We consider the whole-plane SLE conformal map f from the unit disk to the slit plane, and show that its mixed moments, involving a power p of the derivative modulus |f'| and a power q of the map |f| itself, have closed forms along some integrability curves ... More

Logarithmic Coefficients and Generalized Multifractality of Whole-Plane SLEApr 21 2015Apr 21 2017We consider the whole-plane SLE conformal map f from the unit disk to the slit plane, and show that its mixed moments, involving a power p of the derivative modulus |f'| and a power q of the map |f| itself, have closed forms along some integrability curves ... More

Random walks on the random graphApr 08 2015Oct 20 2016We study random walks on the giant component of the Erd\H{o}s-R\'enyi random graph ${\cal G}(n,p)$ where $p=\lambda/n$ for $\lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, ... More

Random walks on the random graphApr 08 2015Nov 25 2015We study random walks on the giant component of the Erd\H{o}s-R\'enyi random graph ${\cal G}(n,p)$ where $p=\lambda/n$ for $\lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, ... More

Edgeworth expansions for profiles of lattice branching random walksMar 16 2015Jun 12 2016Consider a branching random walk on $\mathbb Z$ in discrete time. Denote by $L_n(k)$ the number of particles at site $k\in\mathbb Z$ at time $n\in\mathbb N_0$. By the profile of the branching random walk (at time $n$) we mean the function $k\mapsto L_n(k)$. ... More

A Survey of Manoeuvring Target Tracking MethodsMar 06 2015A comprehensive review of the literature on manoeuvring target tracking for both uncluttered and cluttered measurements is presented. Various discrete-time dynamical models including non-random input, random-input and switching or hybrid system manoeuvre ... More

Linear forms of the telegraph random processes driven by partial differential equationsMar 03 2015Consider $n$ independent Goldstein-Kac telegraph processes $X_1(t), \dots ,X_n(t), \; n\ge 2, \; t\ge 0,$ on the real line $\Bbb R$. Each the process $X_k(t), \; k=1,\dots,n,$ describes a stochastic motion at constant finite speed $c_k>0$ of a particle ... More

Intrinsic random walks and sub-Laplacians in sub-Riemannian geometryMar 02 2015Jan 08 2016On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} $\Delta_\omega$, as the divergence of the horizontal gradient, once a volume $\omega$ is fixed, and the \emph{microscopic Laplacian}, as the operator associated ... More

Intrinsic random walks and sub-Laplacians in sub-Riemannian geometryMar 02 2015May 02 2017On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} $\Delta_\omega$, as the divergence of the horizontal gradient, once a volume $\omega$ is fixed, and the \emph{microscopic Laplacian}, as the operator associated ... More

Invariant measures and error bounds for random walks in the quarter-plane based on sums of geometric termsFeb 25 2015We consider homogeneous random walks in the quarter-plane. The necessary conditions which characterize random walks of which the invariant measure is a sum of geometric terms are provided in [2,3]. Based on these results, we first develop an algorithm ... More

Large Deviations for processes on half-lineFeb 23 2015Nov 27 2015We consider a sequence of processes defined on half-line for all non negative t. We give sufficient conditions for Large Deviation Principle (LDP) to hold in the space of continuous functions with a new metric that is more sensitive to behaviour at infinity ... More

Kendall random walk, Williamson transform and the corresponding Wiener-Hopf factorizationJan 23 2015The paper gives some properties of hitting times and an analogue of the Wiener-Hopf factorization for the Kendall random walk. We show also that the Williamson transform is the best tool for problems connected with the Kendall generalized convolution. ... More

Kendall random walk, Williamson transform and the corresponding Wiener-Hopf factorizationJan 23 2015Dec 09 2016The paper gives some properties of hitting times and an analogue of the Wiener-Hopf factorization for the Kendall random walk. We show also that the Williamson transform is the best tool for problems connected with the Kendall generalized convolution. ... More

Formation of an interface by competitive erosionJan 15 2015In 2006, the fourth author proposed a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle that can be either red or blue. New red and blue particles alternately get emitted from their ... More

Some limit theorems for heights of random walks on spiderDec 17 2014Jul 01 2015A simple symmetric random walk is considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We establish a strong approximation of this random walk by the so-called Brownian spider. Transition probabilities are ... More

Kendall random walksNov 30 2014The paper deals with a new class of random walks strictly connected with the Pareto distribution. We consider stochastic processes in the sense of generalized convolution or weak generalized convolution following the idea given in [1]. The processes are ... More

Quantum factorization of 56153 with only 4 qubitsNov 25 2014Nov 27 2014The largest number factored on a quantum device reported until now was 143. That quantum computation, which used only 4 qubits at 300K, actually also factored much larger numbers such as 3599, 11663, and 56153, without the awareness of the authors of ... More

Quadratic and rate-independent limits for a large-deviations functionalSep 15 2014We construct a stochastic model showing the relationship between noise, gradient flows and rate-independent systems. The model consists of a one-dimensional birth-death process on a lattice, with rates derived from Kramers' law as an approximation of ... More

Warehousing Credit (CVA) Risk, Capital (KVA) and Tax (TVA) ConsequencesJul 11 2014Jan 07 2015Credit risk may be warehoused by choice, or because of limited hedging possibilities. Credit risk warehousing increases capital requirements and leaves open risk. Open risk must be priced in the physical measure, rather than the risk neutral measure, ... More

Some characterizations for Markov processes as mixed renewal processesJul 11 2014Jul 19 2016In this paper the class of mixed renewal processes (MRPs for short) with mixing parameter a random vector from \cite{lm6z3} (enlarging Huang's \cite{hu} original class) is replaced by the strictly more comprising class of all extended MRPs by adding a ... More

Cover time of a random graph with a degree sequence II: Allowing vertices of degree twoJun 04 2014We study the cover time of a random graph chosen uniformly at random from the set of graphs with vertex set $[n]$ and degree sequence $\mathbf{d}=(d_i)_{i=1}^n$. In a previous work, the asymptotic cover time was obtained under a number of assumptions ... More

VAR and ES/CVAR Dependence on data cleaning and Data Models: Analysis and ResolutionMay 29 2014Historical (Stressed-) Value-at-Risk ((S)VAR), and Expected Shortfall (ES), are widely used risk measures in regulatory capital and Initial Margin, i.e. funding, computations. However, whilst the definitions of VAR and ES are unambiguous, they depend ... More

KVA: Capital Valuation AdjustmentMay 02 2014Oct 24 2014Credit (CVA), Debit (DVA) and Funding Valuation Adjustments (FVA) are now familiar valuation adjustments made to the value of a portfolio of derivatives to account for credit risks and funding costs. However, recent changes in the regulatory regime and ... More

How Tall Can Be the Excursions of a Random Walk on a SpiderFeb 23 2014We consider a simple symmetric random walk on a spider, that is a collection of half lines (we call them legs) joined at the origin. Our main question is the following: if the walker makes $n$ steps how high can he go up on all legs. This problem is discussed ... More

Large deviations for random surfaces: the hyperbolic nature of Liouville Field TheoryJan 23 2014Feb 01 2014Liouville Field Theory (LFT) is a model of $2d$ random surfaces involved in $2d$ string theory or in the description of the fluctuations of metrics in $2d$ quantum gravity. This is a probabilistic model that consists in weighting the shifted Free Field ... More

Semiclassical limit of Liouville Field TheoryJan 23 2014Oct 12 2017Liouville Field Theory (LFT for short) is a two dimensional model of random surfaces, which is for instance involved in $2d$ string theory or in the description of the fluctuations of metrics in $2d$ Liouville quantum gravity. This is a probabilistic ... More

Cut-off for lamplighter chains on tori: dimension interpolation and phase transitionDec 16 2013Aug 23 2015Given a finite, connected graph $G$, the lamplighter chain on $G$ is the lazy random walk $X^\diamond$ on the associated lamplighter graph $G^\diamond={\mathbf Z}_2 \wr G$. The mixing time of the lamplighter chain on the torus ${\mathbf Z}_n^d$ is known ... More

Cut-off for lamplighter chains on tori: dimension interpolation and phase transitionDec 16 2013Aug 14 2018Given a finite, connected graph $G$, the lamplighter chain on $G$ is the lazy random walk $X^\diamond$ on the associated lamplighter graph $G^\diamond={\mathbf Z}_2 \wr G$. The mixing time of the lamplighter chain on the torus ${\mathbf Z}_n^d$ is known ... More

On Quantum Markov Chains on Cayley tree III: Ising modelNov 26 2013Nov 27 2013In this paper, we consider the classical Ising model on the Cayley tree of order k and show the existence of the phase transition in the following sense: there exists two quantum Markov states which are not quasi-equivalent. It turns out that the found ... More

Heat kernel and Green function estimates on affine buildingsOct 08 2013We obtain the optimal global upper and lower bounds for the transition density $p_n(x,y)$ of a finite range isotropic random walk on affine buildings. We present also sharp estimates for the corresponding Green function.

Uniformity of the late points of random walk on Z_n^d for d >= 3Sep 12 2013Suppose that $X$ is a simple random walk on $\Z_n^d$ for $d \geq 3$ and, for each $t$, we let $\U(t)$ consist of those $x \in \Z_n^d$ which have not been visited by $X$ by time $t$. Let $\tcov$ be the expected amount of time that it takes for $X$ to visit ... More

Discrete random walk with geometric absorptionSep 04 2013We consider a discrete random walk (RW) in n dimensions . The RW is adapted with a geometric absorption process: at any discrete time there is a constant probability that absorption occurs in the current state. To model the RW with geometric absorption ... More

Harnack Inequalities and Local Central Limit Theorem for the Polynomial Lower Tail Random Conductance ModelAug 05 2013Jun 29 2015We prove upper bounds on the transition probabilities of random walks with i.i.d. random conductances with a polynomial lower tail near $0$. We consider both constant and variable speed models. Our estimates are sharp. As a consequence, we derive local ... More

The maximum of a symmetric next neighbor walk on the non-negative integersMay 23 2013We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the 2- dimensional probability distribution P{Sn = x,max1?j?n Sn = a} of being at position x after n steps, while ... More

Approximately pi proofs that the stock market can approximate piMay 19 2013We give three derivations of Polya's approximation for the expected range of a simple random walk in one dimension. This result allows for an estimation of the volatility of a financial instrument from the difference between the high and low prices, or, ... More

Convex hulls of planar random walks with driftJan 17 2013Denote by $L_n$ the length of the perimeter of the convex hull of $n$ steps of a planar random walk whose increments have finite second moment and non-zero mean. Snyder and Steele showed that $n^{-1} L_n$ converges almost surely to a deterministic limit, ... More

Brauer-Thrall theory for maximal Cohen-Macaulay modulesNov 14 2012The Brauer-Thrall Conjectures, now theorems, were originally stated for finitely generated modules over a finite-dimensional $\sk$-algebra. They say, roughly speaking, that infinite representation type implies the existence of lots of indecomposable modules ... More